Strapdown Associates Inc., Paul G Savage, Books, Strapdown …strapdownassociates.com/Appendices F, G, H to Variation... · 2020-06-24 · CV CV() ×= ×C−1 ⎡⎣()⎤⎦ (F-2)

APPENDICES F, G, AND H

TO

GENERATING STRAPDOWN SPECIFIC-FORCE/ANGULAR-RATE FOR SPECIFIED ATTITUDE/ POSITION VARIATION FROM A REFERENCE TRAJECTORY

Paul G. Savage

Strapdown Associates, Inc. Maple Plain, MN 55359 USA

WBN-14026a

www.strapdowassociates.com April 21, 2020

INTRODUCTION

This document is an addendum to [1], providing derivations in Appendices F, G, and H herein deemed too detailed for inclusion in [1]. Appendix F provides useful formulas applied in Appendices G and H, Appendix G provides derivations of specific-force/velocity/position solutions generated using the Method 1approach under [1] defined test example conditions, and Appendix H provides derivations of specific-force/velocity/position solutions generated using Method 2 under the same defined test example conditions. Definitions for analytical parameters appearing in these appendices are provided in [1] and repeated where used in this addendum.

NOTATION General Notation V = Arbitrary vector without specific coordinate frame component definition.

AV = Column matrix with elements equal to general vector V projections on general coordinate frame A axes.

( AV ×) = Cross-product (or skew symmetric) form of AV defined such that for the cross-product

of V with another arbitrary vector W in the general A frame: ( )A A AV W V W× = × A .

DAC = Generalized direction cosine matrix that transforms vectors from general coordinate

frame A to general coordinate frame D (i.e., D ADAV VC= ).

Coordinate Frames B = “Body” coordinate frame aligned with orthogonal strapdown inertial sensor axes fixed in the

rotating body. E = Earth frame fixed to the rotating earth.

1

E0 = Inertial non-rotating inertial frame aligned with E at trajectory start time t = 0. Trajectory Generator Update Cycle Indices m = Trajectory generator update cycle index ( at trajectory start time t = 0). 0m =n = Trajectory generator even (or alternate) update cycle index (i.e., ). 2m n=Important Note: Each cycle index subscript identifies the m cycle time instant value for that

parameter (e.g., subscript 2n indicates a parameter value as cycle , and 2n-1 indicates a parameter value at cycle .

2m = n2 1m n= −

Trajectory Type Subscripts ref = Parameter or coordinate frame identifier for the variation trajectory. var = Parameter or coordinate frame identifier for the variation trajectory. Parameter Definitions

Parameters are listed next in alphabetical order with Greek letters ordered using the English translation (i.e., Delta under D, muΔ μ under m, omega under o, phi ω φ under p, upsilon under u). Parameters used exclusively in the appendices are defined separately in the appendices where they appear.

υ

BSFa = Specific force acceleration vector of the rotating body (that would be measured by

strapdown accelerometers attached to the rotating body and aligned with body axes).

mDAC = Direction cosine matrix D

AC at the end of trajectory update cycle m. BmαΔ = Integral over an m cycle of B frame measured inertial angular rate Bω (that would be

measured by strapdown gyros attached to the rotating body and aligned with body axes,

i.e., 1

tB Bmm tm

dtα ω−

Δ = ∫ . BvarvarαΔ = Particular value of B

mαΔ defined for the sample to be constant for . 0 9m≥ > −

'BvarvarαΔ = Particular value of B

mαΔ defined for the sample to be constant for . 0m >BmυΔ = Integral over an m cycle of B frame measured specific-force (acceleration) B

SFa (i.e.,

1tB Bm

m SFtmdtaυ

−Δ = ∫ ).

g = Earth’s mass attraction gravity vector (relative to earth’s center) at trajectory position location R.

avgg = Constant average approximation of g to simplify the test example model.

I = Identity matrix. Brefl = Specified position displacement vector of varR relative to refR , a constant in the Bref

frame for the test example.

2

Bω = Rotating body angular rate vector relative to non-rotating inertial space that would be measured by strapdown gyros attached to the body and aligned with rotating body axes.

R = Position vector from earth’s center to the trajectory designated position location (“navigation center” ).

mR = Position vector R at the end of trajectory update cycle m. t = Elapsed time from the start of a trajectory.

mt = Time t at the end of trajectory update cycle m.

mT = Time interval from to (assumed constant for this article). 1mt − mtV = Velocity of trajectory position relative to non-rotating inertial space defined as the time rate

of change of position evaluated in inertially non-rotating 0E coordinates: 0

0E

E d RVdt

≡ .

mV = Velocity vector V at the end of trajectory update cycle m.

APPENDIX F

USEFUL FORMULAS

First order expansion approximations are employed in Appendix G and H for matrices of the form [ ]aM aI E= + where I is the identity matrix a is an arbitrary scalar and matrix aE is

much smaller than I. Additionally, note that and that[ ] [ ]1aa aa II EE

− ++ I=

[ ][ ]a aa aI IE E− +

aaI E+

2a aa E E+

] 1−

I=

]1aa

−

. Equating the previous two expressions and multiplying on

the right by [ yields the useful formula:

(F-1) [ ] [ ] [ ] [1 2a a aa a aa aI II IaE E E EE E

− = − + ≈ −+ + Another useful identity derived in [?, Sect. 3.1.1] is [?, Eq. (3.1.1-38)] is: ( ) ( )1C V C VC−× = ×⎡⎣ ⎤⎦ (F-2) where C is an arbitrary direction cosine matrix, V is an arbitrary vector, and (V ×) is the cross-product skew-symmetric form of V as defined in the Notation section of this article.

APPLICABLE EQUATIONS FROM THE MAIN ARTICLE

The following equations taken from the main article are referenced in Appendices G and H to follow, and repeated next for convenient referencing.

3

( )( ) ( )

( )1

0 00 0 011 1

2

2 2

1 cos sin1 1

12

m mm m

m m

mmm var

mmm m m m m

Bvar Bvarvar varBvar Bvar

Vvar var varBvar mBvar var varvarmtBvar Bvartvar SF

Bvar E EE E EmVvarBvarvar var var var var

IG

dta

g gV V C G T

α αα α

α αα

υ

υ

−

−− −

⎛ ⎞− Δ Δ⎜ ⎟≡ + × + − ×Δ Δ⎜ ⎟Δ Δ⎝ ⎠Δ

≡Δ ∫

= + + +Δ

(3)

( )( )

( ) ( )( )

0 0 0 011 1

2

2

2 2

sin1 1 12

1 cos1 12

mm m

m

m

mmm m m m

Bvarvar Bvar

Rvar varBvarBvar varvarm

Bvarvar Bvar

varmBvar Bvarvar varm m

BvarE E E Em mRvarBvarvar var var var

IG

VR R C GT

αα

αα

αα

α α

υ−− −

⎛ ⎞Δ⎜ ⎟≡ + − ×Δ⎜ ⎟Δ⎝ ⎠Δ

⎡ ⎤⎛ ⎞− Δ⎢ ⎥⎜ ⎟

+ − ×Δ⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟Δ Δ⎢ ⎥⎝ ⎠⎣ ⎦

= + + Δ ( )0 01

212

6 m mE E

mvar varg gT T

−+ +

(4)

( ) ( )0 0 0

1 1

0 01

10

1 2/1

26

m m m

m

m m

E E Emvar var varBvar E

mRvarBvarvar mm E Emvar var

VR R TC G T

g g Tυ

− −

−

−

−

⎡ ⎤− −⎢ ⎥

=Δ ⎢ ⎥− +⎢ ⎥

⎣ ⎦

(9)

4

( )( )

( )

022 1

0 02 1 2 1 2 1 2 12 2 2 2

0 0 0 0 02 1 2 2 2 2 2 2 2 2

102 1 22 1

2 1

2 1

2 1

2

12 6

nn

n n n nn n

n n n n n

EEn Rvar VvarBvar Bvar nn

E En mden Rvar Vvar VvarBvar Bvar

E E E E Em n mnum var var var var var

n

C G C GA

C G G C G

n

B A T

V V VR RB T A

A I

−

− − − −− −

− − − −

−− −

−

−

−

≡

⎡ ⎤≡ + −⎢ ⎥⎣ ⎦

≡ − − − −

⎛+ −⎜⎝

T

( )0 0 02 2 1 2

2 12 12 1

2 1 22 1

1

52 6n n n

nnn

E E Enn mvar var var

Bvarnumdenvar

Ag g gIIA T

B Bυ

− −

−−−

−−

−

⎡ ⎤⎞ ⎛ ⎞+ − + −⎟ ⎜ ⎟⎢ ⎥⎠ ⎝ ⎠⎣ ⎦

=Δ

2

(31)

( ) ( )( )

( )( )

02 1 2 12 2

0 02 2 22 1 2 1

0 0 0 0 02 2 2 2 2 2 2 2 2

02 2

102 2 12 2

2

2

22

2

12 6

n nn

n n nn n

n n n n n

n

EEn Rvar Vvar VvarBvar Bvar nn

E En mden Rvar VvarBvar Bvar

E E E E Em nnum var var var var var

Ennvar

C G G C GA

C G C GB A

V V VR Rn m

T

B T A T

A gI IA

− −−

− −

− − −

−

−−≡ +

≡ −

≡ − − − −

⎛ ⎞+ − + −⎜ ⎟⎝ ⎠

0 01 2

222

2 2

1

52 6n n

nnn

E Enmvar var

Bvarnumdenvar

Ag gI T

B Bυ

− −

−

⎡ ⎤⎛ ⎞+ −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

=Δ

2

( ) ( ) ( )

( ) ( )

0 09

0 0 01 1

0 0 01 1

2

2

For 9 :

For 0 9 :

12

For 0 :

1' '

2

m

m m m

m m m

E EBvar Bvar

Bvar Bvar BvarE E Evar var varBvar Bvar Bvar

Bvar Bvar BvarE E Evar var varBvar Bvar Bvar

m C C

m

I IC C C

m

I IC C C

α α α

α α

−

− −

− −

≤ − =

≥ > −

⎡ ⎤ ⎡ ⎤≈ + × + × ≈ +Δ Δ Δ⎢ ⎥ ×⎢ ⎥⎣ ⎦⎣ ⎦>

⎡ ⎤≈ + × + × ≈ +Δ Δ Δ⎢ ⎥⎣ ⎦

( )'α⎡ ⎤×⎢ ⎥⎣ ⎦

(32)

( ) ( )( ) ( )

1For 9 :2

1 1 1For 0 9 :2 2 3

1 1 1For 0 : ' '2 2 3

m m

m m

m m

Vvar Rvar

Bvar BvarVvar var Rvar var

Bvar BvarVvar var Rvar var

m I IG G

m I IG G

m I IG G

α α

α α

≤ − = =

⎡ ⎤≥ > − ≈ + × ≈ + ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤> ≈ + × ≈ +Δ Δ ×⎢ ⎥⎣ ⎦

(33)

0 0 0

0 mmBrefE E E

mref Bvarvar m VR CT= + l (37)

5

0 Constantm

Eavgvar

g ≈ ≡ 0Eg (38)

APPENDIX G

ANALYTICAL DETAIL FOR METHOD 1 TEST EXAMPLE SOLUTION

This appendix provides analytical detail leading to the Method 1 test example solutions of (9) for specific force and resulting velocity, position thereof. First order expansion approximations are employed in the development, facilitated by approximation methods derived in the previous Appendix F.

Under the (37) and (38) example conditions, Method 1 specific force in (9) becomes

( )

( )( )

00 0 01 1

0 0 00 0

00 01 1

120

1

10

1 2

10

1

1 /2

1/1

2

m mm m

m

m m

Bvar E EE E Em m mRvarBvar var var avgvar varmm

BrefE E Em mref Bvar refE

mRvarBvar mm Bref EE Em mBvar avgvar

EBvarm

gVR RC G T T T

m mV l VCT TC G T

gl VC T T

υ− −

− −

−

−

−

−

−

−

⎛ ⎞= − − −Δ ⎜ ⎟⎝ ⎠

⎛ ⎞+ − −⎜ ⎟= ⎜ ⎟

⎜ ⎟− − −⎝ ⎠

= ( ) ( ) ( ) 00 0 0 01 1 0

21 /2m m m

Bref EE E E Em mRvar Bvar Bvar ref avgvarm

gl V VC CC G T T− −⎡ ⎤− − − −⎢ ⎥⎣ ⎦

mT

(G-1)

Substituting (G-1) in (3) with (38) for gravity then obtains for velocity 0

mEvarV :

( ) ( )( )

( )

00 0 011

0 01 00 0

1100 0

1 0

01

10

1

11

12

mmm m m

m mmmm

m

mm

Bvar EE E EmVvarBvar avgvar var var

BrefE EBvar BvarE EE E m mVvar RvarBvar Bvar avgvar mm

EE Emrefvar

EVvar RvarBvar m

gV V C G T

lC C

T gV C G C G TgV V T

I C G G

υ−−

−

−−

−

−

−

−

−−

= + +Δ

⎡ ⎤−⎢ ⎥

⎢ ⎥= + +⎢ ⎥− − −⎢ ⎥⎣ ⎦

= − ( )

( ) ( ) ( )

01

0 01 00

100

0

01

11 01 1

2

m

m mmm

E EBvar varm

BrefE EBvar BvarE EE m mVvar RvarBvar Bvar avgm m

EEmref

VC

lC C

T gC G G C TgV T

−

−

−

−

−−−

⎡ ⎤⎢ ⎥⎣ ⎦

⎡ ⎤−⎢ ⎥

⎢ ⎥+ +⎢ ⎥

+ −⎢ ⎥⎣ ⎦

(G-2)

6

With (32), the term in (G-1) and (G-2) is to first order: ( )0 01m m

E EBvar BvarC C −

−

( ) ( )

( )

0 01

0 0 0 0 01 1 1 1

0 0 01 1

For 9 :

0

For 0 9 :

For 0 :

'

m m

m m m m m

m m m

E EBvar Bvar

Bvar BvarE E E E Evar varBvar Bvar Bvar Bvar Bvar

BvarE E EvarBvar Bvar Bvar

m

C C

m

IC C C C C

m

C C C

α α

α

−

− − − −

− −

≤ −

− =

≥ > −

⎡ ⎤− ≈ + × − =Δ Δ⎢ ⎥⎣ ⎦>

− ≈ ×Δ

×

)

(G-3)

Applying (33) and (F-1) of Appendix F, the term in (G-2) is to first order

accuracy: ( 1

mVvar RvarmG G−

( )

( ) ( ) ( )( ) ( ) ( )

11

11

For 9 :

1 22

For 0 9 :

1 1 12 32

1 1 1 1 12 2 22 3 2 3 6

m

m

Vvar Rvarm

Bvar BvarVvar Rvar var varm

Bvar Bvar Bvar Bvavar var var var

m

I I IG G

m

I IG G

I I I I

α α

α α α

−−

−−

≤ −

⎛ ⎞= =⎜ ⎟⎝ ⎠

≥ > −

⎧ ⎫⎡ ⎤ ⎡ ⎤≈ + × + ×Δ Δ⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎩ ⎭⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎛ ⎞≈ + × − × ≈ + − × = +Δ Δ Δ⎜ ⎟⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎝ ⎠⎣ ⎦

( )

( ) ( )1

For 0 :12 '6m

r

BvarVvar Rvar varm

m

IG G

α

α−

⎡ ⎤×Δ⎢ ⎥⎣ ⎦>

⎡ ⎤≈ + ×Δ⎢ ⎥⎣ ⎦

(G-4)

With (G-4) and (F-2) of Appendix F, the term is given by ( ) ( )01

11 01mm

EEVvar RvarBvar Bvarm mC G G C−

−−−

7

( ) ( )

( ) ( ) ( ) ( )( )( )

01

0 01 1

0 01 1

11 01

1 11 0 01 1

10

1

For 9 :

2

For 0 9 :

126

1 12 23 3

mm

mm m

m m

EEVvar RvarBvar Bvarm m

BvarE EE EVvar Rvar varBvar BvarBvar Bvarm m m

Bvar BvarEE Evar varBvar BvarBvarm

m

IC G G C

m

IC G G CC C

I IC CC

α

α

−

− −

− −

−−−

− −−− −

−

−

≤ −

=

≥ > −

⎡ ⎤= + ×Δ⎢ ⎥⎣ ⎦

= + × = +Δ ( )

( ) ( ) ( )0 01 1

11 01

For 0 :

12 '3mm m

BvarEE EVvar Rvar varBvar BvarBvarm m

m

IC G G CC

α

α− −

−−−

⎡ ⎤×Δ⎢ ⎥⎣ ⎦>

⎡ ⎤= + ×Δ⎢ ⎥⎣ ⎦

(G-5)

Substituting (G-3) – (G-5) in (G-2) and retaining only first order terms then obtains velocity

0m

EvarV generated by Method 1 under the test example conditions:

( )

0 00

00 0 0 01 10

0 00

For 9 :

For 8, 6, 4, 2, 0 :

126

For 7, 5, 3, 1:

m

m mm

m

E Erefvar

BrefBvar Bvar EE E E E

mvar varref Bvar Bvar avgvarm

E Erefvar

m V V

m

l gV V C C TT

m V V

α α− −

≤ − =

= − − − −

⎛ ⎞≈ + × − ×Δ Δ⎜ ⎟⎜ ⎟

⎝ ⎠

= − − − − =

(G-6) For 1, 3, 5, :m =

( )( )

( )

0 0 010

001

0 0 0 01 10

'2

1'

6For 2, 4, 6, :

12

6

mm

m

m mm

BrefBvar BvarE E Evar varref Bvarvar

m

Bvar Bvar EEmvar varBvar avg

BrefBvar BvarE E E Evar varref Bvar Bvarvar

m

lV V CT

gC T

m

lV V C CT

α α

α α

α α

−

−

− −

⎡ ⎤= + − ×Δ Δ⎢ ⎥

⎢ ⎥⎣ ⎦

⎡ ⎤− − ×Δ Δ⎢ ⎥⎣ ⎦=

⎡ ⎤= + × − ×Δ Δ⎢ ⎥

⎢ ⎥⎣ ⎦

0Emavgg T

Derivation steps that led to (G-6) are discussed at the end of this appendix.

Substituting (G-6) into (G-1) with (33) for yields, to first order accuracy, the

corresponding Method 1specific force profile that generated the (G-6) velocity response: RvarmG

8

( )

( ) ( )

( )

0

0

10

9

10

1

For 9 :

For 8, 6, 4, 2, 0 :

123

For 7, 5, 3, 1:

223

m

m

m

Bvar E EmBvar avgvar

BrefBvar Bvar Bvar E E

mvar var Bvar avgvar mm

BrefBvar Bvar Bvar

var varvarm

gm C T

m

l gI C TT

m

l IT

υ

υ α α

υ α α

−

−

−

−

−

≤ − = −Δ

= − − − −

⎡ ⎤≈ × − − ×Δ Δ Δ⎢ ⎥⎣ ⎦= − − − −

⎡ ⎤= − × − − ×Δ Δ Δ⎢ ⎥⎣ ⎦( ) 0

10

1E E

mBvar avgmgC T−

(G-7) For 1, 3, 5, :m =

( )( ) ( )

( )( )

01

01

1

1

2 2'

1'

3For 2, 4, 6, :

2 2'

1 2 '3

m

m

BrefBvar Bvar Bvar

var varvarm

Bvar Bvar E Emvar var Bvar avgm

BrefBvar Bvar Bvar

var varvarm

Bvar Bvarvar var B m

lT

gI C T

m

lT

I

υ α α

α α

υ α α

α α

−

−

−

−

= − ×Δ Δ Δ

⎧ ⎫⎡ ⎤− − + ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭=

= − − ×Δ Δ Δ

⎧ ⎫⎡ ⎤− − − ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭( ) 00E Emvar avggC T

Derivation steps that led to (G-7) are discussed at the end of this appendix.

Velocity Derivation Steps Leading To (G-6): Applying (G-3) – (G-5) in (G-2) and retaining only first order terms obtains velocity 0

mEvarV

generated by Method 1 under test example conditions for m cycles and lower: 8−

9

( ) ( )

( ) ( ) ( )

0 09 0

0 0 0898 9

0 08 9 00

8900

0

11 08 9

11 08 9

For 8 :

For 8 :

12

E Erefvar

EE E EVvar RvarBvar Bvarvar var

BrefE EBvar BvarE EE m mVvar RvarBvar Bvar avg

EEmref

m V V

m

IV VC G G C

lC C

T gC G G C TgV T

−

−−− −

− −−−

−−− −

−−− −

< − =

= −

⎡ ⎤= −⎢ ⎥⎣ ⎦

⎡ ⎤−⎢ ⎥

⎢ ⎥+ +⎢ ⎥

+ −⎢ ⎥⎣ ⎦

( )09 0

13

BvarEvarBvar refI VC α

−⎡ ⎡= − + ×Δ⎢ ⎢⎣ ⎦⎣ ⎦

0E⎤⎤⎥⎥ (G-8)

( )

( )

0 00 0 09 9 0

00 0 09 90

1 123 2

12

6

BrefBvar Bvar E EE E E

m mvar varBvar Bvar ref avg avgm

BrefBvar Bvar EE E E

mvar varref Bvar Bvar avgm

l g gI VC C T TT

l gV C C TT

α α

α α

− −

− −

⎡ ⎤⎛ ⎞⎧ ⎫⎡ ⎤+ + × × + − +⎢ ⎥Δ Δ⎜ ⎟⎨ ⎬⎢ ⎥ ⎜ ⎟⎣ ⎦⎩ ⎭⎢ ⎥⎝ ⎠⎣ ⎦⎛ ⎞

≈ + × − ×Δ Δ⎜ ⎟⎜ ⎟⎝ ⎠

The next velocity calculations use the following first order approximation development based on

in (32) multiplied by 0m

EBvarC ( 1Bvar

varI α−

⎡ + ×Δ⎢⎣ )⎤⎥⎦ with (F-1) from Appendix F.

( ) ( )( ) ( ) ( ) ( )

( ) ( )

0 0 01

0 0 01

0 01

1

Similarly: ' '

m m m

m m m

m m

Bvar BvarE E Evar varBvar Bvar Bvar

Bvar Bvar Bvar BvarE E Evar var var varBvar Bvar Bvar

Bvar BvarE Evar varBvar Bvar

I IC C C

IC C C

C C

α α

α α α

α α

−

−

−

−⎡ ⎤ ⎡ ⎤= + × ≈ − ×Δ Δ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

⎡ ⎤× = − × × ≈ ×Δ Δ Δ⎢ ⎥⎣ ⎦

× ≈ ×Δ Δ

∴ αΔ (G-9)

Eq. (G-9) is used frequently in this appendix for Method 1 velocity derivations and in Appendix H for the Method 2 solution under test example conditions.

Applying (G-8) - (G-9) and (G-3) – (G-5) in (G-2) finds for : 7m = −

10

( ) ( )

( ) ( ) ( )

0 0 07 78 87 8

0 07 8 00

7 78 800

0

11 0

11 0

For 7 :

12



EEmref avg

m

IV VC G G C

lC C

T gC G G C TgV T

− −− −− −

− −− −− −

−−

−−

= −

⎡ ⎤= −⎢ ⎥⎣ ⎦

⎡ ⎤−⎢ ⎥

⎢ ⎥+ +⎢ ⎥

+ −⎢ ⎥⎣ ⎦

( )( )

0 0900

800

9

213 1

6

BrefBvarE Evarref BvarBvarE m

varBvarBvar EE

mvarBvar avg

lV CTI C

gC T

αα

α

−

−

−

⎡ ⎤⎛ ⎞+ ×⎢ ⎥Δ⎜ ⎟⎜ ⎟⎡ ⎤ ⎢ ⎥⎡ ⎤ ⎝= − + ×Δ⎢ ⎥⎢ ⎥

⎠⎢ ⎥⎣ ⎦⎣ ⎦⎢ ⎥− ×Δ⎢ ⎥⎣ ⎦

(G-10)

( )

( )

0 00 0 08 9 0

00 0 08 80

08

1 123 2

12

6

13





BvaEvarBvar

l g gI VC C T TT

l gV C C TT

C

α α

α α

− −

− −

−

⎡ ⎤⎛ ⎞⎧ ⎫⎡ ⎤+ + × × + − +⎢ ⎥Δ Δ⎜ ⎟⎨ ⎬⎢ ⎥ ⎜ ⎟⎣ ⎦⎩ ⎭⎢ ⎥⎝ ⎠⎣ ⎦⎛ ⎞

≈ − − × + ×Δ Δ⎜ ⎟⎜ ⎟⎝ ⎠

− ( )( ) ( )

0 0 080 0

0 0 00 0 08 80

00

2 2

1 13 6

Brefr BvarE E E

varref Bvar refm

Bvar BvarE EE E Em mvar varBvar ref Bvar avg

Eref

lV VCT

g gVC CT T

V

α α

α α

−

− −

⎛ ⎞× + × +Δ Δ⎜ ⎟⎜ ⎟

⎝ ⎠

− + × − × +Δ Δ

=

Emg T

Similarly, the velocity solution for through is as previously derived for and in (G-8) and (G-10).

6m = − 0m = 8m = −7−

Following the procedure leading to (G-10) then obtains for m cycles 1 and 2:

11

( ) ( )

( ) ( ) ( )

0 0 01 10 01 0

0 01 0 00

1 10 000

0

11 0

11 0

For 1:

12



EEmref avg

m

IV VC G G C

lC C

T gC G G C TgV T

−−

−−

=

⎡ ⎤= −⎢ ⎥⎣ ⎦

⎡ ⎤−⎢ ⎥

⎢ ⎥+ +⎢ ⎥

+ −⎢ ⎥⎣ ⎦

( )( )

0 0100

000

1

21'

3 16

BrefBvarE Evarref BvarBvarE m

varBvarBvar EE

mvarBvar avg

lV CTI C

gC T

αα

α

−

−

⎡ ⎤⎛ ⎞+ ×⎢ ⎥Δ⎜ ⎟⎜ ⎟⎡ ⎤ ⎢ ⎥⎡ ⎤ ⎝= − + ×Δ⎢ ⎥⎢ ⎥

⎠⎢ ⎥⎣ ⎦⎣ ⎦⎢ ⎥− ×Δ⎢ ⎥⎣ ⎦

(G-11)

( )

( )

0 00 0 00 0 0

00 0 00 00

00

1 12 ' '3 2

12

6

13





BvarEvarBvar

l g gI VC C T TT

l gV C C TT

C

α α

α α

α

⎡ ⎤⎛ ⎞⎧ ⎫⎡ ⎤+ + × × + − +⎢ ⎥Δ Δ⎜ ⎟⎨ ⎬⎢ ⎥ ⎜ ⎟⎣ ⎦⎩ ⎭⎢ ⎥⎝ ⎠⎣ ⎦⎛ ⎞

≈ − − × + ×Δ Δ⎜ ⎟⎜ ⎟⎝ ⎠

− Δ( )( ) ( )( )

0 0 000 0

0 0 00 0 00 00

0 0 00 00

2 2' '

1 1' '

3 6

12 ' '6

BrefBvarE E Evarref Bvar ref

m

Bvar BvarE EE E Em mvar varBvar ref Bvar avg

BrefBvar Bvar BvarE E Evar var varref Bvar Bvar

m

lV VCT

g gVC CT T

lV C CT

α

α α

α α α

⎛ ⎞× + × +Δ⎜ ⎟⎜ ⎟

⎝ ⎠

− + × − × +Δ Δ

⎡ ⎤= + − × −Δ Δ Δ⎢ ⎥

⎢ ⎥⎣ ⎦

Emg T

( ) 0Bvar Emvar g Tα⎡ ⎤− ×Δ⎢ ⎥⎣ ⎦

12

( ) ( )

( ) ( ) ( )

0 0 02 21 12 1

0 02 1 00

2 21 100

0

11 0

11 0

For 2 :

12



EEmref avg

m

IV VC G G C

lC C

T gC G G C TgV T

−−

−−

=

⎡ ⎤= −⎢ ⎥⎣ ⎦

⎡ ⎤−⎢ ⎥

⎢ ⎥+ +⎢ ⎥

+ −⎢ ⎥⎣ ⎦

( ) ( )( )

0 0000

100

0

2 '1

'3 1

'6

BrefBvar BvarE Evar varref BvarBvarE m

varBvarBvar Bvar EE

mvar varBvar avg

lV CTI C

gC T

α αα

α α

⎡ ⎤⎡ ⎤+ − ×⎢ ⎥Δ Δ⎢ ⎥

⎡ ⎤ ⎢ ⎥⎡ ⎤ ⎢ ⎥⎣ ⎦= − + ×Δ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎡ ⎤⎢ ⎥− − ×Δ Δ⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦

(G-12)

( )

( ) ( )

0 00 0 01 1 0

00 0 01 10

1 12 ' '3 2

1' '2

6



BrefBvar Bvar Bvar Bvar EE E Evar var var varref Bvar Bvar avg

m

l g gI VC C T TT

l gV C CT

α α

α α α α

⎡ ⎤⎛ ⎞⎧ ⎫⎡ ⎤+ + × × + − +⎢ ⎥Δ Δ⎜ ⎟⎨ ⎬⎢ ⎥ ⎜ ⎟⎣ ⎦⎩ ⎭⎢ ⎥⎝ ⎠⎣ ⎦⎡ ⎤

≈ − − − × + − ×Δ Δ Δ Δ⎢ ⎥⎢ ⎥⎣ ⎦

( )( ) ( )

0 0 0 01 10 0

0 0 00 0 01 10

0 010

1 2 2' '3

1 1' '

3 6

2

m

BrefBvar BvarE E E Evar varBvar ref Bvar ref

m

Bvar BvarE EE E Em mvar varBvar ref Bvaravg avg

BrefBvarE Evarref Bvar

m

T

lV VC CT

g gVC CT T

lV CT

α α

α α

α

Emg T

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞− × + × +Δ Δ⎜ ⎟⎜ ⎟

⎝ ⎠

− + × − × +Δ Δ

⎛ ⎞= + × −Δ⎜ ⎟⎜ ⎟

⎝ ⎠( )00

116

Bvar EEmvarBvar avggC Tα ×Δ

Similarly, the velocity solution for is as previously derived for and 2 in (G-11) and (G-12). The combined result for

2m > 1m =0

mEvarV velocity is then as shown in (G-6).

Specific Force Derivation Steps Leading To (G-7):

Included in the following development for specific force is the observation from (33) with

(H-2) that to first order accuracy, ( ) ( )1 123

BvarRvar varm IG α

− ⎡ ⎤≈ − ×Δ⎢ ⎥⎣ ⎦for 0 9 , and m≥ > −

( ) (1 12 '3

BvarRvar varm IG α

− ⎡≈ − ×Δ⎢⎣ ⎦)⎤⎥ for m > 0. Applying (G-3) – (G-6) in (G-1) and retaining

only first order terms then obtains specific force m

BvarvarυΔ generated by Method 1 under test

example conditions for eac m cycle:

13

( )( )

00 01 0

0 0

10

1

1 10 0

9 9

For 8 :

12

112 2

m mBvar E EE E

mRvar refBvar avgvar varmm

E E E Em mBvar Bvaravg avg

m

gV VC G T

g gIC CT T

υ−

−

−

− −

− −

< −

⎛ ⎞= − − +Δ ⎜ ⎟⎝ ⎠

⎛ ⎞= − = −⎜ ⎟⎝ ⎠

(G-13) For 8, 6, 4, 2, 0 :m = − − − −

( )

( ) ( )

00 0 01 1 0

00 0 01 0 0

10

1

10

1

12

1 123 2

mm m

m

BrefBvar BvarE EE E E

mvarRvar Bvar refBvar avgvar varmm m


mvar varBvar ref refBvar avgm m

l gV VCC G TT

l gI V VCC TT

υ α

α α

− −

−

−

−

−

−

⎡ ⎤⎛ ⎞= × − +⎢ ⎥Δ Δ⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

⎡ ⎛ ⎞⎡ ⎤= − × × − + −⎢Δ Δ⎜ ⎟⎢ ⎥ ⎜ ⎟⎣ ⎦ ⎝ ⎠⎣

−

( ) ( ) 01

01

123

BrefBvar Bvar E E

mvar var Bvar avgmm

l gI C TT

α α−

−

⎤⎥

⎢ ⎥⎦

⎡ ⎤≈ × − − ×Δ Δ⎢ ⎥⎣ ⎦

14

( )

( ) ( )

00 0 01 1 0

0 01 2

10

1

10

1

For 7, 5, 3, 1:

12

2123

mm m

m m



BrefBvar BvarE Evar varBvar BvarBvar E m

var Bvarm

m

l gV VCC G TT

lC C

TI C

υ α

αα

− −

− −

−

−

−

−

= − − − −

⎡ ⎤⎛ ⎞= × − +⎢ ⎥Δ Δ⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

× −Δ Δ⎡ ⎤= − ×Δ⎢ ⎥⎣ ⎦

−

( ) 0 002

1 16 2m

Bref

m

Bvar E EEm mvarBvar avg

lT

g gC T T

α

α−

⎧ ⎫⎛ ⎞×⎪ ⎪⎜ ⎟⎜ ⎟⎪ ⎪⎝ ⎠⎨ ⎬

⎪ ⎪+ × −Δ⎪ ⎪⎩ ⎭

( ) 02

10

12 4

m

Bref BrefBvar BvarE Evar varBvarBvarmm m

l lCC

T Tα

−

−

−≈ × − ×Δ αΔ (G-14)

( ) ( ) ( ) ( )( ) ( )( )

( )( )

0 002

001

0

1 10 0

1 1

1 10 0

1 1

10

1

1 13 3

123

13

m

m

Bvar BvarE E EEm mvar varBvarBvar Bvaravg avgm m

BrefBvar BvarE EE

mvar varBvarBvar Bvar avgm mm

Bvar Evar Bvar avgm

g gICC CT T

l gCC C TT

C

α α

α α

α

−

−

− −

− −

− −

− −

−

−

⎡ ⎤⎡ ⎤+ × − − ×Δ Δ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

⎡ ⎤≈ − × + ×Δ Δ⎢ ⎥

⎣ ⎦

+ ×Δ

E

E

( )( )( ) ( )

( ) ( )

0

0 0

0

10

1

1 10 0

1 1

10

1

223

223

E E Em mBvar avgm

BrefBvar Bvar E E E E

m mvar var Bvar Bvaravg avgm mmBref

Bvar Bvar E Emvar var Bvar avgmm

g gCT T

l g gC CT TT

l gI C TT

α α

α α

−

−

− −

− −

−

−

−

= − × + × −Δ Δ

⎡ ⎤= − × − − ×Δ Δ⎢ ⎥⎣ ⎦

15

( )

( ) ( )

00 0 01 1 0

0 01 2

10

1

10

1

For 1, 3, 5, :

1'

2

2'12 '3

mm m

m m



Bref BrefBvar BvarE Evar varBvar BvarBvar E m m

var Bvarm

m

l gV VCC G TT

l lC C

T TI C

υ α

α αα

− −

− −

−

−

−

−

=

⎛ ⎞= × − +Δ Δ⎜ ⎟⎜ ⎟

⎝ ⎠

× − ×Δ Δ⎡ ⎤= − ×Δ⎢ ⎥⎣ ⎦

−

( ) 0 002

1 16 2m

Bvar E EEm mvarBvar avgg gC T Tα

−

⎧ ⎫⎛ ⎞⎪ ⎪⎜ ⎟⎜ ⎟⎪ ⎪⎝ ⎠⎨ ⎬⎪ ⎪

+ × −Δ⎪ ⎪⎩ ⎭

( ) 02

10

12 4'

m

Bref BrefBvar BvarE Evar varBvarBvarmm m

lCC

T Tα

−

−

−≈ × − ×Δ lαΔ (G-15)

( ) ( ) ( ) ( )( ) ( ) ( )( )

( )

0 002

001

1 10 0

1 1

1 10 0

1 1

1

1 1'

3 3

12 2'3

1'

3

m

m

Bvar BvarE E EEm mvar varBvarBvar Bvaravg avgm m

BrefBvar Bvar BvarE EE

mvar var varBvarBvar Bvar avgm mm

Bvarvar m

g gICC CT T

l gCC C TT

α α

α α α

α

−

−

− −

− −

− −

− −

−

⎡ ⎤⎡ ⎤+ × − − ×Δ Δ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦E

E⎡ ⎤≈ − × + ×Δ Δ Δ⎢ ⎥

⎣ ⎦

+ ×Δ ( ) ( )( ) ( ) ( )

( )( ) ( )

0 0

0

0

10 0

1 1

10

1

10

1

12 2' '3

12 2' '3

E E E Em mBvar Bvaravg avgm

BrefBvar Bvar Bvar Bvar E E

mvar var var var Bvar avgmm

E EmBvar avgm

BrefBvar Bvar Bvar Bvarvar var var var

m

g gC CT T

l gC TT

gC T

l IT

α α α α

α α α α

−

− −

−

−

−

−

−

⎡ ⎤= − × + + ×Δ Δ Δ Δ⎢ ⎥⎣ ⎦

−

⎡= − × − − + ×Δ Δ Δ Δ⎢⎣ ( ) 01

01

E EmBvar avgm

gC T−

−⎧ ⎫⎤⎨ ⎬⎥⎦⎩ ⎭

16

( )

( ) ( )

00 0 011 1 0

01

021

10

10

For 2, 4, 6, :

1'

2

'

12 2' '3

m mmm m

m

mm


mvarRvar Bvar refBvar avgvar varm

BrefBvarEvarBvar

m

Bvar Bvar BvaE Evar var varBvarBvar

m

l gV VCC G TT

lC

T

I CC

υ α

α

α α

−− −

−

−−

−

−

=

⎛ ⎞= × − +Δ Δ⎜ ⎟⎜ ⎟

⎝ ⎠

×Δ

⎡ ⎤= − × − −Δ Δ⎢ ⎥⎣ ⎦

−

( )( )

( ) ( )

0 002

021

10

1 1'

6 2

2 4' '

m

mm

Brefr

m

Bvar Bvar E EEm mvar varBvar avg avg

Bref BrefBvar Bvar BvarE Evar var varBvarBvar

m m

lT

g gC T T

l lCC

T T

α

α α

α α α

−

−−

−

⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪⎡ ⎤⎪ ⎪×Δ⎢ ⎥⎨ ⎬

⎢ ⎥⎪ ⎪⎣ ⎦⎪ ⎪

⎡ ⎤⎪ ⎪+ − × −Δ Δ⎢ ⎥⎣ ⎦⎪ ⎪⎩ ⎭⎡ ⎤

≈ × − − ×Δ Δ Δ⎢ ⎥⎢ ⎥⎣ ⎦

( ) ( ){ }0021

101

'3 mm

Bvar BvarE EEmvar varBvarBvar avggCC Tα α

−−

−⎡+ −Δ Δ⎢⎣

⎤×⎥⎦ (G-16)

( ) ( )( )

( ) ( ) ( )( )( ) ( )

01

0011 1

01

10

1 10 0

1 10 0

1

1'

3

2 2'

1'

3

1'

3

m

mm m

m

Bvar E Emvar Bvar avg

BrefBvar Bvarvar var

m

Bvar BvarE EEmvar varBvarBvar Bvar avg

Bvar E E Emvar Bvar Bvaravg am

gI C T

lT

gCC C T

gC CT

α

α α

α α

α

−

−− −

−

−

− −

− −

−

⎡ ⎤− − ×Δ⎢ ⎥⎣ ⎦

≈ − − ×Δ Δ

⎧ ⎫⎡ ⎤+ − ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

+ × −Δ

E

( ) ( ) ( )( )

( ) ( ) ( )

0

01

01

0

10

10

10

1

12 2 2' '3

12 2 2' '3

m

m

Emvg


mvar var var var Bvar avgm

E EmBvar avg

BrefBvar Bvar Bvar Bvar E Evar var var var Bvar avgmm

g T

l gC TT

gC T

l gI CT

α α α α

α α α α

−

−

−

−

−

−

⎡ ⎤= − − × + − ×Δ Δ Δ Δ⎢ ⎥⎣ ⎦

−

⎧ ⎫⎡ ⎤= − − × − − − ×Δ Δ Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭mT

The combined (G-13) – (G-16) result for

mBvarvarυΔ specific force is then as shown in (G-7).

17

APPENDIX H

ANALYTICAL DETAIL FOR METHOD 2 TEST EXAMPLE SOLUTION

This appendix provides analytical detail leading to the Method 2 test example solution of Eqs. (31) for specific force and resulting velocity, position thereof. SPECIFIC FORCE

Under test example conditions, Method 2 performance is demonstrated with gravity

approximated to be constant 0Eg as in (38). Then (31) simplifies to

( )( )

( )( )

022 1

0 02 1 2 1 2 1 2 12 2 2 2

0 0 0 0 02 1 2 2 2 2 2 2 2 2

0

102 1 22 1

2 1

2 1

2 1

2

2

nn

n n n nn n

n n n n n




En avg

C G C GA

C G G C G

n

B A T

V V VR RB T A

IA

−

− − − −− −

− − − −

−− −

−

−

−

≡

⎡ ⎤≡ + −⎢ ⎥⎣ ⎦

≡ − − − −

+ −

T

2 12 12 1

2

1nnn

m

Bvarnumdenvar

g T

B Bυ −−−−=Δ

(H-1)

( ) ( )( )

( )( )

02 1 2 12 2

0 02 2 22 1 2 1

0 0 0 0 02 2 2 2 2 2 2 2 2

0

22

102 2 12 2

2

2

22

2

2

n nn

n n nn n

n n n n n

nn




En mavg

Bvarvar

C G G C GA

C G C GB A

V V VR Rn m

T

B T A T

gIA T

υ

− −−

− −

− − −

−

−−≡ +

≡ −

≡ − − − −

+ −

=Δ 21

nnumdenB B−

The development sequence for deriving

2 1nBvarvarυ

−Δ and

2nBvarvarυΔ in (H-1) for the test example is,

based on (32) and (33), first finding , 2 1nA − 2nA , then , , then 2 1ndenB − 2ndenB 02n

EvarR , 0

2nEvarV ,

then , , and from those, 2nnumB 2 1nnumB − 2nBvarvarυ

1−Δ and

2nBvarvarυΔ .

Finding 2 1nA − And 2nA

To derive , 2 1nA − 2nA , first expand their equations in (H-1):

18

(H-2)

( )( ) ( )

( ) ( )( )

022 1

022 1

02 1 2 12 2

02 12 2

022 2

102 1 22 1

11 02 2 1

102 2 12 2

10

2 12 2

nn

nn

n nn

nn

nn


EERvar VvarBvar Bvarn n


EERvar VvarBvar Bvar nn

EBvar

C G C GA

C G G C

C G G C GA

IC G C G

C

−

−

− −−

−−

−

−− −

−−−

−

−−−

−−

≡

=

≡ +

= +

= ( ) ( )1

11 02 1 2 2

ERvar Vvar Bvarn n

IG G C−

−−− −

+

To obtain in (H-2), apply (33) for and , yielding to first

order accuracy: (2

12nRvar Vvar nG G

−) 2Vvar nG 2nRvarG

( )

( ) ( ) ( )( ) ( ) ( )

( )

2 2

2 2

2 2

1

11

1

1For 5 :2

For 0 4 :

1 1 122 3

1 1 1 1 12 3 2 2 6

1 1For 0 :2 6

n n

n n

n n

Rvar Vvar

Bvar BvarRvar var varVvar

Bvar Bvar Bvarvar var var

Rvar vaVvar

n IG G

n

I IG G

I I I

n IG G

α α

α α α

−

−−

−

≤ − =

≥ ≥ −

⎡ ⎤ ⎡ ⎤≈ + × + ×Δ Δ⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦⎡ ⎤ ⎡ ⎤ ⎡= + × − × ≈ − ×Δ Δ Δ⎢ ⎥ ⎢ ⎥ ⎢⎣ ⎦ ⎣ ⎦ ⎣

> ≈ −

⎤⎥⎦

( )'Bvarrα⎡ ⎤×Δ⎢ ⎥⎣ ⎦

(H-3)

Substituting (H-3) in (H-2) and dropping higher order terms then obtains , 2 1nA − 2nA :

19

( ) ( )( )( ) ( )

( )

02 22 1 2 1

02 1 2 1

02 1 2 12 2 2 2

02 2 2 2

022 1

11 02 1

10

11 02

10

2 1

For 5 :

1 12 2

1 32 2For 4 :

n nn n

n n

n nn n

n n

nn

EEn Rvar VvarBvar Bvar

EEBvar Bvar


EEBvar Bvar

En RBvar

n

C G G CA

I IC C

IC G G CA

I I IC C

n

CA

− −

− −

− −− −

− −

−

−−−

−

−−

−

−

≤ −

=

= =

= +

= + =

= −

= ( ) ( )( ) ( )( )( )

2 1

02 1 2 1

02 1 2 1

11 02

10

10

1 12 6

1 12 6

n

n n

n n

Evar Vvar Bvarn

Bvar EEvarBvar Bvar

Bvar EEvarBvar Bvar

G G C

IC C

I C C

α

α

−

− −

− −

−−

−

−

⎡ ⎤= − ×Δ⎢ ⎥⎣ ⎦⎡ ⎤

= − ×Δ⎢ ⎥⎣ ⎦

( ) ( )02 1 2 12 2 2 2

11 0232n nn n

EEn Rvar VvarBvar Bvar I IC G G CA − −− −

−−= + = (H-4)

( )( )( ) ( )

( ) ( )( )

02 1 2 1

02 1 2 12 2 2 2

02 2 2 2

02 2 2 2

102 1

11 02

10

1

For 0 3 :

1 12 6

1 12 6

3 12 18

n n

n nn n

n n

n n

Bvar EEn varBvar Bvar


Bvar EEvarBvar Bvar

BvarEvarBvar Bva

n

I C CA

IC G G CA

I IC C

I C

α

α

α

− −

− −− −

− −

− −

−−

−−

−

−

≥ ≥ −

⎡ ⎤= − ×Δ⎢ ⎥

⎣ ⎦

= +

⎡ ⎤= − ×Δ⎢ ⎥⎣ ⎦

= − ×Δ ( )+

( )( )( )( )

02 1 2 1

02 2 2 2

0

102 1

102

For 0 :

1 1'

2 6

3 1'

2 18

n n

n n

Er



C

n

I C CA

I C CA

α

α

− −

− −

−−

−

⎡ ⎤⎢ ⎥⎣ ⎦

>

⎡ ⎤= − ×Δ⎢ ⎥

⎣ ⎦⎡ ⎤

= − ×Δ⎢ ⎥⎣ ⎦

Summary of (H-4) Results:

20

( )( )02 1 2 1

2 1 2

102 1 2

1 3For 5 :2 2

For 4 :

1 1 32 6 2n n

n n

Bvar EEn nvarBvar Bvar

n I IA A

n

I IC CA Aα− −

−

−−

≤ − = =

= −

⎡ ⎤= − × =Δ⎢ ⎥

⎣ ⎦

( )( )( )( )

02 1 2 1

02 2 2 2

102 1

10

2

For 0 3 :

1 12 6

3 12 18

n n

n n



n

I C CA

I C CA

α

α

− −

− −

−−

−

≥ ≥ −

⎡ ⎤= − ×Δ⎢ ⎥

⎣ ⎦⎡ ⎤

= − ×Δ⎢ ⎥⎣ ⎦

(H-5)

( )( )( )( )

02 1 2 1

02 2 2 2

102 1

10

2

For 0 :

1 1'

2 6

3 1'

2 18

n n

n n



n

I C CA

I C CA

α

α

− −

− −

−−

−

>

⎡ ⎤= − ×Δ⎢ ⎥

⎣ ⎦⎡ ⎤

= − ×Δ⎢ ⎥⎣ ⎦

Finding 2 1ndenB − And 2ndenB

To determine and , substitute (33) and (H-4) into the , expressions in (H-2), and drop higher order terms:

2 1ndenB − 2ndenB 2 1ndenB − 2ndenB

( )0 02 1 2 1 2 1 2 12 2 2 2

0 0 02 2 2 2 2 2 9

0 02 2 22 1 2 1

2 1

2

For 4 :

1 12 2

n n n nn n

n n n

n n nn n


E E E Em mBvar Bvar Bvar Bvar

E Enden Rvar VvaBvar Bvar

n

C G G C G

0 m

B A T

I I I IC C C CT T

C G CB A

− − − −− −

− − −

− −

−

< −

⎡ ⎤= + −⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞= + − = =⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

= −( )T−

0 0 0 02 1 2 1 2 1 9

1 32 2n n n

mr

E E E Em mBvar Bvar Bvar Bvar

G T

I I IC C C CT T− − − −⎛ ⎞= − = − = −⎜ ⎟⎝ ⎠

mT

(H-6)

21

( )

( )( )0 0

2 1 2 1 2 1 2 12 2 2 2

0 02 2 2 1 2 22 1

02 2 2 2

2 1

10

1

For 4 :

1 1 12 2 6

112

n n n nn n

n n n

n n


Bvar EE E EmvarBvar Bvar BvarBvar

EBvar B

n

C G G C GB A

I I I IC C CC T

IC

α

− − − −− −

− − −

− −

−

−

−

= −

⎡ ⎤= + −⎢ ⎥⎣ ⎦⎧ ⎫⎡ ⎤⎪ ⎪⎛ ⎞= + − − ×Δ⎨ ⎬⎢ ⎥⎜ ⎟

⎝ ⎠⎪ ⎪⎣ ⎦⎩ ⎭

= + ( )

0n

T

−

( )( )0 02 1 2 22 1

10 0

n nnBvarE EE E

mvarBvar Bvarvar BvarC CC C Tα− −−

−⎡ ⎤×Δ⎢ ⎥

⎣ ⎦

( )09

112

BvarEmvarBvar IC Tα

−⎡= + ×Δ⎢⎣ ⎦

⎤⎥ (H-7)

( )( ) ( )

( )

0 02 2 22 1 2 1

0 02 1 2 1

09

2

1 1 3 12 3 2 2

712

n n nn n

n n


Bvar BvarE Emvar varBvar Bvar

BvarEmvarBvar

C G C GB A

I I IC C T

IC T

α α

α

− −

− −

−

= −

⎧ ⎫⎡ ⎤ ⎡= + × − + ×Δ Δ⎨ ⎬⎢ ⎥ ⎢⎣ ⎦ ⎣⎩ ⎭⎡ ⎤= − + ×Δ⎢ ⎥⎣ ⎦

T

⎤⎥⎦

22

( )

( ) ( )( )( )

0 02 1 2 1 2 1 2 12 2 2 2

02 2

02 1 2 22 1

2 1

10

For 0 4 :

1 1 12 3 2

1 12 6

n n n nn n

n

n nn


Bvar BvarEvar varBvar

Bvar EEvarBvar BvarBvar

n

C G G C GB A

I IC

I C C

α α

α

− − − −− −

−

− −−

−

−

≥ > −

⎡ ⎤= + −⎢ ⎥⎣ ⎦

⎧ ⎫⎡ ⎤ ⎡ ⎤+ × + + ×Δ Δ⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎩ ⎭=

⎡ ⎤− − ×Δ⎢ ⎥

⎣ ⎦

T

( )0 12

mBvarEvar

TIC α⎡ ⎤+ ×Δ⎢ ⎥⎣ ⎦

( )( )( )

( )

02 2

02 1 2 22 1

02 2

10

512

112

12

n

n nn

n

BvarEvarBvar

mBvar EEvarBvar BvarBvar

BvarEmvarBvar

ICT

C C

IC T

α

α

α

−

− −−

−

−

⎧ ⎫⎡ ⎤+ ×Δ⎪ ⎪⎢ ⎥⎪ ⎪⎣ ⎦= ⎨ ⎬⎪ ⎪+ ×Δ⎪ ⎪⎩ ⎭

⎡ ⎤= + ×Δ⎢ ⎥⎣ ⎦

0EC (H-8)

( )( )

( )( ) ( )

0 02 2 22 1 2 1

02 1

0 02 2 2 12 2

02 1 2 1

2

10

1 12 3

3 1 12 18 2

16

n n nn n

n

n nn

n n


BvarEvarBvar

mBvar BvarEE Evar varBvar BvarBvar

EBvar Bv

C G C GB A T

IC

TI IC CC

C

α

α α

− −

−

− −−

− −

−

= −

⎧ ⎫⎡ ⎤+ ×Δ⎪ ⎪⎢ ⎥⎣ ⎦⎪ ⎪= ⎨ ⎬⎡ ⎤ ⎡ ⎤⎪ ⎪− − × + ×Δ Δ⎢ ⎥ ⎢ ⎥⎪ ⎪⎣ ⎦⎣ ⎦⎩ ⎭

− +=

( ) ( )( )

( )

0 02 1

02 2

02 1

34

112

12

n

n

n

Bvar BvarE Evar varar Bvar

mBvarEvarBvar

BvarEmvarBvar

C CT

C

IC T

α α

α

α

−

−

−

⎧ ⎫× − ×Δ Δ⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪+ ×Δ⎪ ⎪⎩ ⎭

⎡ ⎤= − + ×Δ⎢ ⎥⎣ ⎦

23

( )

( ) ( )( )( )

0 02 1 2 1 2 1 2 12 2 2 2

02 2

02 1 2 22 1

2 1

10

For 0 :

1 1 1' '

2 3 2

1 1'

2 6

n n n nn n

n

n nn



Bvar EEvarBvar BvarBvar

n

C G G C GB A

I IC

I C C

α α

α

− − − −− −

−

− −−

−

−

>

⎡ ⎤= + −⎢ ⎥⎣ ⎦

⎧ ⎫⎡ ⎤ ⎡ ⎤+ × + + ×Δ Δ⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎩ ⎭=

⎡ ⎤− − ×Δ⎢ ⎥

⎣ ⎦

T

( )( ) ( )

( )( )

0

0 0 02 2 2 2 2 2

0 02 1 2 22 1

10

1'

2

2 1' '

3 41

'12

n n n

n nn

mBvarEvar


mBvar EE EvarBvar BvarBvar

TIC

C C CT

C CC

α

α α

α

− − −

− −−

−

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎡ ⎤⎢ ⎥+ ×Δ⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦

⎡ ⎤+ × − ×Δ Δ⎢ ⎥⎢ ⎥=⎢ ⎥+ ×Δ⎢ ⎥⎣ ⎦

( )02 2

1'

2nBvarE

mvarBvar IC Tα−⎡= + ×Δ⎢⎣ ⎦

⎤⎥ (H-9)

( )( )

( )( ) ( )

0 02 2 22 1 2 1

02 1

0 02 2 2 12 2

02 1 2

2

10

1 1'

2 3

3 1 1' '

2 18 2

16

n n nn n

n

n nn

n n


BvarEvarBvar

mBvar BvarEE Evar varBvar BvarBvar

EBvar

C G C GB A T

IC

TI IC CC

C

α

α α

− −

−

− −−

− −

−

= −

⎧ ⎫⎡ ⎤+ ×Δ⎪ ⎪⎢ ⎥⎣ ⎦⎪ ⎪= ⎨ ⎬⎡ ⎤ ⎡ ⎤⎪ ⎪− − × + ×Δ Δ⎢ ⎥ ⎢ ⎥⎪ ⎪⎣ ⎦⎣ ⎦⎩ ⎭

− +=

( ) ( )( )

( )

0 01 2 1

02 2

02 1

3' '

41

'12

1'

2

n

n

n


mBvarEvarBvar

BvarEmvarBvar

C CT

C

IC T

α α

α

α

−

−

−

⎧ ⎫× − ×Δ Δ⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪+ ×Δ⎪ ⎪⎩ ⎭

⎡ ⎤= − + ×Δ⎢ ⎥⎣ ⎦

Summary of (H-6) to (H-9) Results:

24

( ) ( )

0 02 1 29 9

0 02 1 29 9

For 4 :

For 4 :1 7

12 12

n n

n n

E Em mden denBvar Bvar

Bvar BvarE Em mden denvar varBvar Bvar

n

C CB T B T

n

I IC CB T B Tα α

− − −

− − −

< −

= = −

= −

⎡ ⎤ ⎡ ⎤= + × = − + ×Δ Δ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

(H-10) For 0 4 :n≥ > −

( ) ( )

( ) ( )

0 02 1 22 2 2 1

0 02 1 22 2 2 1

1 12 2

For 0 :1 1

' '2 2

n nn n

n nn n



I IC CB T B T

n

I IC CB T B T

α α

α α

− − −

− − −

⎡ ⎤ ⎡= + × = − + ×Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣>

⎡ ⎤ ⎡= + × = − + ×Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣

⎤⎥⎦

⎤⎥⎦

Finding The 0

2nEvarR And 0

2nEvarV Terms

Eq. (37) for 0

mEvarR with is used as the basis for evaluating the 2m = n 0

2nEvarR , 0

2nEvarV terms

in (H-1) under the example conditions.

( ) ( )

( )

0 0 022 0

0 01 10 0

2

0 0 0 01 10 0

0 02 1 2 10

0

2

2

1 1

2

2

nn

m mn m

m m

n n

BrefE E Emref Bvarvar

E Evar varE E

var varm

Bref BrefE E E Em mref Bvar ref Bvar

mBrefE E

Bvar BvarEref

m

n V lR CT

R RV V

T

m mV l VC CT T

T

lC CV

T

+ −

+ −

+ −

= +

−= =

+ + − − −=

−= +

l (H-11)

From (H-11) with (32) for , the 0

mEBvarC 0 0

2 2nE Evar varR R

−−

2nposition change in (H-1) becomes:

25

( ) ( )( )

0 0 02 2 2 0

0 0 0 0 02 8 9 2

0 0 0 0 02 8 9 9

2

2

For 4 : 2

For 4 :

12

12

n n

n n

n n

E E Emrefvar var



n VR R T

n

IC C C C C

C C C C C

α α

α

−

− − −

− − − −

< − − =

= −

⎡ ⎤= = + × + × =Δ Δ⎢ ⎥⎣ ⎦

− = − = × +Δ Δ

9−

( )( )

( ) ( )

0 0 0 0 0 0 08 92 2 2 8 10 0

0 090

2

2

122

For 0 4 :

n nBrefE E E E E E E

mref Bvar Bvarvar var var var

Bvar Bvar BrefE Em var varref Bvar

V lR R R R C CT

V lCT

n

α

α α

− −− − −

−

⎡ ⎤×⎢ ⎥⎣ ⎦

− = − = + −

⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦

≥ > −

( ) ( )( ) ( )

0 02 2 1

0 02 1 2 2

2

2

12

12

n n

n n



IC C

IC C

α α

α α

−

− −

⎡ ⎤= + × +Δ Δ ×⎢ ⎥⎣ ⎦⎡ ⎤= + × +Δ Δ ×⎢ ⎥⎣ ⎦

(H-12)

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( )

0 02 2 2

02 2

02 2

2 2

2 2

2

1 12 2

1 12 2

2 2

n n

n

n

Bvar Bvar Bvar BvarE Evar var var varBvar Bvar

Bvar Bvar Bvar Bvar BvarEvar var var var varBvar

Bvar BvaEvar varBvar

I IC C

IC

IC

α α α α

α α α α α

α

−

−

−

⎡ ⎤ ⎡ ⎤= + × + × + × + ×Δ Δ Δ Δ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

⎡ ⎤= + × + × + × + × + ×Δ Δ Δ Δ Δ⎢ ⎥⎣ ⎦

= + × +Δ

2

( )( ) ( )

( ) ( )( ) ( )

0 0 02 2 2 2 2

02 2

0 0 0 02 22 2 2 0

2

2

2

2 2

2

2 2

Fo

n n n

n

nn n

r



Bvar Bvar BrefE E E Em var varref Bvarvar var

I IC C C

C

V lR R CT

α

α α

α α

α α

− −

−

−−

⎡ ⎤× +Δ⎢ ⎥⎣ ⎦

⎡ ⎤− = + × + × + −Δ Δ⎢ ⎥⎣ ⎦

⎡ ⎤≈ × + ×Δ Δ⎢ ⎥⎣ ⎦

⎡ ⎤− ≈ + × + ×Δ Δ⎢ ⎥⎣ ⎦

( ) ( )0 0 0 02 22 2 2 0

2

r 0 :

2 2 ' 'nn n


n

V lR R CT α α−−

>

⎡ ⎤− ≈ + × + ×Δ Δ⎢ ⎥⎣ ⎦


26

( ) ( )

( ) ( )

0 0 02 2 2 0

0 0 0 098 10 0

0 0 0 02 22 2 2 0

2

2

For 4 : 2

For 4 :

122

For 0 4 :

2 2

n n

nn n

E E Emrefvar var


Bvar BvarE E E Em var varref Bvarvar var

n VR R T

n

V lR R CT

n

VR R CT

α α

α α

−

−− −

−−

< − − =

= −

⎡ ⎤− = + × + ×Δ Δ⎢ ⎥⎣ ⎦

≥ > −

⎡− ≈ + × + ×Δ Δ⎣

( ) ( )0 0 0 02 22 2 2 0

2

For 0 :

2 2 ' 'nn n

Bref


l

n

V lR R CT α α−−

⎤⎢ ⎥

⎦>

⎡ ⎤− ≈ + × + ×Δ Δ⎢ ⎥⎣ ⎦

(H-13)

The 0

2 2nEvarV

− and 0 0

2 2nE Evar varV V

−−

2nvelocity terms in (H-1) are obtained from (H-11) with

(32) for . First, 0m

EBvarC 0

2nEvarV is obtained. Then what follows is modified to get 0

2 2nEvarV

− and

0 02 2n n

EvarV V

−−

2Evar .

27

( )( )

( )

0 0 0 0 02 1 2 12 1 2 1 0

0 0 0 09 90 0

0 02 1 2 10 0

10 0

0 08 9

For 4 :

2

2 2

2For 4 :

n nn n

n n

BrefE E E E Emref Bvar Bvarvar var

BrefE E E Em mref Bvar Bvar ref

E Evar varE E

refvarm

BvarE EvarBvar Bvar

n

V lR R C CT

V l VC CT T

R RV V

Tn

IC C α

+ −+ −

− −

+ −−

− −

< −

− = + −

= + − =

−= =

= −

= + × +Δ ( )( ) ( )

( ) ( ) ( ) ( )

0 07 8

09

2

2

2 2

12

12

1 12 2

Bvarvar


Bvar Bvar Bvar BvarEvar var var varBvar

IC C

I IC

α

α α

α α α α

− −

−

⎡ ⎤×Δ⎢ ⎥⎣ ⎦⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦

⎡ ⎤ ⎡ ⎤= + × + × + × + ×Δ Δ Δ Δ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

( ) ( )09

22 2Bvar BvarE

var varBvar IC α α−⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦

+ (H-14)

( ) ( )( ) ( )

( )( )

0 0 07 9 9

09

0 0 0 0 07 97 9 0

0 090

2

2

2

2 2

2

2

2 2




Bvar BvarE Em var varref Bvar

I IC C C

C

V lR R C CT

V CT

α α

α α

α α

− − −

−

− −− −

−

⎡ ⎤− = + × + × + −Δ Δ⎢ ⎥⎣ ⎦

⎡ ⎤≈ × + ×Δ Δ⎢ ⎥⎣ ⎦

− = + −

= + × +Δ Δ( )

( ) ( )0 0

7 90 0 098 0

2

2

Bref

E E Brefvar var Bvar BvarE E E

var varref Bvarvarm m

l

R R lV V CT T

α α− −−−

⎡ ⎤×⎢ ⎥⎣ ⎦

− ⎡ ⎤= = + × + ×Δ Δ⎢ ⎥⎣ ⎦

28

( ) ( )( ) ( )( ) ( )

( ) ( )

0 07 8

0 06 7

0 05 6

07

2

2

2

2

For 3 :

12

12

12

12





n

IC C

IC C

IC C

IC

α α

α α

α α

α α

− −

− −

− −

−

= −

⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦

⎡= + × + ×Δ Δ⎣

( ) ( )( ) ( ) ( ) ( )0

8

2

2 2

12

1 2 22

Bvar Bvarvar var


I

I IC

α α

α α α α−

⎤ ⎡ ⎤+ × + ×Δ Δ⎢ ⎥ ⎢ ⎥⎦ ⎣ ⎦

⎡ ⎤ ⎡= + × + × + × + ×Δ Δ Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣

⎤+ ⎥⎦

( ) ( )08

2932

Bvar BvarEvar varBvar IC α α

−⎡= + × + ×Δ Δ⎢⎣ ⎦

⎤⎥ (H-15)

( ) ( )( )

( ) ( )

0 0 05 7 8

0 0 0 0 05 75 7 0

0 080

0 05 70

6 0

2

2

2 2

2

2 2 2

2




E Evar varE

rvarm

C C C

V lR R C CT

V lCT

R RV

T

α α

α α

− − −

− −− −

−

− −−

⎡ ⎤− ≈ × + ×Δ Δ⎢ ⎥⎣ ⎦

− = + −

⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦

−= = ( ) ( )0 0

8

22

BrefBvar BvarE Evar varef Bvar

m

lV CT

α α−⎡ ⎤+ × + ×Δ Δ⎢ ⎥⎣ ⎦

29

( ) ( )( ) ( )( ) ( )

( ) ( )

0 05 6

0 04 5

0 03 4

05

2

2

2

2

For 2 :

12

12

12

12





n

IC C

IC C

IC C

IC

α α

α α

α α

α α

− −

− −

− −

−

= −

⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦

⎡= + × + ×Δ Δ⎣

( ) ( )( ) ( ) ( ) ( )0

6

2

2 2

12

1 2 22

Bvar Bvarvar var


I

I IC

α α

α α α α−

⎤ ⎡ ⎤+ × + ×Δ Δ⎢ ⎥ ⎢ ⎥⎦ ⎣ ⎦

⎡ ⎤ ⎡= + × + × + × + ×Δ Δ Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣

⎤+ ⎥⎦

( ) ( )06

2932


−⎡= + × + ×Δ Δ⎢⎣ ⎦

⎤⎥ (H-16)

( ) ( )( ) ( )

( )

0 0 03 5 6

0 0 0 063 5 0

0 03 50 0 0

64 0

2

2

2

2 2

2 2 2

22



E Evar var Bvar BvarE E E

var varref Bvarvarm

C C C

V lR R CT

R RV V C

T

α α

α α

α

− − −

−− −

− −−−

⎡ ⎤− ≈ × + ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤− = + × + ×Δ Δ⎢ ⎥⎣ ⎦

−= = + × +Δ Δ( )

Bref

m

lT

α⎡ ⎤×⎢ ⎥⎣ ⎦

30

( ) ( )( ) ( )( ) ( )

( ) ( )

0 03 4

0 02 3

0 01 2

03

2

2

2

2

For 1:

12

12

12

12





n

IC C

IC C

IC C

IC

α α

α α

α α

α α

− −

− −

− −

−

= −

⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦

⎡= + × + ×Δ Δ⎣

( ) ( )( ) ( ) ( ) ( )0

4

2

2 2

12

1 2 22

Bvar Bvarvar var


I

I IC

α α

α α α α−

⎤ ⎡ ⎤+ × + ×Δ Δ⎢ ⎥ ⎢ ⎥⎦ ⎣ ⎦

⎡ ⎤ ⎡= + × + × + × + ×Δ Δ Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣

⎤+ ⎥⎦

( ) ( )04

2932


−⎡= + × + ×Δ Δ⎢⎣ ⎦

⎤⎥ (H-17)

( ) ( )( ) ( )

( )

0 0 01 3 4

0 0 0 041 3 0

0 01 30 0 0

42 0

2

2

2

2 2

2 2 2

22



E Evar var Bvar BvarE E E

var varref Bvarvarm

C C C

V lR R CT

R RV V C

T

α α

α α

α

− − −

−− −

− −−−

⎡ ⎤− ≈ × + ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤− = + × + ×Δ Δ⎢ ⎥⎣ ⎦

−= = + × +Δ Δ( )

Bref

m

lT

α⎡ ⎤×⎢ ⎥⎣ ⎦

31

( ) ( )( ) ( )

( ) ( )( ) ( )

0 01 2

0 00 1

0 01 0

01

2

2

2

2

For 0 :

12

12

1' '

2

12





n

IC C

IC C

IC C

IC

α α

α α

α α

α α

− −

−

−

=

⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦

⎡ ⎤= + × + ×Δ Δ⎢⎣

( ) ( )( ) ( )

( ) ( )( )

01

02

2

2

22

1' '

2

1' '

2

'1

12 '2

Bvar Bvarvar var


Bvar Bvarvar var

Bvar BvarEvar varBvar Bvar

var va

I

IC

IIC

α α

α α α α

α αα α

α

−

−

⎡ ⎤+ × +Δ Δ ×⎥ ⎢ ⎥⎦ ⎣ ⎦

⎧ ⎫⎡ ⎤ ⎡= + + × + + ×Δ Δ Δ Δ⎨ ⎬⎢ ⎥ ⎢⎣ ⎦ ⎣⎩ ⎭⎤⎥⎦

⎡ ⎤+ + ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤≈ + × + ×Δ Δ⎢ ⎥⎣ ⎦ + +Δ( )Bvar

rα

⎧ ⎫⎪ ⎪⎪ ⎪⎨ ⎬

⎡ ⎤⎪ ⎪×Δ⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭

( ) (02

212 2' '2

Bvar Bvar Bvar BvarEvar var var varBvar IC α α α α

−

⎧ ⎫⎡ ⎤ ⎡= + + × + +Δ Δ Δ Δ⎨ ⎬⎢ ⎥ ⎢⎣ ⎦ ⎣⎩ ⎭) ⎤×⎥⎦ (H-18)

( )( ) ( )

( )( )

0 0 01 1 2

0 0 0 0 01 11 1 0

0 020

2 2

'

1 2'2

2

'2

Bvar Bvarvar var

E E EBvar Bvar Bvar Bvar Bvar Bvar

var var var


Bvar Bvarvar var

E Emref Bvar

C C C

V lR R C CT

V CT

α α

α α α

α α

− −

−−

−

⎡ ⎤+ ×Δ Δ⎢ ⎥⎣ ⎦− ≈ ⎧ ⎫⎡ ⎤+ + × − ×Δ Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

− = + −

⎡ + ×Δ Δ⎣

= +( ) ( )

( )( ) ( )

0 01 10

0

0 020

2 2

2 2

1 2'2

2

'1

12 2'2

BrefBvar Bvar Bvarvar var var

E Evar varE

varm

Bvar Bvarvar var Bref

E Eref Bvar Bvar Bvar Bvar mvar var var

l

R RV

T

lV CT

α α α

α α

α α α

−

−

⎤⎢ ⎥⎦

⎧ ⎫⎡ ⎤+ + × − ×Δ Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

−=

⎡ ⎤+ ×Δ Δ⎢ ⎥⎣ ⎦= + ⎧ ⎫⎡ ⎤+ + × − ×Δ Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

32

( ) ( )( ) ( )( ) ( )

( ) ( )

0 01 0

0 02 1

0 03 2

01

2

2

2

2

For 1:

1' '

2

1' '

2

1' '

2

1' '

2





n

IC C

IC C

IC C

IC

α α

α α

α α

α α

=

⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦

⎡= + × + ×Δ Δ⎣

( ) ( )21' '

2Bvar Bvarvar varI α α⎤ ⎡ ⎤+ × +Δ Δ⎢ ⎥ ⎢ ⎥

⎦ ⎣ ⎦×

( ) (01

22 2' 'Bvar BvarE

var varBvar IC α α⎡= + × + ×Δ Δ⎢⎣ ⎦

) ⎤⎥ (H-19)

( ) ( ) ( ) ( )( ) ( )

( ) ( )

00

00

0 0 03 1 0

0 03 1 0

2 2

2

2

1 2 2' ' ' '2

93 ' '2

2 2' '

2




E Erefvar var

I IC

IC

C C C

R R

α α α α

α α

α α

⎡ ⎤ ⎡= + × + × + × + ×Δ Δ Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣

⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦

⎡ ⎤− ≈ × + ×Δ Δ⎢ ⎥⎣ ⎦

− =

⎤⎥⎦

( )( ) ( )

( ) ( )

0 0 03 1

0 000

0 03 10 0 0

02 0

2

2

2 2 2' '

2' '2

BrefE E Em Bvar Bvar


E E Brefvar var Bvar BvarE E E

var varref Bvarvarm m

V lC CT

V lCT

R R lV V CT T

α α

α α

+ −

⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦

− ⎡ ⎤= = + × + ×Δ Δ⎢ ⎥⎣ ⎦

33

( ) ( )( ) ( )( ) ( )

( ) ( )

0 03 2

0 04 3

0 05 4

03

2

2

2

2

For 2 :

1' '

2

1' '

2

1' '

2

1' '

2





n

IC C

IC C

IC C

IC

α α

α α

α α

α α

=

⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦

⎡= + × + ×Δ Δ⎣

( ) ( )21' '

2Bvar Bvarvar varI α α⎤ ⎡ ⎤+ × +Δ Δ⎢ ⎥ ⎢ ⎥

⎦ ⎣ ⎦×

( ) (03

22 2' 'Bvar BvarE

var varBvar IC α α⎡= + × +Δ Δ⎢⎣ ⎦

) ⎤× ⎥ (H-20)

( ) ( ) ( ) ( )( ) ( )

( ) ( )

02

02

0 0 05 3 2

0 05 3 0

2 2

2

2

1 2 2' ' ' '2

93 ' '2

2 2' '

2




E Erefvar var

I IC

IC

C C C

R R

α α α α

α α

α α

⎡ ⎤ ⎡= + × + × + × + ×Δ Δ Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣

⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦

⎡ ⎤− ≈ × + ×Δ Δ⎢ ⎥⎣ ⎦

− =

⎤⎥⎦

( )( ) ( )

( ) ( )

0 0 05 3

0 020

0 05 30 0 0

24 0

2

2

2 2 2' '

2' '2

BrefE E Em Bvar Bvar


E Evar var Bvar Bvar BrefE E E

var varref Bvarvarm

V lC CT

V lCT

R RV V C

T

α α

α α

+ −

⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦

− ⎡ ⎤= = + × + ×Δ Δ⎢ ⎥⎣ ⎦

l

Summary of (H-14) to (H-20) Results:

34

( ) ( )

( ) ( )

( ) ( )

0 010 0

0 0 098 0

0 0 086 0

0 0 064 0

2

2

2

2

2

E Erefvar


mBref

Bvar BvarE E Evar varref Bvarvar

mBref

Bvar BvarE E Evar varref Bvarvar

V V

lV V CT

lV V CT

lV V C

α α

α α

α α

−

−−

−−

−−

=

⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦

⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦

⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦ mT

( ) ( )0 0 042 0

22


m

lV V CT

α α−−⎡= + × + ×Δ Δ⎢⎣ ⎦

⎤⎥ (H-21)

( )( ) ( )

( ) ( )

0 0 020 0

0 0 002 0

0 0 024 0

2 2

2

'1

12 2'2

2' '


E E Eref Bvarvar Bvar Bvar Bvar m

var var var


m

E E Evarref Bvarvar

lV V CT

lV V CT

V V C

α α

α α α

α α

−

⎡ ⎤+ ×Δ Δ⎢ ⎥⎣ ⎦= + ⎧ ⎫⎡ ⎤+ + × − ×Δ Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦

= + ( ) ( )22' 'Bvar Bvar Bref

var lα α⎡ ⎤× + ×Δ Δ⎢ ⎥⎣ ⎦

To obtain 0

2 2nEvarV

− in (H-1) as a function of as is 0

2 2nEBvarC −

02n

EvarV in (H-21), find

as a function of . First note that

02n

EBvarC

02 2n

EBvarC −

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( )( )

2 2

2 2

2 3 4

4

1 12 2

1 12 2

1 1 12 2 4

14

Bvar Bvar Bvar Bvarvar var var var

Bvar Bvar Bvar Bvar Bvarvar var var var var


Bvarvar

I I

I

I

I

α α α α

α α α α α

α α α

α

⎡ ⎤ ⎡+ × + × − × + ×Δ Δ Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣

⎡ ⎤− × + × + × − × + ×Δ Δ Δ Δ Δ⎢ ⎥⎢ ⎥=⎢ ⎥+ × − × + ×Δ Δ Δ⎢ ⎥⎣ ⎦

= + ×Δ

+

3

⎤⎥⎦

( ) ( ) ( ) ( ) ( )2 21 1' ' ' ' '

2 2Bvar Bvar Bvar Bvar Bvarvar var var var varI Iα α α α α⎡ ⎤ ⎡ ⎤× + × − × + × = + ×Δ Δ Δ Δ Δ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦

414

(H-22)

But

35

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

12 2

12 2

1 122

1 1' ' ' '22



I I I

I I I

α α α α

α α α α

−

−

⎡ ⎤ ⎡+ × + × + × + ×Δ Δ Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣

⎡ ⎤ ⎡+ × + × + × + ×Δ Δ Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣

⎤ =⎥⎦

⎤ =⎥⎦

(H-23)

Thus, to third order accuracy:

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

12 2

12 2

112 2

11' ' ' '2 2



I I

I I

α α α α

α α α α

−

−

⎡ ⎤ ⎡+ × + × ≈ − × + ×Δ Δ Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣

⎡ ⎤ ⎡+ × + × ≈ − × + ×Δ Δ Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣

⎤⎥⎦

⎤⎥⎦

(H-24)

Applying (H-24) to (32) obtains:

( ) ( )( ) ( )

( ) ( )

0 01 9

0 01

0

0 01

12

2

2

For 9 :

For 0 9 :

12

12

1For 0 : ' '2

m

m m

m

m m

E EBvar Bvar




m C C

m

IC C

IC

m IC C

α α

α α

α α

− −

−

−

−

≤ − =

≥ > −

⎡ ⎤≈ + × + ×Δ Δ⎢ ⎥⎣ ⎦

⎡ ⎤≈ − × + ×Δ Δ⎢ ⎥⎣ ⎦

⎡ ⎤> ≈ − × +Δ Δ ×⎢ ⎥⎣ ⎦

(H-25)

Successive application of (H-23) then finds as a function of : 0

2mEBvarC −

0m

EBvarC

36

( ) ( )

( ) ( )( ) ( ) ( )

0 01 9

0 09 8

0 08 7

06

2

2

2

For 8 :

12

For 2 8 :

12

1 12 2

mE EBvar Bvar



Bvar Bvar BvarEvar var varBvar

m C C

IC C

m

IC C

I IC

α α

α α

α α α

− −

− −

− −

−

< − =

⎡ ⎤≈ − × + ×Δ Δ⎢ ⎥⎣ ⎦

− ≥ > −

⎡ ⎤≈ − × + ×Δ Δ⎢ ⎥⎣ ⎦

⎡ ⎤= − × + × − × +Δ Δ Δ⎢ ⎥⎣ ⎦

( )( ) ( ) ( ) ( ) ( )

( ) ( )

06

06

2

2 2

2

1 12 2

2 2

Bvarvar

Bvar Bvar Bvar Bvar BvarEvar var var var varBvar


IC

IC

α

α α α α α

α α

−

−

⎡ ⎤×Δ⎢ ⎥⎣ ⎦

⎡ ⎤≈ − × + × − × + × + ×Δ Δ Δ Δ Δ⎢ ⎥⎣ ⎦

⎡ ⎤= − × + ×Δ Δ⎢ ⎥⎣ ⎦

2

( ) (0 02

22 2

m mBvar BvarE Evar varBvar Bvar IC C α α

+ )⎡ ⎤= − × +Δ Δ ×⎢ ⎥⎣ ⎦

(H-26)

( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( )( )

0 01 0

01

01

2

2 2

2

For 1:

12

1 1' '

2 2

1'

2

'



Bvar Bvar Bvarvar var varE

BvarBvar Bvarvar var

m

IC C

I IC

IC

α α

α α α α

α α α

α

−

= −

⎡ ⎤≈ − × + ×Δ Δ⎢ ⎥⎣ ⎦

⎡ ⎤ ⎡= − × + × − × + ×Δ Δ Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣

− × + × − ×Δ Δ Δ=

+ ×Δ

⎤⎥⎦

( ) ( )( ) ( )

( ) ( )

01

0 02

2

2

2

1'

21

' '2

For 0 :

2 2' 'm m

Bvarvar



IC

m

IC C

α α

α α α α

α α+

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥× + ×Δ Δ⎢ ⎥⎣ ⎦

⎧ ⎫⎡ ⎤ ⎡= − + × + + ×Δ Δ Δ Δ⎨ ⎬⎢ ⎥ ⎢⎣ ⎦ ⎣⎩ ⎭≥

⎡ ⎤= − × + ×Δ Δ⎢ ⎥⎣ ⎦

⎤⎥⎦

With (H-26) and (H-21), 0

2 2nEvarV

− can be found from (H-1) as a function of .0

2 2nEBvarC −

37

( ) ( )

( ) ( ) ( ) ( )

0 02 2 0

0 0 0 092 2 8 0

0 080

00

2

2 2

For 4 :

For 3 :

12

n

n

E Erefvar

BrefBvar BvarE E E Evar varref Bvarvar var

mBref

Bvar Bvar Bvar BvarE Evar var var varref Bvar

m

Eref

n V V

n

lV V V CT

lIV CT

V

α α

α α α α

−

−− −

−

≤ − =

= −

⎡ ⎤= = + × + ×Δ Δ⎢ ⎥⎣ ⎦

⎡ ⎤ ⎡= + − × + × × + ×Δ Δ Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣

≈ +

⎤⎥⎦

( )08

higher than second order termsBref

BvarEvarBvar

m

lC

Tα

−⎡ ⎤× +Δ⎢ ⎥⎣ ⎦

(H-27) For 2 :n = −

( ) ( )

( ) ( ) ( ) ( )

( )

0 0 0 082 2 6 0

0 060

0 060

2

2 2

2

2 2 2

higher than

n


mBref


m

BvarE Evarref Bvar

lV V V CT

lIV CT

V C

α α

α α α α

α

−− −

−

−

⎡ ⎤= = + × + ×Δ Δ⎢ ⎥⎣ ⎦

⎡ ⎤ ⎡= + − × + × × + ×Δ Δ Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣

≈ + × +Δ

⎤⎥⎦

( )0 0 0 042 2 4 0

second order terms

For 1:

higher than second order termsn

Bref

m

BrefBvarE E E Evarref Bvarvar var

m

lT

n

lV V V CT

α−− −

⎡ ⎤⎢ ⎥⎣ ⎦

= −

⎡ ⎤= ≈ + × +Δ⎢ ⎥⎣ ⎦

(Continued)

38

(H-27) Continued

( )( ) ( )

( )( )

( )

0 0 020 0

0 000

2 2

2

For 0 :

'1

12 2'2

'212 2


E E Eref Bvarvar Bvar Bvar Bvar m

var var var

Bvar BvarBvar var varvarE Eref Bvar Bvar

var

n

lV V CT

IV C

α α

α α α

α αα

α

−

=

⎡ ⎤+ ×Δ Δ⎢ ⎥⎣ ⎦= + ⎧ ⎫⎡ ⎤+ + × − ×Δ Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

⎡ ⎤+ ×⎡ ⎤ Δ Δ− ×Δ ⎢ ⎥⎣ ⎦⎢ ⎥= + ⎢ ⎥

⎢ ⎥+ ×Δ⎣ ⎦ ( ) ( )

( )( ) ( )( ) ( )

0 000

2 2

2 2

1 2'2

'

1 2'12

22 '

+ higher

Bref

Bvar Bvar Bvar mvar var var

Bvar Bvarvar var

Bvar Bvar BvarE E var var varref Bvar


lT

V C

α α α

α α

α α α

α α α

⎧ ⎫⎡ ⎤+ + × − ×Δ Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

⎡ ⎤+ ×Δ Δ⎢ ⎥⎣ ⎦⎧ ⎫⎡ ⎤+ + × − ×Δ Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦= + ⎩ ⎭

⎡ ⎤− × + ×Δ Δ Δ⎢ ⎥⎣ ⎦

( )( ) ( )( ) ( )

( ) ( ) ( )0 0

00

2 2

2

than second order terms

'

4 4' '1 12 2 4 '

+ highe

Bref

m

Bvar Bvarvar var

Bvar Bvar Bvar Bvarvar var var varE E

ref BvarBvar Bvar Bvar Bvarvar var var var

lT

V C

α α

α α α α

α α α α

⎡ ⎤+ ×Δ Δ⎢ ⎥⎣ ⎦⎧ ⎫× + × × + ×Δ Δ Δ Δ⎪ ⎪⎪ ⎪= + + ⎨ ⎬⎪ ⎪⎡ ⎤− × − × + ×Δ Δ Δ Δ⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭

( ) ( ) ( )0 000

2 2

r than second order terms

1' '1

22

+ higher than second order terms

Bref

m

Bvar Bvar Bvar BvarE E var var var varref Bvar

lT

V Cα α α α

⎧ ⎫⎡ ⎤⎡ ⎤+ × + × − ×Δ Δ Δ Δ⎪ ⎪⎢ ⎥⎢ ⎥⎣ ⎦= + ⎨ ⎬⎣ ⎦⎪ ⎪⎩ ⎭

(Continued)

39

(H-27) Continued

( )( ) ( )

( )( )

0 0 0 022 2 0 0

0 000

2 2

2

For 1:

'1

12 2'2

'212 2

n


E E E Eref Bvarvar var Bvar Bvar Bvar m

var var var

Bvar BvBvar var varvarE Eref Bvar Bvar

var

n

lV V V CT

IV C

α α

α α α

αα

α

−−

=

⎡ ⎤+ ×Δ Δ⎢ ⎥⎣ ⎦= = + ⎧ ⎫⎡ ⎤+ + × − ×Δ Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

+⎡ ⎤ Δ− ×Δ⎢ ⎥

= + ⎢ ⎥⎢ ⎥+ ×Δ⎣ ⎦

( )( ) ( )

( ) ( ) ( )( ) ( )

0 000

2 2

2 2

1 2'2

1 2' '1 22

2 '

arBref

Bvar Bvar Bvar mvar var var

Bvar Bvar Bvar Bvar Bvarvar var var var varE E

ref BvarBvar Bvar Bvarvar var var

lT

V C

α

α α α

α α α α α

α α α

⎡ ⎤×Δ⎢ ⎥⎣ ⎦⎧ ⎫⎡ ⎤+ + × − ×Δ Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

⎧ ⎫⎡ ⎤ ⎡ ⎤+ × + + × − ×Δ Δ Δ Δ Δ⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎩ ⎭= +⎡− × + ×Δ Δ Δ⎢⎣

( )( ) ( )( ) ( )

( ) ( ) ( )0 0

00

2 2

2


'

4 4' '1 12 2 4 '

Bref

m

Bvar Bvarvar var

Bvar Bvar Bvar Bvarvar var var varE E

ref BvarBvar Bvar Bvar Bvarvar var var var

lT

V C

α α

α α α α

α α α α

⎤⎥⎦

⎡ ⎤+ ×Δ Δ⎢ ⎥⎣ ⎦⎧ ⎫× + × × + ×Δ Δ Δ Δ⎪ ⎪⎪ ⎪= + + ⎨ ⎬⎪ ⎡ ⎤− × − × + ×Δ Δ Δ Δ⎢ ⎥⎪ ⎣ ⎦⎩

( ) ( ) ( )0 000

0 0 0 002 2 2 0

2 2

+higher than second order terms

1' '1

22


For 2 :

n

Bref

m

Bvar Bvar Bvar Bvar BrefE E var var var varref Bvar

m

E E E Eref Bvarvar var

lT

lV CT

n

V V V C

α α α α

−

⎪⎪⎭

⎧ ⎫⎡ ⎤+ × + × − ×Δ Δ Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦= + ⎩ ⎭

=

= = + ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

0 020

0 020

2

2 2

2 2

2' '

2 2 2' ' ' '

2 2 + higher than sec' ' '


mBref


m

Bvar Bvar BvarE Evar var varref Bvar

lT

lIV CT

V C

α α

α α α α

α α α

⎡ ⎤× + ×Δ Δ⎢ ⎥⎣ ⎦

⎡ ⎤ ⎡= + − × + × × + ×Δ Δ Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣

= + × + × − ×Δ Δ Δ

⎤⎥⎦

( )0 020

ond order terms

+ higher than second order terms'

Bref

mBref

BvarE Evarref Bvar

m

lT

lV CT

α

⎡ ⎤⎢ ⎥⎣ ⎦

⎡ ⎤= + ×Δ⎢ ⎥⎣ ⎦ (Continued)

40

(H-27) Concluded

( ) ( )

( ) ( ) ( ) ( )

( )

0 0 0 022 2 4 0

0 040

0 040

2

2 2

For 3 :

2' '

2 2 2' ' ' '

hi'

n


mBref


m

BvarE Evarref Bvar

n

lV V V CT

lIV CT

V C

α α

α α α α

α

−

=

⎡ ⎤= = + × + ×Δ Δ⎢ ⎥⎣ ⎦

⎡ ⎤ ⎡= + − × + × × + ×Δ Δ Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣

≈ + × +Δ

⎤⎥⎦

( )0 0 02 22 2 0

gher than second order terms

For 3 :

higher than second order terms'nn

Bref

m

BrefBvarE E Evarref Bvarvar

m

lT

n

lV V CT

α−−

⎡ ⎤⎢ ⎥⎣ ⎦

>

⎡ ⎤≈ + × +Δ⎢ ⎥⎣ ⎦


( )

( )

0 02 2 0

0 0 088 0

0 0 066 0

0 04 0

For 4 :

higher than second order terms


nE E

refvar


mBref

BvarE E Evarref Bvarvar

m

Erefvar

n V V

lV V CT

lV V CT

V

α

α

−

−−

−−

−

≤ − =

⎡ ⎤≈ + × +Δ⎢ ⎥⎣ ⎦

⎡ ⎤≈ + × +Δ⎢ ⎥⎣ ⎦

≈ ( )04


BvarE EvarBvar

m

lV CT

α−⎡ ⎤+ × +Δ⎢ ⎥⎣ ⎦

( )0 0 022 0



m

lV V CT

α−−⎡ ⎤≈ + × +Δ⎢ ⎥⎣ ⎦

(H-28)

( ) ( ) ( )

( )

0 0 000 0

0 0 022 0

2 21' '1

22


+ higher than second order term'

Bvar Bvar Bvar Bvar BrefE E E var var var var

ref Bvarvarm


lV V CT

V V C

α α α α

α

⎧ ⎫⎡ ⎤⎡ ⎤+ × + × − ×Δ Δ Δ Δ⎪ ⎪⎢ ⎥⎢ ⎥⎣ ⎦= + ⎨ ⎬⎣ ⎦⎪ ⎪⎩ ⎭

= + ×Δ

( )

( )

0 0 044 0

0 0 02 22 2 0

s

higher than second order terms'

For 3 :

higher than second order terms'nn

Bref

mBref


m


m

lT

lV V CT

n

lV V CT

α

α−−

⎡ ⎤⎢ ⎥⎣ ⎦

⎡ ⎤≈ + × +Δ⎢ ⎥⎣ ⎦

>

⎡ ⎤≈ + × +Δ⎢ ⎥⎣ ⎦

To find 0 02 2n

E Evar varV V

−−

2n for (H-1), subtract (H-28) from (H-21):

41

( ) ( )

( ) ( )

0 0 0 010 12 0 0

0 0 0 098 10 0 0

09

0 0 0 086 8 0

2

2

0E E E Eref refvar var

BrefBvar BvarE E E E Evar varref Bvar refvar var

mBref


m

BE E E Evarref Bvarvar var

V V V V

lV V V VCT

lC

T

V V V C

α α

α α

− −

−− −

−

−− −

− = − =

⎡ ⎤− = + × + × −Δ Δ⎢ ⎥⎣ ⎦

⎡ ⎤= × + ×Δ Δ⎢ ⎥⎣ ⎦

− = +

0

( ) ( )

( )

( )

0 080

08

0 0 064 6 0

2

2

2


2 higher than second order terms

Brefvar Bvar

varm


mBref

BvarEvarBvar

m

E E Eref Bvvar var

lT

lV CT

lC

T

V V V

α α

α

α

−

−

−− −

⎡ ⎤× + ×Δ Δ⎢ ⎥⎣ ⎦

⎡ ⎤− − × +Δ⎢ ⎥⎣ ⎦

⎡ ⎤= × −Δ⎢ ⎥⎣ ⎦

− = + ( ) ( )02

2Bref

Bvar BvarEvar varar

m

lC

Tα α⎡ ⎤× + ×Δ Δ⎢ ⎥

⎣ ⎦

( )0 060


BvarE Evarref Bvar

m

lV CT

α−⎡ ⎤− − × +Δ⎢ ⎥⎣ ⎦

(H-29)

( )

( ) ( )

( )

06

0 0 0 042 4 0

0 040

2

2

2 higher than second order term

2


BrefBvarEvarBvar

mBref

Bvar BvarE E E Evar varref Bvarvar var

m

BvarE Evarref Bvar

lC

T

lV V V CT

V C

α

α α

α

−

−− −

−

⎡ ⎤= × −Δ⎢ ⎥⎣ ⎦

⎡ ⎤− = + × + ×Δ Δ⎢ ⎥⎣ ⎦

⎡ ⎤− − × +Δ⎢⎣ ⎦

( )( )

( ) ( )

04

0 0 0 020 2 0

00

2

2 2


'1

12 2'2

Bref

mBref

BvarEvarBvar

m


E E E Eref Bvarvar var Bvar Bvar Bvar m

var var var

ref

lT

lC

T

lV V V CT

α

α α

α α α

−

−−

⎥

⎡ ⎤= × −Δ⎢ ⎥⎣ ⎦

⎡ ⎤+ ×Δ Δ⎢ ⎥⎣ ⎦− = + ⎧ ⎫⎡ ⎤+ + × − ×Δ Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

− ( )

( )

02

02


1 higher than second order term'2

BrefBvarE EvarBvar

mBref


m

lV CT

lC

T

α

α α

−

−

⎡ ⎤− × +Δ⎢ ⎥⎣ ⎦

⎡ ⎤= − × −Δ Δ⎢ ⎥⎣ ⎦

(Continued)

42

(H-29) Concluded

( ) ( )( ) ( ) ( )

0 0 0 002 0 0

0 000

0

2

2 2

2' '

1' '1

22



m

Bvar Bvar Bvar Bvar BrefE E var var var varref Bvar

m

B

lV V V CT

lV CT

α α

α α α α

⎡ ⎤− = + × + ×Δ Δ⎢ ⎥⎣ ⎦

⎧ ⎫⎡ ⎤⎡ ⎤+ × + × − ×Δ Δ Δ Δ⎪ ⎪⎢ ⎥⎢ ⎥⎣ ⎦− − ⎨ ⎬⎣ ⎦⎪ ⎪⎩ ⎭

=

( ) ( )( ) ( ) ( )

( )

0

00

2

2 2

2

2' '

1 1' '

2 4higher than second order terms

1 7'

2 4

Bvar Bvarvar var

BrefBvar Bvar Bvar BvarEvar var var varvar

m

Bvar Bvar BvarE var var varBvar

lC

T

C

α α

α α α α

α α

⎧ ⎫× + ×Δ Δ⎪ ⎪⎪ ⎪

⎡ ⎤⎪ ⎪⎡ ⎤− + × − × − ×Δ Δ Δ Δ⎨ ⎬⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎪ ⎪⎪ ⎪−⎪ ⎪⎩ ⎭

⎡ ⎤− × +Δ Δ Δ⎢ ⎥⎣ ⎦= ( ) ( )

( ) ( )

( )

0 0 0 024 2 0

0 020

2

2

1'

4higher than second order term

2' '

+ higher than second order terms'

Bvar Brefvar

m


m

BvarE Evarref Bvar

lT

lV V V CT

V C

α α

α α

α

⎡ ⎤× + ×Δ⎢ ⎥⎢ ⎥

−⎢ ⎥⎣ ⎦

⎡ ⎤− = + × + ×Δ Δ⎢ ⎥⎣ ⎦

⎡ ⎤− − ×Δ⎢ ⎥⎣ ⎦

( )

( )

( ) ( )

( )

02

02

0 0 0 046 4 0

0 040

2

2

2

2 higher than second order term'

2 '

2' '

'

Bref

mBref

BvarEvarBvar

mBref

BvarEvarBvar

mBref

Bvar BvarE E E Evar varref Bvarvar var

m

BvarE Evarref Bvar

lT

lC

T

lC

T

lV V V CT

V C

α

α

α α

α

⎡ ⎤= × −Δ⎢ ⎥⎣ ⎦

≈ ×Δ

⎡ ⎤− = + × + ×Δ Δ⎢ ⎥⎣ ⎦

− − × +Δ

( )04

2


2 '

Bref

mBref

BvarEvarBvar

m

lT

lC

Tα

⎡ ⎤⎢ ⎥⎣ ⎦

⎡ ⎤≈ ×Δ⎢ ⎥⎣ ⎦


43

( ) ( )

( )

0 0 0 010 12 0 0

0 0 098 10

0 0 086 8

0 04 6

2

2

0


E E E Eref refvar var

BrefBvar BvarE E Evar varBvarvar var

mBref

BvarE E EvarBvarvar var

m

E Evar var

V V V V

lV V CT

lV V CT

V V

α α

α

− −

−− −

−− −

− −

− = − =

⎡ ⎤− = × + ×Δ Δ⎢ ⎥⎣ ⎦

⎡ ⎤− = × −Δ⎢ ⎥⎣ ⎦

− ( )06

22 higher than second order term

BrefBvarEvarBvar

m

lC

Tα

−⎡ ⎤= × −Δ⎢ ⎥⎣ ⎦

( )0 0 042 4


BrefBvarE E EvarBvarvar var

m

lV V CT

α−− −⎡ ⎤− = × −Δ⎢ ⎥⎣ ⎦

(H-30)

( )( ) ( ) ( )

0 0 020 2

0 0 002 0

2 2

1 higher than second order term'2

1 7 1' '

2 4 4higher than second order

BrefBvar BvarE E Evar varBvarvar var

m

Bvar Bvar Bvar BvarE E E var var var var

Bvarvar var

lV V CT

V V C

α α

α α α α

−−⎡ ⎤− = − × −Δ Δ⎢ ⎥⎣ ⎦

⎡ ⎤− × + × + ×Δ Δ Δ Δ⎢ ⎥⎣ ⎦− =−

( )

( )

0 0 024 2

0 0 046 4

2

2

term

2 higher than second order term'

2 higher than second order terms'

Bref

m

BrefBvarE E EvarBvarvar var

mBref

BvarE E EvarBvarvar var

m

lT

lV V CT

lV V CT

α

α

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

⎡ ⎤− = × −Δ⎢ ⎥⎣ ⎦

⎡ ⎤− = × −Δ⎢ ⎥⎣ ⎦

Finding 2 1nnumB − And 2nnumB

To find and , substitute , 2 1nnumB − 2nnumB 2 1nA − 2nA from (H-5), 0 02 2n

E Evar varR R

−−

2n from (H-

13), 02 2n−

EvarV from (H-28), and 0 0

2nEar −2n

Evar v−

2V V from (H-30) into the (H-1) and

equations: 2 1nnumB − 2nnumB

( )( )

( )( )

0 0 0 0 02 1 2 2 2 2 2 2 2 2

0 0

0 0 0 0 02 2 2 2 2 2 2 2 2

0 0

2 1

2 22 1

2

2 22

For 4 :

2

2

2

2

n n n n n

n n n n m n


E En m mavg


E En m mavg

n

V V VR Rn mB T A T

g gIA T T

V V VR RB T A

g gIA T T

− − − −

− − −

−

−

< −

≡ − − − −

+ − = −

≡ − − − −

+ − =

T

(H-31)

44

( )( )

( ) ( )( )( )

0 0 0 0 02 1 2 2 2 2 2 2 2 2

0

0 0 090 0

09 9

2 1

22 1

2

10

For 4 :

2

2

2 2

1 12 6

n n n n n nE E E E E

m n mnum var var var var var

En mavg

Bvar Bvar BrefE E Em mvar varref Bvar ref

Bvar EEvarBvar Bvar

n

V V VR RB T A

gIA T

V lCT T

I C C

α α

α

− − − −

−

− −

−

−

−

= −

≡ − − − −

+ −

⎡ ⎤= + × + × −Δ Δ⎢ ⎥⎣ ⎦

⎡− − ×Δ

⎣

T

V

( ) ( )

( )( )( ) ( )

( )

09

009 9

0 09 9

009

2

120

2

16

12

16

Bvar Bvar BrefEvar varBvar

Bvar E EEmvarBvar Bvar avg

Bvar Bref Bvar BrefE Evar varBvar Bvar

Bvar EEmvarBvar avg

lC

gI C C T

l lC C

gI C T

α α

α

α α

α

−

− −

− −

−

−

⎤ ⎡ ⎤× + ×Δ Δ⎢ ⎥ ⎢ ⎥⎣ ⎦⎦

⎡ ⎤− + ×Δ⎢ ⎥⎣ ⎦

≈ × − ×Δ Δ

⎧ ⎫⎡ ⎤− + ×Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

( ) ( ) 00 09 9

21 12 6

Bvar Bref Bvar EE Emvar varBvar Bvar avggIlC C Tα α

− −⎧ ⎡ ⎤= × − + ×Δ Δ⎨ ⎢ ⎥⎣ ⎦⎩ ⎭

⎫⎬ (H-32)

( )( )

( ) ( ) ( ) ( )

( )

0 0 0 0 02 2 2 2 2 2 2 2 2

0

0 09 9

0

09

2

22

2 2

2

2

2

32

n n n n m nE E E E E


En mavg

Bvar Bvar Bref Bvar Bvar BrefE Evar var var varBvar Bvar

Emavg

Bvar BEvarBvar

V V VR RB T A

gIA T

l lC C

g T

C

α α α α

α

− − −

− −

−

≡ − − − −

+ −

⎡ ⎤ ⎡= × + × − × + ×Δ Δ Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣

+

≈ ×Δ

T

⎤⎥⎦

( )( )

009

009

2

2

32

12

ref Bvar Bref EEmvarBvar avg

Bvar Bref EEmvarBvar avg

gl lC T

glC T

α

α

−

−

− × +Δ

= − × +Δ

45

( )( )

( ) ( )

( )

0 0 0 0 02 1 2 2 2 2 2 2 2 2

0

0 0 080 0

08

7

2 1

22 1

2

For 3 :

2

2

2 2 2

2

16

n n n n nE E E E E

m nnum var var var var var

En mavg


BrefBvarEvarBvar

m

Bva

n

V V VR Rn mB T A T

gIA T

V lCT T

lC

T

I

α α

α

− − − −

−

−

−

−

−

= −

≡ − − − −

+ −

⎡ ⎤= + × + × −Δ Δ⎢ ⎥⎣ ⎦

− ×Δ

− −

V

( )( ) ( )

( )( )( ) ( )

0 087

007 7

00 08 8

1 20

120

2 2

16

16

Bvar Bvar BrefEE Evar varr BvarBvar


Bvar Bref Bvar EE Emvar varBvar Bvar avg

lC CC

gI C C T

gIlC C T

α α

α

α α

−−

− −

− −

−

−

⎡ ⎤× ×Δ Δ⎢ ⎥

⎣ ⎦⎡ ⎤

− + ×Δ⎢ ⎥⎣ ⎦

⎧ ⎫⎡ ⎤= × − + ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭ (H-33)

( )( )

( ) ( )( )

( )

0 0 0 0 02 2 2 2 2 2 2 2 2

0

0 0 080 0

08

08 2

2

22

2

1

2

2

2 2 2

2

1318

n n n n m

n


En m


Bvar BrefEvarBvar

BvarEvarBvar

V V VR Rn mB T A T

gIA T

V lCT T

lC

I C

α α

α

α

− − −

−

−

−

−

≡ − − − −

+ −

⎡ ⎤= + × + × −Δ Δ⎢ ⎥⎣ ⎦

− ×Δ

− − ×Δ ( )

V

( )

( )( )( ) ( )

082

008 8

00 08 8

20

120

2 2

16

16

Bvar BrefE EvarBvarBvar



lCC

gI C C T

gIlC C T

α

α

α α

−−

− −

− −

−

⎡ ⎤×Δ⎢ ⎥

⎣ ⎦⎡ ⎤

+ − ×Δ⎢ ⎥⎣ ⎦

⎧ ⎫⎡ ⎤= − × + − ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

46

( )( )

( ) ( )

( )

0 0 0 0 02 1 2 2 2 2 2 2 2 2

0

0 0 02 20 0

02 2

2 1

22 1

2

For 0 > 3 :

2

2

2 2 2

2

1

n n n n n

n

n


En mavg


BrefBvarEvarBvar

m

n

V V VR Rn mB T A T

gIA T

V lCT T

lC

T

I

α α

α

− − − −

−

−

−

−

> −

≡ − − − −

+ −

⎡ ⎤= + × + × −Δ Δ⎢ ⎥⎣ ⎦

− ×Δ

− −

V

( )( ) ( )

( )( )( ) ( )

0 02 1 2 22 1

002 1 2 1

00 02 2 2 2

1 20

120

2 2

6

16

16

n nn

n n

n n

Bvar Bvar BrefEE Evar varBvar BvarBvar



lC CC

gI C C T

gIlC C T

α α

α

α α

− −−

− −

− −

−

−

⎡ ⎤× ×Δ Δ⎢ ⎥

⎣ ⎦⎡ ⎤

− + ×Δ⎢ ⎥⎣ ⎦

⎧ ⎫⎡ ⎤≈ × − + ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭ (H-34)

( )( )

( ) ( )( )

0 0 0 0 02 2 2 2 2 2 2 2 2

0

0 0 02 20 0

02 2

02 2

2

22

2

2

2

2 2 2

2

1318

n n n n m

n

n

n


En mavg


Bvar BrefEvarBvar

BvEvarBvar

V V VR Rn mB T A T

gIA T

V lCT T

lC

I C

α α

α

− − −

−

−

−

≡ − − − −

+ −

⎡ ⎤= + × + × −Δ Δ⎢ ⎥⎣ ⎦

− ×Δ

− −

V

( )( ) ( )

( )( )( ) ( )

02 22 2

002 2 2 2

00 02 2 2 2

1 20

120

2 2

16

16

nn

n n

n n

ar Bvar BrefE EvarBvarBvar



lCC

gI C C T

gIlC C T

α α

α

α α

−−

− −

− −

−

−

⎡ ⎤× ×Δ Δ⎢ ⎥

⎣ ⎦⎡ ⎤

+ − ×Δ⎢ ⎥⎣ ⎦

⎧ ⎫⎡ ⎤= − × + − ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

47

( )( )

( ) ( )( )

0 0 0 0 02 1 2 2 2 2 2 2 2 2

0

0 0 020 0

02

1

2 1

22 1

2

For 0 :

2

2

2 2 2

2

1 14 6

n n n n n nE E E E E


En mavg


Bvar BrefEvarBvar

Bvar

n

V V VR RB T A

gIA T

V lCT T

lC

I

α α

α

− − − −

−

−

−

−

−

=

≡ − − − −

+ −

⎡ ⎤= + × + × −Δ Δ⎢ ⎥⎣ ⎦

− ×Δ

− −

T

V

( )( ) ( )( )

( ) ( )

0 021

001

0 02 2

10

2

'

16

1 1'

4 6

BrefBvar Bvar BvarEE Evar var varBvarBvar

m

Bvar EEmvarBvar avg

Bvar Bvar Bref BvarE Evar var varBvar Bvar a

lC CC

T

gI C T

IlC C

α α α

α

α α α

−−

−

− −

−⎡ ⎤ ⎡ ⎤× −Δ Δ Δ⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦⎧ ⎫⎡ ⎤− + ×Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

⎧ ⎫⎡ ⎤⎡ ⎤≈ − − × − + ×Δ Δ Δ⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎩ ⎭

×

0 2Emvgg T

(H-35)

( )( )

( ) ( )( )

( )

0 0 0 0 02 2 2 2 2 2 2 2 2

0

0 0 020 0

02

02

2

22

2

2

2

2 2 2

2

3 14 18



En mavg


Bvar BrefEvarBvar

BvarEvarBvar

V V VR RB T A

gIA T

V lCT T

lC

I C

α α

α

α

− − −

−

−

−

≡ − − − −

+ −

⎡ ⎤= + × + × −Δ Δ⎢ ⎥⎣ ⎦

− ×Δ

− − ×Δ ( )

T

V

( )( )

( )( )( )

022

002 2

02

1 20

2

120

'

12'12

2

16

3'

4

Bvar Bvarvar var

BrefE E Bvar BvarBvarBvar var var

Bvarvar



lCC

gI C C T

C

α α

α α

α

α

α

−−

− −

−

−

−

⎡ ⎤− ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤ ⎧ ⎫⎡ ⎤+ ×Δ Δ⎢ ⎥ ⎪ ⎪⎢ ⎥⎪ ⎪⎣ ⎦⎣ ⎦ + ⎨ ⎬

⎪ ⎪− ×Δ⎪ ⎪⎩ ⎭

⎡ ⎤+ − ×Δ⎢ ⎥⎣ ⎦

≈ − −Δ Δ( ) ( ) 002

216

Bref Bvar EEmvarBvar avggIl C Tα α

−⎧ ⎫⎡ ⎤⎡ ⎤× + − ×Δ⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎩ ⎭

Note: Second order terms in (H-35) are neglected. If second order terms were included,

( )( 'Bvar Bvarvar varα α×Δ Δ )× products would appear which are too difficult to explain. For simplification, this

article only carries the highest order ( )Bvarvarα ×Δ or ( 'Bvar

varα ×Δ ) products.

48

( )( )

( ) ( )( )

0 0 0 0 02 1 2 2 2 2 2 2 2 2

0

0 0 000 0

00

2 1

22 1

2

For 1:

2

2

2 2 2' '

'

n n n n nE E E E E

m nnum var var var var var

En mavg


Bvar Bvarvar var v

EBvar

n

V V VR Rn mB T A T

gIA T

V lCT T

C

α α

α α

− − − −−

−

=

≡ − − − −

+ −

⎡ ⎤= + × + × −Δ Δ⎢ ⎥⎣ ⎦

⎡ ⎤+ × −Δ Δ⎢ ⎥⎣ ⎦−

V

( ) ( )( ) 2

'

1'

2

Bvar Bvar Bvarar var var

Bref

Bvar Bvarvar var

lα α α

α α

⎧ ⎫⎡ ⎤× + ×Δ Δ Δ⎢ ⎥⎪ ⎪⎣ ⎦⎪ ⎪⎨ ⎬

⎡ ⎤⎪ ⎪+ + ×Δ Δ⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭

( )( )( )

( )( )

0 01 01

1 20

2

'

1 1 7' '

4 6 212

Bvar Bvarvar var


Bvarvar

I lC CC

α α

α α

α

−

⎧ ⎫⎡ ⎤− ×Δ Δ⎪ ⎪⎢ ⎥⎣ ⎦⎪ ⎪⎡ ⎤ ⎪ ⎪− − × + ×Δ Δ⎨ ⎬⎢ ⎥

⎣ ⎦ ⎪ ⎪⎪ ⎪

+ ×Δ⎪ ⎪⎩ ⎭

(H-36)

( )( )( ) ( )

( ) ( )

001 1

0 00 0

00 00 1

120

2

1'

6

2 ' '

1 1' '

4 6


Bvar Bref Bvar Bvar BrefE Evar var varBvar Bvar

Bvar Bvar Bref Bvar EE Emvar var varBvar Bvar

gI C C T

l lC C

gIlC C

α

α α α

α α α

−⎡ ⎤− + ×Δ⎢ ⎥⎣ ⎦

⎡ ⎤≈ × − + ×Δ Δ Δ⎢ ⎥⎣ ⎦⎧ ⎫⎡ ⎤⎡ ⎤− − × − + ×Δ Δ Δ⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎩ ⎭

( ) ( ) 00 00 0

23 1' '

4 6Bvar Bvar Bref Bvar EE E

mvar var varBvar Bvar

T

gIlC C Tα α α⎧ ⎫⎡ ⎤⎡ ⎤= − × − + ×Δ Δ Δ⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎩ ⎭

(Continued)

49

(H-36) Concluded

( )( )

( ) ( )( ) ( )

0 0 0 0 02 2 2 2 2 2 2 2 2

0

0 0 000 0

00

2

22

2

2

2

2 2 2' '

'



En mavg


Bvar Bvar Bvar Bvvar var var var

EBvar

V V VR RB T A

gIA T

V lCT T

C

α α

α α α

− − −≡ − − − −

+ −

⎡ ⎤= + × + × −Δ Δ⎢ ⎥⎣ ⎦

⎡ ⎤+ × − ×Δ Δ Δ⎢ ⎥⎣ ⎦−

T

V

( )( )

( )( )( )

( )( )

0 00 00

2

1 20

2

'

1'

2

'

3 1 1 7' '

2 18 2 212

ar Bvarvar

Bref

Bvar Bvarvar var

Bvar Bvarvar var

Bvar Bvar BEE Evar varBvar BvarBvar

Bvarvar

l

I C CC

α α

α α

α α

α α

α

−

⎧ ⎫⎡ ⎤+ ×Δ Δ⎢ ⎥⎪ ⎪⎣ ⎦⎪ ⎪⎨ ⎬

⎡ ⎤⎪ ⎪+ + ×Δ Δ⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭⎧ ⎫⎡ ⎤− ×Δ Δ⎪ ⎪⎢ ⎥⎣ ⎦⎪ ⎪⎡ ⎤ ⎪ ⎪− − × + ×Δ Δ⎨ ⎬⎢ ⎥

⎣ ⎦ ⎪ ⎪⎪ ⎪

+ ×Δ⎪ ⎪⎩ ⎭

( )( ) ( )

( ) ( )

000

0 00 0

00 00 0

21'

6

2 ' '

3 1' '

4 6

ref

Bvar EEmvarBvar avg

Bvar Bref Bvar Bvar BrefE Evar var varBvar Bvar

Bvar Bvar Bref Bvar EE Evar var varBvar Bvar avg

l

gI C T

l lC C

IlC C

α

α α α

α α α

⎧ ⎫⎡ ⎤+ − ×Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎡ ⎤≈ × − + ×Δ Δ Δ⎢ ⎥⎣ ⎦

⎧ ⎫⎡ ⎤⎡ ⎤− − × + − ×Δ Δ Δ⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎩ ⎭

( ) ( ) 00 00 0

2

21 1' '

4 6

m

Bvar Bvar Bref Bvar EE Emvar var varBvar Bvar avg

g T

gIlC C Tα α α⎧ ⎫⎡ ⎤⎡ ⎤= − × + − ×Δ Δ Δ⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎩ ⎭

50

( )( )

( ) ( )( )

0 0 0 0 02 1 2 2 2 2 2 2 2 2

0

0 0 02 20 0

02 2

2 1

22 1

2

For 1:

2

2

2 2 2' '

2 '

1 12

n n n n n

n

n


En mavg


Bvar BrefEvarBvar

n

V V VR Rn mB T A T

gIA T

V lCT T

lC

I

α α

α

− − − −

−

−

−

−

>

≡ − − − −

+ −

⎡ ⎤= + × + × −Δ Δ⎢ ⎥⎣ ⎦

− ×Δ

− −

V

( )( ) ( )( )

( ) ( )

0 02 1 2 22 1

002 1

00 02 2 2 1

2

1 20

2

2 2

2' '6

1'

61

' '6

n nn

n

n n

n


Bvar EEmvarBvar avg


lC CC

gI C T

gIlC C T

α α

α

α α

− −−

−

− −

−

−⎡ ⎤× ×Δ Δ⎢ ⎥

⎣ ⎦⎧ ⎫⎡ ⎤− + ×Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

⎧ ⎫⎡ ⎤≈ × − + ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

≈ ( ) ( ) 00 02 2 2

2 21' '

6 nBvar Bref Bvar EE E

mvar varBvar Bvar avggIlC C Tα α−

⎧ ⎡ ⎤× − + ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎫

(H-37)

( )( )

( ) ( )( )

0 0 0 0 02 2 2 2 2 2 2 2 2

0

0 0 02 20 0

02 2

02 2

2

22

2

2

2

2 2 2' '

2 '

3 12 18

n n n n m n

n

n

n


En mavg


Bvar BrefEvarBvar

EvBvar

V V VR RB T A

gIA T

V lCT T

lC

I C

α α

α

− − −

−

−

−

≡ − − − −

+ −

⎡ ⎤= + × + × −Δ Δ⎢ ⎥⎣ ⎦

− ×Δ

− −

T

V

( )( ) ( )( )

( ) ( )

02 22 2

002 2

00 02 2 2 2

1 20

2

2 2

2' '

1'

61

' '6

nn

n

n n

Bvar Bvar BrefE Ear varBvarBvar

Bvar EEmvarBvar avg


lCC

gI C T

gIlC C T

α α

α

α α

−−

−

− −

−⎡ ⎤× ×Δ Δ⎢ ⎥

⎣ ⎦⎧ ⎫⎡ ⎤+ − ×Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

⎧ ⎫⎡ ⎤≈ − × + − ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

Summary of (H-31) – (H-37) Results:

51

( ) ( )( )

0 02 1 2

00 02 1 9 9

002 9

02 1 8

2 2

2

2

2

For 4 :

For 4 :1 12 6

12

For 3 :

n n

n

n

n

E Em mnum num

Bvar Bref Bvar EE Emnum var varBvar Bvar avg

Bvar Bref EEmnum varBvar avg

Enum vBvar

n

g gB T B Tn

gIlC CB T

glCB T

n

CB

α α

α

−

− − −

−

− −

< −

= − =

= −

⎧ ⎫⎡ ⎤= × − + ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

= − × +Δ

= −

= ( ) ( )( ) ( )

008

00 02 8 8

2

2 2

1616n

Bvar Bref Bvar EEmar varBvar avg


gIl C T

gIlC CB T

α α

α α

−

− −

⎧ ⎫⎡ ⎤× − + ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎧ ⎫⎡ ⎤= − × + − ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

(H-38) For 0 > 3 :n > −

( ) ( )( ) ( )

00 02 1 2 2 2 2

00 02 2 2 2 2

02 1 2

2 2

2 2

1616

For 0 :14

n n n

n n n

n



BvaEnum varBvar

gIlC CB T

gIlC CB T

n

CB

α α

α α

− − −

− −

− −

⎧ ⎫⎡ ⎤= × − + ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎧ ⎫⎡ ⎤= − × + − ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭=

≈ − ( ) ( )( ) ( )

002

00 02 2 2

02 1 0

2

2

1'

63 1

'4 6

For 1:34

n

n

r Bvar Bref Bvar EEmvar varBvar avg

Bvar Bvar Bref Bvar EE Emnum var var varBvar Bvar avg

BvaEnum varBvar

gIl C T

gIlC CB T

n

CB

α α α

α α α

−

− −

−

⎧ ⎫⎡ ⎤⎡ ⎤− × − + ×Δ Δ Δ⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎩ ⎭⎧ ⎫⎡ ⎤⎡ ⎤≈ − − × + − ×Δ Δ Δ⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎩ ⎭

=

= ( ) ( )( ) ( )

000

00 02 0 0

02 1 2 2

2

2

2

1' '

61 1

' '4 6

For 1:

n

n n

r Bvar Bref Bvar EEmvar varBvar avg

Bvar Bvar Bref Bvar EE Emnum var var varBvar Bvar avg

BvaEnum varBvar

gIl C T

gIlC CB T

n

CB

α α α

α α α

− −

⎧ ⎫⎡ ⎤⎡ ⎤− × − + ×Δ Δ Δ⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎩ ⎭⎧ ⎫⎡ ⎤⎡ ⎤= − × + − ×Δ Δ Δ⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎩ ⎭

>

≈ ( ) ( )( ) ( )

002 2

00 02 2 2 2 2

2

2 2

1' '

61

' '6

n

n n n

r Bref Bvar EEmvarBvar avg


gIl C T

gIlC CB T

α α

α α

−

− −

⎧ ⎫⎡ ⎤× − + ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎧ ⎫⎡ ⎤≈ − × + − ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

52

Finding Specific Force 2 1n

Bvarvarυ

−Δ And

2nBvarvarυΔ

Specific force

2 1nBvarvarυ

−Δ and

2nBvarvarυΔ are obtained by substituting , from (H-10)

and , from (H-38) into the (H-1) 2 1

1ndenB −

−2

1ndenB−

2 1nnumB − 2nnumB 2 1nBvarvarυ

−Δ ,

2nBvarvarυΔ formulas:

( )( )

02 12 1 92 1

022 92

11 0

11 0

For 4 :

nnn

nnn

Bvar E Emnumden Bvar avgvar

Bvar E Emnumden Bvar avgvar

n

gCB B T

gCB B T

υ

υ

−− −−

−

−−

−−

< −

= = −Δ

= = −Δ

(H-39)

53

( ) ( ) ( )( )

( ) ( ) ( )

2 12 12 1

09

900

9

099

1

10

10

For 4 :

11 2

12 16

1 12 6

nnnBvar

numdenvarBref

BvarEvarBvarBvar E m

var BvarBvar EE

mvarBvar avg

BrefBvar BvarE Evar varBvarBvar

m

n

B B

lC

TI CgI C T

l I CCT

υ

αα

α

α α

−−−

−

−

−

−−

−

−

−

= −

=Δ

×Δ⎡ ⎤= − ×Δ⎢ ⎥⎣ ⎦ ⎧ ⎫⎡ ⎤− + ×Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

≈ × − +Δ Δ

( )( )( ) ( ) ( ) ( )( )

( )( )( )

0

09

0 0099 9 9

09

10

1 10 0 0

10

1

112

1 12 6

112

12

Emavg


BrefBvar BvarE E E E EE

m mvar varBvarBvar Bvar Bvar avgm


BrefBvarvar

m

g T

gC T

l g gCC C CT TT

gC T

lT

α

α α

α

α

−

−− − −

−

−

−

− −

−

−

⎧ ⎫⎡ ⎤×⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

+ ×Δ

= × − − ×Δ Δ

+ ×Δ

= × −Δ ( )

1−

( )( )( ) ( ) ( )

( ) ( ) ( )

0 09 9

09

09

10 0

10

10

112

1 12 12

1 1 12 4 3

BvarE E E Em mvarBvar Bvaravg avg

BrefBvar Bvar E E

mvar var Bvar avgm

BrefBvar Bvar E E

mvar var Bvar avgm

g gC CT T

l gI C TT

l gI C TT

α

α α

α α

−

−

−

−

−

−

− ×Δ

⎡ ⎤= × − + ×Δ Δ⎢ ⎥⎣ ⎦

⎡ ⎤⎛ ⎞= × − − − ×Δ Δ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦ (H-40)

( ) ( ) ( )( ) ( ) ( )( )

( )

222

0099

0 09 9

1

10

1 10 0

7 112 2

1 72 12

1 72 12

nnnBvar

numdenvar

Bvar Bvar BrefE EEvar varBvarBvar avg

BrefBvar BvarE E E E

mvar varBvar Bvaravg avgm

BrefBvarvar var

m

B B

gI lCC

l g gC C TT

l IT

υ

α α

α α

α

−−

− −

−

−

− −

=Δ

⎡ ⎤ ⎡= − − × − × +Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣

≈ × − + ×Δ Δ

= × − −Δ

⎤⎥⎦

( ) ( )( ) ( ) ( )

09

09

10

101 1 1

2 4 3

Bvar E EmBvar avg

BrefBvar Bvar E E

mvar var Bvar avgm

gC T

l gI C TT

α

α α

−

−

−

−

⎡ ⎤×Δ⎢ ⎥⎣ ⎦

⎡ ⎤⎛ ⎞= × − − + ×Δ Δ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

54

( ) ( ) ( )( )

( ) ( )

2 12 12 1

02 2

2 200

2 2

02 22 2

1

21

0

12 0

For 0 4 :

12 1

6

16

nnn

n

n

n

nn

Bvarnumdenvar

BrefBvarEvarBvarBvar E m

var BvarBvar EE

mvarBvar avg

BrefBvar E Evar BvarBvar

m

n

B B

lC

TI CgI C T

l I CCT

υ

αα

α

α

−−−

−

−

−

−−

−

−

−

> > −

=Δ

×Δ⎡ ⎤= − ×Δ⎢ ⎥⎣ ⎦ ⎧ ⎫⎡ ⎤− + ×Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

≈ × − +Δ ( )( )( )

( ) ( ) ( )( )( ) ( )

0

02 2

0 02 2 2 2

2 2

10

1 12 0 0

12

12

1 12 6

1 12 6

n

n n

n

Bvar Emvar avg



m mvar varBvar Bvar avgm


m

g T

gC T

l g gC CT TT

l IT

α

α

α α

α α

−

− −

−

−

− −

−

⎧ ⎫⎡ ⎤×Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

+ ×Δ

⎛ ⎞= × − + − ×Δ Δ⎜ ⎟⎝ ⎠

⎡ ⎤⎛ ⎞= × − − − ×Δ Δ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦( ) 00E E

mBvar avggC T

(H-41)

( ) ( ) ( )( )

( ) ( ) ( )

222

02 2

2 100

2 2

02 22 1

1

21

0

12 0

12 1

6

16

nnn

n

n

n

nn

Bvarnumdenvar

BrefBvarEvarBvarBvar E m

var BvarBvar EE

mvarBvar


m

B B

lC

TI CgI C T

l I CCT

υ

αα

α

α α

−

−

−

−−

−

−

−

=Δ

− ×Δ⎡ ⎤= − − ×Δ⎢ ⎥⎣ ⎦ ⎧ ⎫⎡ ⎤+ − ×Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

⎧ ⎡ ⎤≈ × − − ×Δ Δ⎨ ⎢ ⎥⎣ ⎦

( )( )( ) ( ) ( )( )

( ) ( ) ( )

0

02 1

0 02 1 2 1

02 1

10

1 12 0 0

12 0

12

1 12 6

1 12 6

n

n n

n

Emavg



m mvar varBvar Bvaravg avgm

BrefBvar Bvar E E

mvar var Bvar avgm

g T

gC T

l g gC CT TT

l gI C TT

α

α α

α α

−

− −

−

−

− −

−

⎫⎬

⎩ ⎭

+ ×Δ

⎛ ⎞≈ × − + + ×Δ Δ⎜ ⎟⎝ ⎠

⎡ ⎤⎛ ⎞= × − − + ×Δ Δ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

55

( ) ( )( )

( )( )

2 12 12 1

02

2 002

1

10

2

For 0 :

1'

1 4 /126

1'

4

nnnBvar

numdenvar


Bvar Emvar Bvar Bvar EE

mvarBvar avg


n

B B

lCI C T

gI C T

l

υ

α αα

α

α α

−−−

−

−

−

−

−

=

=Δ

⎡ ⎤− − ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤= − ×Δ⎢ ⎥ ⎧ ⎫⎣ ⎦ ⎡ ⎤− + ×Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

⎡ ⎤≈ − − ×Δ Δ⎢ ⎥⎣ ⎦

( ) ( ) ( )( )0 0022 2

1 10 01 1

6 2

m

Bvar BvarE EEm mvar varBvarBvar Bvaravg avg

T

g gI CC CT Tα α−− −

− −⎧ ⎫⎡ ⎤− + × + ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭E E

( ) ( )( ) ( ) ( )( )

02

0 0022 2

10

1 10 0

1'

4

1 16 2

BrefBvar Bvar E E

mvar var Bvar avgm

Bvar BvarE E EEm mvar varBvarBvar Bvar avg

l gC TT

g gCC CT T

α α

α α

−

−− −

−

− −

⎡ ⎤≈ − − × −Δ Δ⎢ ⎥⎣ ⎦

⎡ ⎤− × + ×Δ Δ⎢ ⎥⎣ ⎦E

(H-42)

( ) ( )( ) ( )( ) ( )( )

( ) ( )

02

0 0022 2

02

10

1 10 0 0

10

1'

4

1 16 2

1'

4

16

BrefBvar Bvar E E

mvar var Bvar avgm

Bvar BvarE E E EEm mvar varBvarBvar Bvar Bvaravg avg

BrefBvar Bvar E E

mvar var Bvarm

l gC TT

g gCC C CT T

l gC TT

α α

α α

α α

−

−− −

−

−

− −

−

⎡ ⎤≈ − − × −Δ Δ⎢ ⎥⎣ ⎦

− × + ×Δ Δ

⎡ ⎤≈ − − × −Δ Δ⎢ ⎥⎣ ⎦

−

2

1 E−

−

( )( ) ( )( )( ) ( ) ( )

0 02 2

02

1 10 0

10

12

1 1 1'

4 2 6

Bvar BvarE E E Em mvar varBvar Bvaravg avg


mvar var var Bvar avgm

g gC CT T

l gI C TT

α α

α α α

− −

−

− −

−

× + ×Δ Δ

⎡ ⎤⎛ ⎞⎡ ⎤= − − × − − − ×Δ Δ Δ⎜ ⎟⎢ ⎥⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦

(Continued)

56

(H-42) Concluded

( ) ( )( )

( )( ) ( )

222

02

1 002

021

1

10

2

10

3'

1 4 /126

3'

4

nnnBvar

numdenvar



mvarBvar

BrBvar BvarE Evar varBvarBvar

B B

lCI C T

gI C T

CC

υ

α αα

α

α α

−

−

−

−−

−

−

−

=Δ

⎡ ⎤− − ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤= − − ×Δ⎢ ⎥ ⎧ ⎫⎣ ⎦ ⎡ ⎤+ − ×Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

⎡ ⎤= − ×Δ Δ⎢ ⎥⎣ ⎦

( ) ( ) ( )( )( ) ( )

( )( ) ( )

0 0021 1

01

01 1

1 10 0

10

1 10

1 16 2

3'

4

1 16 2

ef

m

Bvar BvarE EEm mvar varBvarBvar Bvaravg avg

BrefBvar Bvar E E

mvar var Bvar avgm

Bvar BvarE Emvar varBvar Bvavg

lT

g gI CC CT T

l gC TT

gC T

α α

α α

α α

−− −

−

− −

− −

−

− −

⎧ ⎫⎡ ⎤− − × + ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

⎡ ⎤≈ − × −Δ Δ⎢ ⎥⎣ ⎦

+ × + ×Δ Δ ( )

E E

( ) ( ) ( )

0

01

0

103 1 1

'4 2 6

E Emar avg



gC T

l gI C TT

α α α−

−⎡ ⎤⎛ ⎞⎡ ⎤= − × − − + ×Δ Δ Δ⎜ ⎟⎢ ⎥⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦

57

( ) ( )( )

( )( )

2 12 12 1

00

0 000

1

10

2

For 1:

3'

1 4 /'12 '6

3'

4

nnnBvar

numdenvar



mvarBvar avg


m

n

B B

lCI C T

gI C T

lT

υ

α αα

α

α α

−−−−

−

=

=Δ

⎡ ⎤− ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤= − ×Δ⎢ ⎥ ⎧ ⎫⎣ ⎦ ⎡ ⎤− + ×Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

⎡ ⎤≈ − × +Δ Δ⎢ ⎥⎣ ⎦ ( )( )( ) ( )

( ) ( )( )( ) ( )( )

00

0000

00

00 0

10

10

10

1 10 0

1'

2

1'

6

3 1' '

4 2

1'

6


BvarE EEmvarBvarBvar avg



BvarE E Em varBvar Bvaravg avg

gC T

gI CC T

l gC TT

gC CT

α

α

α α α

α

−

−

−

− −

×Δ

⎧ ⎫⎡ ⎤− + ×Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

⎡ ⎤≈ − × + ×Δ Δ Δ⎢ ⎥⎣ ⎦

− − ×Δ

( ) ( ) ( )

0

00

103 1 1

' '4 2 6

Em



g T

l gI C TT

α α α−⎡ ⎤⎛ ⎞⎡ ⎤= − × − − − ×Δ Δ Δ⎜ ⎟⎢ ⎥⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦

(H-43)

( ) ( )( )

( )( )

222

00

1 000

1

10

2

1'

1 4 /'12 '6

1 1' '

4 2

nnnBvar

numdenvar



mvarBvar avg

BrefBvar Bvar Bvarvar var var

m

B B

lCI C T

gI C T

lT

υ

α αα

α

α α α

−

−

=Δ

⎡ ⎤− ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤= − − ×Δ⎢ ⎥ ⎧ ⎫⎣ ⎦ ⎡ ⎤+ − ×Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

⎡ ⎤≈ − − × +Δ Δ Δ⎢ ⎥⎣ ⎦ ( )( )( ) ( )

( ) ( ) ( )

01

0001

01

10

10

10

1'

6

1 1 1' '

4 2 6

E EmBvar avg

BvarE EEmvarBvarBvar avg



gC T

gI CC T

l gI C TT

α

α α α

−

−

−

×

⎧ ⎫⎡ ⎤− − ×Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

⎡ ⎤⎛ ⎞⎡ ⎤≈ − − × − − + ×Δ Δ Δ⎜ ⎟⎢ ⎥⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦

58

( ) ( ) ( )( )

( ) ( )

2 12 12 1

02 2

2 2 002 1

2 12 2

1

21

02

12 0

For 1:

'1 /' 12 '6

1'

6

nnn

n

n

n

nn

Bvarnumdenvar

Bvar BrefEvarBvarBvar E

mvar Bvar Bvar EEmvarBvar avg

BrefBvar Evar BvarBvar

m

n

B B

lCI C T

gI C T

l ICT

υ

αα

α

α

−−−

−

−

−

−−

−

−

−

>

=Δ

×Δ⎡ ⎤= − ×Δ⎢ ⎥ ⎧ ⎫⎣ ⎦ ⎡ ⎤− + ×Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

≈ × − +Δ ( )( )( )

( ) ( ) ( )

00

02 2

02 2

10

12 0

'

1'

21 1 /' '2 6

n

n

Bvar EEmvar avg


BrefBvar Bvar E E

mvar var Bvar avgm

gC T

gC T

l gI C TT

α

α

α α

−

−

−

−

⎧ ⎫⎡ ⎤×Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

+ ×Δ

⎡ ⎤⎛ ⎞≈ × − − − ×Δ Δ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

(H-44)

( ) ( ) ( )( )

( ) ( )

222

02 2

2 1 002 2

02 22 1

1

21

02

12 0

'1 /' 12 '6

1'

6

nnn

n

n

n

nn

Bvarnumdenvar

Bvar BrefEvarBvarBvar E

mvar Bvar Bvar EEmvarBvar avg


m

B B

lCI C T

gI C T

l I CCT

υ

αα

α

α α

−

−

−

−−

−

−

−

=Δ

− ×Δ⎡ ⎤= − − ×Δ⎢ ⎥ ⎧ ⎫⎣ ⎦ ⎡ ⎤+ − ×Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

≈ × − −Δ Δ( )( )( )

( ) ( ) ( )

0

02 1

02 1

10

12 0

'

1'

21 1

' '2 6

n

n

Emavg


BrefBvar Bvar E E

mvar var Bvar avgm

g T

gC T

l gI C TT

α

α α

−

−

−

−

⎧ ⎫⎡ ⎤×⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

+ ×Δ

⎡ ⎤⎛ ⎞≈ × − − + ×Δ Δ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

Summary of (H-39) – (H-44) Specific Force Results:

59

( ) ( )

( ) ( ) ( )( )

0 09 92 1 2

092 1

2

1 10 0

10

For 4 :

For 4 :

1 1 12 4 3

12

n n

n

n

Bvar BvarE E E Em mBvar Bvaravg avgvar var


mvar var Bvar avgvarm

BrefBvar Bvar

varvarm

n

g gC CT T

n

l gI C TT

l IT

υ υ

υ α α

υ α

− −−

−−

− −

−

< −

= − = −Δ Δ

= −

⎡ ⎤⎛ ⎞= × − − − ×Δ Δ Δ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

= × − −Δ Δ ( ) ( )

( ) ( ) ( )( ) ( )

09

02 22 1

2

10

12 0

2

1 14 3

For 0 4 :

1 12 6

1 12 6

nn

n




BrefBvar Bvar Bvar

var varvarm

gC T

n

l gI C TT

l IT

α

υ α α

υ α α

−

−−

−

−

⎡ ⎤⎛ ⎞+ ×Δ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦> > −

⎡ ⎤⎛ ⎞= × − − − ×Δ Δ Δ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞= × − − + ×Δ Δ Δ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦( )

( ) ( ) ( )

02 1

022 1

10

10

For 0 :

1 1 1'

4 2 6

n

n

E EmBvar avg


mvar var var Bvar avgvarm

gC T

n

l gI C TT

υ α α α

−

−−

−

−

=

⎡ ⎤⎛ ⎞⎡ ⎤= − − × − − − ×Δ Δ Δ Δ⎜ ⎟⎢ ⎥⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦

( ) ( ) ( ) 012

103 1 1

'4 2 6n



l gI C TT

υ α α α−

−⎡ ⎤⎛ ⎞⎡ ⎤= − × − − + ×Δ Δ Δ Δ⎜ ⎟⎢ ⎥⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦ (H-45)

( ) ( ) ( )( ) ( ) ( )

002 1

12

10

10

For 1:

3 1 1' '

4 2 6

1 1 1' '

4 2 6

n

n



BrefBvar Bvar Bvar Bvar E

var var var Bvar avgvarm

n

l gI C TT

l I CT

υ α α α

υ α α α

−

−

−

=

⎡ ⎤⎛ ⎞⎡ ⎤= − × − − − ×Δ Δ Δ Δ⎜ ⎟⎢ ⎥⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞⎡ ⎤= − − × − − + ×Δ Δ Δ Δ⎜ ⎟⎢ ⎥⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦

( ) ( ) ( )( ) ( ) ( )

0

02 22 1

02 12

12 0

12 0

For 1:

1 1' '

2 6

1 1' '

2 6

nn

nn

Em





g T

n

l gI C TT

l gI C TT

υ α α

υ α α

−−

−

−

−

>

⎡ ⎤⎛ ⎞= × − − − ×Δ Δ Δ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞= × − − + ×Δ Δ Δ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

Recognizing that the inverse of a direction cosine matrix equals its transpose, the (

terms in (H-45) become , generating (49) in [1].

) 10

nEBvarC

−

0nBvarEC

60

VELOCITY DETERMINATION

Approximating gravity for test example conditions as the constant 0Eavgg , velocity from (3)

with becomes at time instants m-1 and m: 2m = n

00 0 02 12 22 1 2 2 2 1

0 0 0 022 12 2 1 2

nnn n n

nnn n n

EBvarE E EmVvar avgBvarvar var var

BvarE E E EmVvarBvarvar var var avg

gV V C G T

gV V C G T

υ

υ

−−− − −

−−

= + +Δ

= + +Δ (H-46)

With initial velocity at 00

ErefV , (H-45) for Bvar

varυΔ specific force, (32) for attitude, and

(33) for , (H-46) becomes to first order accuracy for velocity:

0EBvarC

VvarG

( )00 0 0

2 12 22 1 2 2 2 1

0 00 0 09 90 0

00 0 022 12 2 1 2

0 02 10

10

For 4 :

nnn n n

nnn n n

n


E E EE E Em mref Bvar refBvar


E Eref Bvar

n

gV V C G T

g gIV VC C T T

gV V C G T

V C

υ

υ

−−− − −

− −

−−

−

−

< −

= + +Δ

= − + =

= + +Δ

= − ( ) 0 0 09 0

10E E E E

m m refBvar avgg gI VC T T−

−+ =

(H-47)

61

( )( )

00 0 02 12 22 1 2 2 2 1

00 0 09109 10 9

0 090 1

For 4 :

12

1 14 3

nnn n nBvar EE E E

mVvarBvar avgvar var var


BrefBvarvar

mE Eref Bvar

Bvarvar

n

gV V C G T

gV V C G T

lTIV C

I

υ

υ

α

α

−−− − −

−−− − −

− −

= −

= + +Δ

= = + +Δ

×Δ= +

⎡ ⎤⎛ ⎞− − − ×Δ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦( )

( ) ( )( )( ) ( )( )

0

09

00 0 09 9 90

00 0 09 9 90

0

10

10

1 1 12 4 3

1 12 12

Emavg

E EmBvar


mvar varref Bvar Bvar Bvar avgmBref

Bvar Bvar E EE E Evar varref Bvar Bvar Bvar avg

m

g TgC T

l gV C C C TT

lV C C CT

α α

α α

−

− − −

− − −

−

−

⎧ ⎫⎪ ⎪⎪ ⎪ +⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭

⎛ ⎞= + × + − ×Δ Δ⎜ ⎟⎝ ⎠

= + × − ×Δ Δ mg T

(H-48)

( ) ( )( )

( )( )

00 0 0 0892 8 9 2

00 0 09 9 90

09

101 1

2 12

121

2 1 14 3

n nBvar EE E E E

mVvarBvar avgvar var var varBref

Bvar Bvar E EE E Emvar varref Bvar Bvar Bvar avg

mBref

Bvarvar

mBvarEvarBvar

gV V V C G T

l gV C C C TT

lTIC

I

υ

α α

αα

−−− −

− − −

−

−

= = + +Δ

= + × − ×Δ Δ

×Δ⎡ ⎤+ + ×Δ⎢ ⎥⎣ ⎦ ⎛− − + ( ) ( )

( ) ( )( )( ) ( )

0

09

00 0 09 9 90

09

10

10

1

1 12 12

1 1 12 4 3

Emavg

Bvar E Emvar Bvar


mvar varref Bvar Bvar Bvar avgm

BrefBvar BvarEvar varBvar

m

g TgC T

l gV C C C TT

l ICT

α

α α

α α

−

− − −

−

−

−

−

⎧ ⎫⎪ ⎪⎪ ⎪ +⎨ ⎬

⎡ ⎤⎞⎪ ⎪×Δ⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭

= + × − ×Δ Δ

⎡ ⎤⎛ ⎞+ × − − + ×Δ Δ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦( )

( )( )( )

09

0 009 9

0 090

0

101

2

E EmBvar avg

Bvar E E EEm mvarBvar Bvar avg


m

gC T

g gC C T T

lV CT

α

α

−

− −

−

−

⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭

− × +Δ

= + ×Δ

62

( )( ) ( ) ( )

00 0 02 12 22 1 2 2 2 1

0 02 30

0 002 2 2 2

10

For 0 4 :

1 1 12 2 6

nnn n n

n

n n



m

Bvar Bvar E E EEmvar varBvar Bvar avg avg

n

gV V C G T

lV CT

gI IC C T

υ

α

α α

−−− − −

−

− −

−

> > −

= + +Δ

= + ×Δ

⎡ ⎤⎡ ⎤ ⎛ ⎞− + × − − × +Δ Δ⎜ ⎟⎢ ⎥⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦

( ) ( )( )( ) ( )

( )

00 0 02 2 2 2 2 20

002 2 2 2

0 0 02 2 2 20

10

10

12

1 12 6

16

n n n

n n

n n

m


mvar varref Bvar Bvar Bvarm


BrefBvar BvaE E Evar varref Bvar Bvar

m

g T

l gV C C C TT

gC C T

lV C CT

α α

α

α

− − −

− −

− −

−

−

≈ + × − ×Δ Δ

⎡ ⎤⎛ ⎞− − − ×Δ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

= + × −Δ ( )( ) 02 2

10

nr E E

mBvar avggC Tα−

−×Δ

(H-49)

( ) ( )( )( ) ( )

00 0 022 12 2 1 2

00 0 02 2 2 2 2 20

02 1

101

6

1 1 12 2 6

nnn n n

n n n

n





gV V C G T

l gV C C C TT

I IC

υ

α α

α α

−−

− − −

−

−

= + +Δ

≈ + × − ×Δ Δ

⎡ ⎤⎡ ⎤ ⎛ ⎞− + × − + ×Δ Δ⎜ ⎟⎢⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦( )

( ) ( )( )( )( )

( )

0 02 1

00 0 02 1 2 1 2 10

002 1 2 1

02 1 2

10

10

10

1

16

12

1 12 6

n

n n n

n n

n n

E E Em mBvar avg




BvarEvarBvar

g gC T T

l gV C C C TT

gC C T

C

α α

α

α

−

− − −

− −

− −

−

−

−

−

+⎥

≈ + × − ×Δ Δ

− ×Δ

⎡ ⎤⎛ ⎞− − + ×Δ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦( )

( )

01

0 02 10

0

n

E EmBvar avg


m

gC T

lV CT

α−

= + ×Δ

63

( )

( )

00 0 0 02 12 22 1 1 2 2 2 1

00 0122 1

0 030

02 2

For 0 :

141

2

nnn n n

n

Bvar EE E E EmVvarBvar avgvar var var var

Bvar EE EmVvarBvar avgvar var


m

varBvarEvarBvar

n

gV V V C G T

gV C G T

lV CT

IC

υ

υ

α

α

−−− − − −

−−− −

−

−

=

= = + +Δ

= + +Δ

= + ×Δ

−⎡ ⎤+ + ×Δ⎢ ⎥⎣ ⎦

( )( )( )

( ) ( )( )

0

02

00 0 02 2 20

02

10

120

'

1 12 6

12

1'

4

Bvar Bvar Brefvar

EmavgBvar E E

mvar Bvar avg



BvarEvarBvar

lg T

gI C T

l gV C C C TT

C

α α

α

α α

α

−

− − −

−

−

−

⎧ ⎫⎡ ⎤− ×Δ Δ⎪ ⎪⎢ ⎥⎣ ⎦⎪ ⎪ +⎨ ⎬⎡ ⎤⎛ ⎞⎪ ⎪− − − ×Δ⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭

≈ + × − ×Δ Δ

− Δ( ) ( )( )( ) ( )( )

02

00 0 02 2 20

10

10

1 12 6

1 15 '4 6

Bvar Bref Bvar E Emvar var Bvar avg

BrefBvar Bvar Bvar E EE E E

mvar var varref Bvar Bvar Bvar avgm

gl C T

l gV C C C TT

α α

α α α

−

− − −

−

−

⎧ ⎫⎛ ⎞⎡ ⎤− × − − ×Δ Δ⎨ ⎬⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠⎩ ⎭

⎡ ⎤= + − × − ×Δ Δ Δ⎢ ⎥⎣ ⎦

(H-50)

( ) ( )( )

00 0 0 022 12 0 2 1 2

00 0011 0

00 0 02 2 20

101 15 '

4 6

nnn n nBvar EE E E E

mVvarBvar avgvar var var var


BrefBvar Bvar Bvar E EE E Evar var varref Bvar Bvar Bvar

m

gV V V C G T

gV C G T

lV C CT

υ

υ

α α α

−−

−−

− − −

−

= = + +Δ

= + +Δ

⎡ ⎤= + − × − ×Δ Δ Δ⎢ ⎥⎣ ⎦ C

( )( )

( ) ( )( )

001 0

1

0 01 10

2

120

3'

411 122 6

1 15 '4 6

m

Bvar Bvar Brefvar var

Bvar EEmvarBvar avgBvar E E

mvar Bvar

BrefBvar BvarE Evar varref Bvar Bv

m

g T

lgIC T

gI C T

lV CT

α αα

α

α α

−

−

− −

−

⎧ ⎫⎡ ⎤− ×Δ Δ⎪ ⎪⎢ ⎥⎣ ⎦⎪ ⎪⎡ ⎤+ + × +Δ ⎨ ⎬⎢ ⎥ ⎡ ⎤⎛ ⎞⎣ ⎦ ⎪ ⎪− − + ×Δ⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭

⎡ ⎤≈ + − × −Δ Δ⎢ ⎥⎣ ⎦ ( )( )( )( ) ( )

( )( )

001

00 01 11

001 1

0 010

120

120

120

1 3'

2 41 12 6

12

Bvar E EEmvarar Bvar avg

Bvar Bvar Bvar BrefE EE Emvar var varBvar BvarBvar


BvarE Evar vref Bvar

gC C T

g lC CC T

gC C T

V C

α

α α

α

α

−

− −−

− −

−

−

−

−

×Δ

⎡ ⎤− × + − ×Δ Δ Δ⎢ ⎥⎣ ⎦

⎛ ⎞+ + ×Δ⎜ ⎟⎝ ⎠

= + +Δ

α

( )'Bref

Bvarar

m

lT

α⎡ ⎤×Δ⎢ ⎥⎣ ⎦

64

( )

( )

00 0 0 02 12 22 1 1 2 2 2 1

00 0100 1

0 010

00

For 1:

1'

2

1'

2



Bvar EE EmVvarBvar avgvar varBref

Bvar BvarE Evar varref Bvar

m

BvarEvarBvar

n

gV V V C G T

gV C G T

lV CT

IC

υ

υ

α α

α

−−− − −

−

=

= = + +Δ

= + +Δ

⎡ ⎤= + + ×Δ Δ⎢ ⎥⎣ ⎦

+ + ×Δ( )

( )( )( ) ( )( )

0

00

00 0 00 0 00

10

10

3'

4

1 1'

2 6

1 1' '

2 2


m Emavg

Bvar E Emvar Bvar

BrefBvar Bvar Bvar EE E Evar var varref Bvar Bvar Bvar avg

m

lT g T

gI C T

lV C CT

α α

α

α α α

−

−

⎧ ⎫⎡ ⎤− ×Δ Δ⎪ ⎪⎢ ⎥⎣ ⎦⎪ ⎪⎡ ⎤ +⎨ ⎬⎢ ⎥⎣ ⎦ ⎡ ⎤⎛ ⎞⎪ ⎪− − − ×Δ⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭

⎡ ⎤≈ + + × − ×Δ Δ Δ⎢ ⎥⎣ ⎦ C

( ) ( )( )( ) ( )( )

00 00 0 0

00 0 00 0 00

10

10

3 1 1' '

4 2 6

1 15 ' '4 6

Em

BrefBvar Bvar Bvar E EE E

mvar var varBvar Bvar Bvar avgm

BrefBvar Bvar Bvar E EE E E

mvar var varref Bvar Bvar Bvar avgm

g T

l gC C C TT

l gV C C TT

α α α

α α α

−

−

⎛ ⎞⎡ ⎤+ − × + − ×Δ Δ Δ⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠

⎡ ⎤= + − × − ×Δ Δ Δ⎢ ⎥⎣ ⎦ C

(H-51)

( ) ( )( )

00 0 0 022 12 2 2 1 2

00 0211 2

00 0 00 0 00

101 15 ' '

4 6




BrefBvar Bvar Bvar E EE E Evar var varref Bvar Bvar Bvar avg

m

gV V V C G T

gV C G T

l gV C C CT

υ

υ

α α α

−−

−

= = + +Δ

= + +Δ

⎡ ⎤= + − × − ×Δ Δ Δ⎢ ⎥⎣ ⎦

( )( )

( )( )( )

001

01

0 01 10

10

1'

41'

2 1 1'

2 6

1 15 '4 6

m


mBvar EEmvarBvar avg

Bvar E Emvar Bvar

BrefBvar BvarE Evar varref Bvar Bvar

m

T

lT gIC T

gI C T

lV CT

α αα

α

α α

−

⎧ ⎫⎡ ⎤− − ×Δ Δ⎪ ⎪⎢ ⎥⎣ ⎦⎪ ⎪⎡ ⎤+ + × +Δ ⎨ ⎬⎢ ⎥⎣ ⎦ ⎡ ⎤⎛ ⎞⎪ ⎪− − + ×Δ⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭

⎡ ⎤≈ + − × −Δ Δ⎢ ⎥⎣ ⎦ ( )( )( )( ) ( )

( )( )( )

001

00 01 11

001 2 1

0 010

10

10

10

'

1 1' '

2 4

1 1'

2 6

'

n

Bvar E EEmvar Bvar avg

BrefBvar Bvar BvarE EE E

mvar var varBvar BvarBvar avgm


BreBvarE Evarref Bvar

gC C T

lgC CC TT

gC C T

V C

α

α α α

α

α

−

−

−

−

×Δ

⎡ ⎤− × − − ×Δ Δ Δ⎢ ⎥⎣ ⎦

⎛ ⎞+ + ×Δ⎜ ⎟⎝ ⎠

= + ×Δf

m

lT

65

( )

( )( )

( )

00 0 02 12 22 1 2 2 2 1

0 02 30

02 2

2 2

2

1

For 1:

'

'1

'2 1 1

'2 6

nnn n n

n

n

n



mBref

Bvarvar

mBvarEvarBvar

Bvarvar B

n

gV V C G T

lV CT

lTIC

I

υ

α

αα

α

−−− − −

−

−

−

−

>

= + +Δ

= + ×Δ

×Δ⎡ ⎤+ + ×Δ⎢ ⎥⎣ ⎦ ⎡ ⎤⎛ ⎞− − − ×Δ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

( )( ) ( )( )

0

0

00 0 02 2 2 2 2 20

0

101

' '6n n n

Emavg

E Emvar



g TgC T

l gV C C C TT

α α− − −

−

⎧ ⎫⎪ ⎪⎪ ⎪ +⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭

≈ + × − ×Δ Δ

(H-52)

( ) ( )( )

( )( )

00 0 022 12 2 1 2

00 0 02 2 2 2 2 20

02 1

10

2

1' '

6

'1

'2 1

2

nnn n n

n n n

n




BrefBvarvar

mBvarEvarBvar

gV V C G T

l gV C C C TT

lTIC

I

υ

α α

αα

−−

− − −

−

−

= + +Δ

= + × − ×Δ Δ

×Δ⎡ ⎤+ + ×Δ⎢ ⎥⎣ ⎦

− − + ( ) ( )( ) ( )( )

( )( )

0

02 1

00 0 02 1 2 1 2 10

002 1 2 1

2

10

10

10

1'

6

1' '

6

1'

2

n

n n n

n n

n

Emavg

Bvar E Emvar Bvar




g TgC T

l gV C C C TT

gC C T

α

α α

α

−

− − −

− −

−

−

−

−

⎧ ⎫⎪ ⎪⎪ ⎪ +⎨ ⎬

⎡ ⎤⎛ ⎞⎪ ⎪×Δ⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭

≈ + × − ×Δ Δ

− ×Δ

+ ( )( )( )

001 2 1

0 02 10

101 1

'2 6

'

n

n



m

gC C T

lV CT

α

α

−

−

−⎛ ⎞+ ×Δ⎜ ⎟⎝ ⎠

= + ×Δ

Summary of (H-47) to (H-52) Velocity Results:

66

( ) ( )( )( )

0 0 0 02 1 20 0

00 0 0 09 9 99 0

0 0 098 0

2 1

10

For 4 :

For 4 :

1 12 12

For 0 4 :

n n

n

E E E Eref refvar var

BrefBvar Bvar E EE E E E

mvar varref Bvar Bvar Bvar avgvarm


m

var

n V V V V

n

l gV V C C C TT

lV V CT

n

α α

α

−

− − −−

−−

−

−

< − = =

= −

= + × − ×Δ Δ

= + ×Δ

> > −

( ) ( )( )( )

00 0 0 02 2 2 2 2 20

0 0 02 12 0

101

6n n n

nn

BrefBvar Bvar E EE E E E



m

l gV V C C C TT

lV V CT

α α

α

− − −

−

−= + × − ×Δ Δ

= + ×Δ

( )( )( )

( )

0 0 021 0

002 2

0 0 010 0

10

For 0 :

1 5 '4

16

1'

2


m



m

n

lV V CT

gC C T

lV V CT

α α

α

α α

−−

− −

−

−

=

⎡ ⎤= + − ×Δ Δ⎢ ⎥⎣ ⎦

− ×Δ

⎡ ⎤= + + ×Δ Δ⎢ ⎥⎣ ⎦

(H-53)

( ) ( )( )( )

( )

00 0 0 00 0 01 0

0 0 012 0

0 0 02 22 1 0

10

For 1:

1 15 ' '4 6

'

For 1:

'nn

BrefBvar Bvar Bvar E EE E E E

mvar var varref Bvar Bvar Bvar avgvarm


m


n

l gV V C C C TT

lV V CT

n

V V C

α α α

α

α−−

−

=

⎡ ⎤= + − × − ×Δ Δ Δ⎢ ⎥⎣ ⎦

= + ×Δ

>

= + ×Δ ( )( )( )

002 2 2 2

0 0 02 12 0

101

'6

'

n n

nn

BrefBvar E EE

mvarBvar Bvar avgm


m

l gC C TT

lV V CT

α

α

− −

−

−− ×Δ

= + ×Δ

Recognizing that the inverse of a direction cosine matrix equals its transpose, the (


) 10

nEBvarC

−

0nBvarEC

67

POSITION DETERMINATION

Approximating gravity for test example conditions as the constant 0Eavgg , position from (4)

with becomes at time instants m-1 and m: 2m = n

00 0 0 02 12 22 1 2 2 2 2 2 1

00 0 0 022 12 2 1 2 1 2

2

2

12

12

nnn n n n

nnn n n n

EBvarE E E Em mRvar avgBvarvar var var var

Bvar EE E E Em mRvarBvarvar var avgvar var

gVR R C GT T

gVR R C GT T

υ

υ

−−− − − −

−− −

= + + +Δ

= + + +Δ

m

m

T

T

(H-54)

Initializing 0

2 1nEvarR

− position (i.e., 0

2 2nEvarR

−) for derives directly from (37) with

, and from (32):

4n < −

2m n= − 2 0 09m

E EBvar BvarC C −

=

( )0 0 0

92 2 0For 4 : 2 2

nE E E Bref

mref Bvarvarn n VR C lT −−< − = − + (H-55)

With 0

2 2nEvarR

− from (H-55) for , (H-53) for 4n < − 0E

varV velocity, (H-45) for BvarvarυΔ

specific force, (32) for attitude, and (33) for , (H-54) becomes to first order accuracy for position:

0EBvarC RvarG

( ) ( )00 0 0 0

2 12 22 1 2 2 2 2 2 1

0 00 0 0 09 2 2 90 0

2

12 20

For 4 :12

1 12 22 2

nnn n n n

n


E E EE E E EBrefm m mref Bvar ref Bvar Bvar avg avg

n

gVR R C GT T

g gn V VC l C CT T T

υ−−− − − −

− − −

−

< −

= + + +Δ

= − + + − +

m

m

T

T

( ) 0 0

90

00 0 0 022 12 2 1 2 1 2

2

2 1

12nnn n n n

E E Brefmref Bvar


n V C lT

gVR R C GT Tυ

−

−− −

= − +

= + + +Δ mT (H-56)

( ) ( ) 0 00 0 0 09 9 90 0

0 090

1201 12 1

2 2

2

E E EE E E EBrefm m m mref Bvar ref Bvar Bvar avg avg

E E Brefmref Bvar

g gn V VC l C CT T T T

n V C lT

− − −

−

−= − + + − +

= +

mT

68

( )( ) ( )

00 0 0 09109 10 10 9

0 0 090 0

09

09

2

10

For 4 :12

10

121

2 1 14 3


E E EBrefm mref Bvar ref

BrefBvarvar

mEBvar

Bvar E Emvar Bvar

n

gVR R C GT T

V VC lT T

lT

CgI C T

υ

α

α

−−− − − −

−

−

−

−

= −

= + + +Δ

= − + +

⎧×Δ⎪

+ ⎨⎡ ⎤⎛ ⎞− − − ×Δ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

mT

( )( )( )

0

0 090

009 9

2

120

12

194

1 1 12 4 3

Em avg

Bvar BrefE Em varref Bvar


gT

IV lCT

gC C T

α

α

−

− −

−

⎫⎪

⎪ ⎪ +⎬⎪ ⎪⎪ ⎪⎩ ⎭

⎡ ⎤= − + + ×Δ⎢ ⎥⎣ ⎦

⎛ ⎞+ − ×Δ⎜ ⎟⎝ ⎠

mT (H-57)

(Continued)

69

(H-57) Concluded

( )( )( )

00 0 0 0898 9 9 8

0 0 090 0

009 9

09

2

120

12

194

1 1 12 4 3

12

Bvar EE E E Em m mRvarBvarvar var avgvar var

Bvar BrefE E Em mvarref Bvar ref


BvarEvarBvar

gVR R C GT T T

IV l VCT T

gC C T

C

υ

α

α

−−− − − −

−

− −

−

−

= + + +Δ

⎡ ⎤= − + + × +Δ⎢ ⎥⎣ ⎦

⎛ ⎞+ − ×Δ⎜ ⎟⎝ ⎠

+ Δ( ) ( )( )

( )( )

( ) ( )

009 9

009

09

0 090

120

21

0

112

121 1 1

2 3 21 14 3

18

Bref Bvar E EEmvarBvar Bvar avg

BrefBvarvar

mBvar EEm mvarBvar avg

Bvar E Emvar Bvar

E Emref Bvar

gl C C T

lT gIC T T

gI C T

IV CT

α α

αα

α

− −

−

−

−

−

−

× − ×Δ

⎧ ⎫×Δ⎪ ⎪

⎪ ⎪⎡ ⎤+ + × +Δ ⎨ ⎬⎢ ⎥⎣ ⎦ ⎡ ⎤⎛ ⎞⎪ ⎪− − + ×Δ⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭

= − + + ( ) ( )( )( ) ( )( )

( )( )

009 9

00 09 9 9

009 9

09

120

120

10

1 1 14 2 4 3

1 12 12

16

14

Bvar Bref Bvar E EEmvar varBvar Bvar avg

Bvar Bref Bvar E EE Emvar varBvar Bvar Bvar avg

Bvar E EEvarBvar Bvar avg

BEvarBvar

gl C C T

glC C C T

gC C

C

α α

α α

α

− −

− − −

− −

−

−

−

−

⎡ ⎤ ⎛ ⎞× + − ×Δ Δ⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠

+ × − ×Δ Δ

− ×Δ

+ ( ) ( )( )( ) ( )( )

( )( )

009 9

00 0 09 9 90

009 9

120

120

10

1 1 12 4 3

1 1 182 4 3

112

var Bref Bvar E EEmvarBvar Bvar avg

Bvar Bref Bvar E EE E Em mvar varref Bvar Bvar Bvar


gl C C T

gIV lC C CT T

gC C

α α

α α

α

− −

− − −

− −

−

−

−

⎛ ⎞× + + ×Δ Δ⎜ ⎟⎝ ⎠

⎛ ⎞⎡ ⎤= − + + × + − ×Δ Δ⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠

− ×Δ ( )( )( )( )

( )

009 9

009 9

0 090

0 080

12 0

120

16

1 1 12 4 3

8

8

Bvar E EEvarBvar Bvar avg



BrefE Emref Bvar

gC CT

gC C T

IV lCT

V lCT

α

α

α

− −

− −

−

−

−

−

− ×Δ

⎛ ⎞+ + ×Δ⎜ ⎟⎝ ⎠

⎡ ⎤= − + + ×Δ⎢ ⎥⎣ ⎦

= − +

70

( ) ( )

( )

00 0 0 02 12 22 1 2 2 2 2 2 1

0 0 0 02 2 2 30 0

02 2

2

2

For 0 4 :12

2 2

1 12 3

nnn n n n

n n

n


Bref Bvar BrefE E E Em m varref Bvar ref Bvar

Bvar

BvarEvarBvar

n

gVR R C GT T

n V l V lC CT T

IC

υ

α

α

−−− − − −

− −

−

> > −

= + + +Δ

= − + + + ×Δ

⎡ ⎤+ + ×Δ⎢ ⎥⎣ ⎦

mT

( )( ) ( ) 0

2 2

101 1

2 6 n

Brefvar

mm

Bvar E Emvar Bvar

lT

TgI C T

α

α−

−

⎧ ⎫×Δ⎪ ⎪

⎪ ⎪⎨ ⎬

⎡ ⎤⎛ ⎞⎪ ⎪− − − ×Δ⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭

0 212

Emavgg T+ (H-58)

( ) ( )( )( )

( )( )( ) ( )

0 0 02 2 2 20

002 2 2 2

002 2 2 2

0 02 20

120

120

2 1

16

1 1 12 2 6

2 1

n n

n n

n n

n

Bref Bvar BrefE E Em varref Bvar Bvar



BvarE Em varref Bvar

n V lC CT

gC C T

gC C T

n IV CT

α

α

α

α

− −

− −

− −

−

−

−

≈ − + + ×Δ

− ×Δ

⎛ ⎞+ − ×Δ⎜ ⎟⎝ ⎠

⎡= − + + ×Δ

l

( ) 0 02 10

2 1n

Bref

BrefE Emref Bvar

l

n V lCT −

⎤⎢ ⎥⎣ ⎦

= − +

(Continued)

71

(H-58) Concluded

( )

( )( )( )

00 0 0 022 12 2 1 2 1 2

0 02 10

02 2

00

002 2 2 2

2

10

12

2 1

16

nnn n n n

n

n

n n


BrefE Emref Bvar

BrefBvarEvarBvarE m

mrefBvar E EE

mvarBvar Bvar avg

gVR R C GT T

n V lCT

lC

TV TgC C T

υ

α

α

−− −

−

−

− −

−

= + + +Δ

= − +

⎧ ⎫×Δ⎪

⎪+ + ⎨ ⎬⎪− ×Δ⎪⎩

mT

( )( )

( ) ( )( )

002 1

02 1

0 0 02 1 2 10

2

2

21

0

1 1 12 3 21 1

2 6

2

16

n

n

n n

n

m

BrefBvarvar




T

lT gIC T T

gI C T

n V l lC CT

αα

α

α

−

−

− −

−

−

⎪⎪

⎪⎪⎭

⎧ ⎫×Δ⎪ ⎪

⎪ ⎪⎡ ⎤+ + × +Δ ⎨ ⎬⎢ ⎥⎣ ⎦ ⎡ ⎤⎛ ⎞⎪ ⎪− − + ×Δ⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭

≈ + + ×Δ

− ( )( )( )( )

( )( )( )

002 2 2

002 1 2 1

0 002 1 2 1

0 02 10

120

120

12 20

16

1 1 1 12 2 6 2

2

n

n n

n n

n



Bvar E E EEm mvarBvar Bvar avg avg

Bvar BreE Em varref Bvar

gC C T

gC C T

g gC C T T

n IV CT

α

α

α

α

−

− −

− −

−

−

−

−

×Δ

− ×Δ

⎛ ⎞+ + × +Δ⎜ ⎟⎝ ⎠

⎡ ⎤= + + ×Δ⎢ ⎥⎣ ⎦0 0

202

n

f

BrefE Emref Bvar

l

n V lCT= +

72

( )

( )( )

00 0 0 0121 2 2 1

0 0 0 02 30 0

02

2

For 0 :12

2

1'

41 12 3


BrefBref BvarE E E E

m m varref Bvar ref Bvarm

Bvar Bvarvar var

BvarEvarBvar

n

gVR R C GT T

lV l VC CT TT

IC

υ

α

α αα

−−− − − −

− −

−

=

= + + +Δ

= − + + + ×Δ

− − ×Δ Δ⎡ ⎤+ + ×Δ⎢ ⎥⎣ ⎦

m

m

T

T

( )( ) 02

101 1

2 6

Bref

mm


lT

TgI C Tα

−

−

⎧ ⎫⎡ ⎤⎪ ⎪⎢ ⎥⎣ ⎦⎪ ⎪⎨ ⎬

⎡ ⎤⎛ ⎞⎪ ⎪− − − ×Δ⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭

0 212

Emavgg T+ (H-59)

( )( )( )

( ) ( )( )

0 0 02 20

002 2

00 02 2 2

00

120

120

16

1 1 1'

8 4 12



Bvar Bvar Bref Bvar E EE Emvar var varBvar Bvar Bvar avg

Emref

V l lC CT

gC C T

glC C C T

V T

α

α

α α α

− −

− −

− − −

−

−

≈ − + + ×Δ

− ×Δ

⎛ ⎞⎡ ⎤− − × + − ×Δ Δ Δ⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠

= − ( ) ( )( )

( )

0 0 02 2 2

0 0 01 10

0 010

1'

81

'8

1'

8

Bref Bvar Bref Bvar Bvar BrefE E Evar var varBvar Bvar Bvar

Bref Bvar Bvar BrefE E Em var varref Bvar Bvar

Bvar BvarE Em var varref Bvar

l lC C C

V l lC CT

IV CT

α α α

α α

α α

− − −

− −

−

⎡ ⎤+ + × − − ×Δ Δ Δ⎢ ⎥⎣ ⎦

⎡ ⎤≈ − + − − ×Δ Δ⎢ ⎥⎣ ⎦

= − + − − ×Δ Δ

l

Brefl⎧ ⎫⎡ ⎤⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭ (Continued)

73

(H-59) Concluded

( )

( )

00 0 0 0010 1 1 0

0 010

0 0 02 20

212

1'

8

1 15 '4 6



BrefBvar BvarE E E

m var var vref Bvar Bvarm

gVR R C GT T T

IV lCT

lV C CTT

υ

α α

α α

−− −

−

− −

= + + +Δ

⎧ ⎫⎡ ⎤= − + − − ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

⎡ ⎤+ + − × −Δ Δ⎢ ⎥⎣ ⎦ ( )( )

( )( )

( ) ( )

02

001

01

01

10

21

0

3'

41 1 12 3 21 1

2 6

18

Bvar E Emar Bvar avg



Bvar E Emvar Bvar

BvaEvarBvar

gC T

lT gIC T T

gI C T

IC

α

α αα

α

−

−

−

−

−

−

⎧ ⎫⎪ ⎪×Δ⎨ ⎬⎪ ⎪⎩ ⎭

⎧ ⎫⎡ ⎤− ×Δ Δ⎪ ⎪⎢ ⎥⎣ ⎦⎪ ⎪⎡ ⎤+ + × +Δ ⎨ ⎬⎢ ⎥⎣ ⎦ ⎡ ⎤⎛ ⎞⎪ ⎪− − + ×Δ⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭

≈ − ( )( ) ( )( )( )( ) ( )

00 01 1 1

00 01 11

10

10

'

1 15 '4 61 3

'6 8

r Bvar Brefvar


Bvar Bvar Bvar BrefE EE Emvar var varBvar BvarBvar

l

glC C C T

g lC CC T

α α

α α α

α α α

− − −

− −−

−

−

⎧ ⎫⎡ ⎤− ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

⎡ ⎤+ − × − ×Δ Δ Δ⎢ ⎥⎣ ⎦

⎡ ⎤− × + − ×Δ Δ Δ⎢ ⎥⎣ ⎦

( )( )( )

001 1

01

00

1201 1 1

2 2 6Bvar E EE

mvarBvar Bvar avg

Bvar BrefEvarBvar

BrefEBvar

gC C T

I lC

lC

α

α

− −

−

−⎛ ⎞+ + ×Δ⎜ ⎟⎝ ⎠

⎡ ⎤= + ×Δ⎢ ⎥⎣ ⎦

=

74

( )

( )( )

00 0 0 0101 0 0 1

0 0 00 10

00

2

For 1:12

1'

23

'41 1

'2 3


Bref Bvar Bvar BrefE E Em var varref Bvar Bvar

Bvar Bvarvar var

BvarEvarBvar

n

gVR R C GT T T

V l lC CT

IC

υ

α α

α αα

−

=

= + + +Δ

⎡ ⎤= + + + ×Δ Δ⎢ ⎥⎣ ⎦

⎡ ⎤− ×Δ Δ⎢⎣ ⎦⎡ ⎤+ + ×Δ⎢ ⎥⎣ ⎦ ( )( )0

00

21

20

11 1 2'2 6

Bref

EmavgBvar E E

mvar Bvar

lg T

gI C Tα−

⎧ ⎫⎪ ⎪⎥⎪ ⎪ +⎨ ⎬⎡ ⎤⎛ ⎞⎪ ⎪− − − ×Δ⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭

( )0 0 00 00

1'

2Bref Bvar Bvar BrefE E E

m var varref Bvar BvarV lC CT α α⎡≈ + + + ×Δ Δ⎢⎣l⎤⎥⎦

(H-60)

( )( )( ) ( )( )

( )

000 0

00 00 0 2 2

0 0 00 00

120

120

1'

63 1 1 1

' '8 2 2 6

1'

8

n



Bvar Bref BvarE E Em var varref Bvar Bvar

gC C T

glC C T

IV lC CT

α

α α α

α

−

−

−

− ×Δ

⎛ ⎞⎡ ⎤+ − × + − ×Δ Δ Δ⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠

⎡ ⎤= + + × −Δ⎢ ⎥⎣ ⎦

C

( )( )0 0

10

'

1'

8

Bvar Brefvar


l

IV lCT

α α

α α

⎡ ⎤− ×Δ Δ⎢ ⎥⎣ ⎦

⎧ ⎫⎡ ⎤≈ + − − ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

(Continued)

75

(H-60) Concluded

( )( )

00 0 0 0212 1 1 2

0 0 010 0

00

00

212

1'

81 5 '41

'6




BvarEvarBvar

gVR R C GT T T

IV l VCT T

lC

C

υ

α α

α α

α

= + + +Δ

⎧ ⎫⎡ ⎤= + − − × +Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

⎡ ⎤− ×Δ Δ⎢ ⎥⎣ ⎦+

− Δ( )( )

( )( )

( )( )

00

001 0

1

0 010

120

21

20

1'

41 1 1'

1 12 3 2'2 6

128

E EmBvar

Bvar Bvar Brefvar var

Bvar EEmvarBvar avgBvar E E

mvar Bvar avg

E Em varef Bvar

gC T

lgIC T

gI C T

IV CT

α αα

α

−

−

⎧ ⎫⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪×⎪ ⎪⎩ ⎭

⎧ ⎫⎡ ⎤− − ×Δ Δ⎪ ⎪⎢ ⎥⎣ ⎦⎪ ⎪⎡ ⎤+ + × +Δ ⎨ ⎬⎢ ⎥ ⎡ ⎤⎛ ⎞⎣ ⎦ ⎪ ⎪− − + ×Δ⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭

≈ + − ( ) ( )( )( ) ( )( )

( )

01

0 00 01 11 1

01

1 12 20 0

1 5' '4

1 1' '

6 61

'8

Bvar Bvar Bref Bvar Bvar BrefEr var var varBvar

Bvar BvarE E E EE Em mvar varBvar BvarBvar Bvaravg avg


l lC

g gC CC CT T

C

α α α α

α α

α α

− −

⎧ ⎫⎡ ⎤ ⎡− × + − ×Δ Δ Δ Δ⎨ ⎬⎢ ⎥ ⎢⎣ ⎦ ⎣⎩ ⎭

− × − ×Δ Δ

⎡ ⎤− − ×Δ Δ⎢ ⎥⎣ ⎦

⎤⎥⎦

( )( )( )

001 1

0 010

0 020

1201 1 1

'2 2 6

2 '

2

Bref



BrefE Emref Bvar

l

gC C T

IV lCT

V lCT

α

α

−⎛ ⎞+ + ×Δ⎜ ⎟⎝ ⎠

⎡ ⎤= + + ×Δ⎢ ⎥⎣ ⎦

= +

76

( ) ( )

( )

00 0 0 02 12 22 1 2 2 2 2 2 1

0 0 0 02 2 30 0

02 2

2

2

For 1:12

2 2 '

1 1'

2 3

nnn n n n

n n

n



Bvavar

BvarEvarBvar

n

gVR R C GT T

n V l V lC CT T

IC

υ

α

α

−−− − − −

− −

−

>

= + + +Δ

= − + + + ×Δ

⎡ ⎤+ + ×Δ⎢ ⎥⎣ ⎦

mT

( )( ) ( )

( ) ( )( )

0

02 2

0 0 02 2 20

0 02 10

21

20

'1

1 1 2'2 6

2 1 '

2 1

n

n n

n

r Bref

EmavgBvar E E

mvar Bvar


BrefE Emref Bvar

lg T

gI C T

n V l lC CT

n V lCT

α

α

α

−

− −

−

−

⎧ ⎫×Δ⎪ ⎪⎪ ⎪ +⎨ ⎬⎡ ⎤⎛ ⎞⎪ ⎪− − − ×Δ⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭

≈ − + + ×Δ

= − +

(H-61)

( ) ( )( )( )

00 0 0 022 12 2 1 2 1 2

0 0 0 02 1 2 20 0

002 2 2 2

2 1

2

120

12

2 1 '

1'

6

nnn n n n

n n

n n

n




Bvar

gVR R C GT T

n V l V lC CT T

gC C T

υ

α

α

−−

− −

− −

−

−

= + + +Δ

= − + + + ×Δ

− ×Δ

+

mT

( )( )

( ) ( )( )

00

02 1

0 0 02 1 2 10

02 1

2

21

20

'1 1 1

'1 12 3 2'2 6

2 '

16

n

n n

n

Bvar Brefvar

Bvar EEmvar avgBvar E E

mvar Bvar avg


BvarEvarBvar

lgIC T

gI C T

n V l lC CT

C

αα

α

α

−

− −

−

−

⎧ ⎫×Δ⎪ ⎪⎪ ⎪⎡ ⎤+ × +Δ ⎨ ⎬⎢ ⎥ ⎡ ⎤⎛ ⎞⎣ ⎦ ⎪ ⎪− − + ×Δ⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭

≈ + + ×Δ

− ( )( )( )( )

( )( )( )

02 1

002 1 2 1

002 1 2 1

0 0 02 1 2 10

00

120

120

120

'

1'

61 1 1

'2 2 6

2 '

2

n

n n

n n

n n

E EmBvar avg




Emref

gC T

gC C T

gC C T

n V l lC CT

n V T

α

α

α

α

−

− −

− −

− −

−

−

−

×Δ

− ×Δ

⎛ ⎞+ + ×Δ⎜ ⎟⎝ ⎠

= + + ×Δ

= 02n

BrefEBvar lC+

Summary of (H-56) of (H-61) Position Results:

77

( )

( )( )( )

0 0 0 0 0 09 92 1 20 0

0 0 099 0

009 9

120

For 4 :

2 1 2

For 4 :194

1 1 12 4 3

n nE E E E E EBref Bref

m mref Bvar ref Bvarvar var

Bvar BrefE E Em varref Bvarvar


n

n nV VR RC l C lT T

n

IV lR CT

gC C T

α

α

− −−

−−

− −

−

< −

= − + = +

= −

⎡ ⎤= − + + ×Δ⎢ ⎥⎣ ⎦

⎛ ⎞+ − ×Δ⎜ ⎟⎝ ⎠

0 0 088 0

8

For 0 4 :

BrefE E Emref Bvarvar V lR CT

n−−

= − +

> > −

( )0 0 02 12 1 0

2 1nn

BrefE E Emref Bvarvar n VR CT −−

= − + l (H-62)

( )

( )

0 0 022 0

0 0 0 0 01 01 00

0 0 011 0

2

For 0 :1

'8

For 1:1

'8

nnBrefE E E

mref Bvarvar

Bvar Bvar Bref BrefE E E E Em var varref Bvar Bvarvar var

Bvar BvarE E Em var varref Bvarvar

n V lR CT

n

IV lR RC CT

n

IVR CT

α α

α α

−−

= +

=

⎧ ⎫⎡ ⎤= − + − − × =Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭=

⎡ ⎤= + − − ×Δ Δ⎢ ⎥⎣ ⎦

l

( )

0 0 022 0

0 0 0 0 0 02 1 22 1 20 0

2

For 1:

2 1 2n nn n

Bref

BrefE E Emref Bvarvar

Bref BrefE E E E E Em mref Bvar ref Bvarvar var

l

V lR CT

n

n nV l VR RC CT T−−

⎧ ⎫⎨ ⎬⎩ ⎭

= +

>

= − + = + l

Recognizing that the inverse of a direction cosine matrix equals its transpose, the


( ) 10

nEBvarC

−

0nBvarEC

REFERENCES

[1] Savage, Paul G., “Generating Strapdown Specific-Force/Angular-Rate For Specified Attitude/ Position Variation From A Reference Trajectory”, SAI WBN-14026, www.strapdownassociates.com, April 21, 2020. http://www.strapdownassociates.com/Variation%20Trajectory%20Generator.pdf

78

http://www.strapdownassociates.com/Variation%20Trajectory%20Generator.pdf

Documents

Strapdown Associates Inc., Paul G Savage, Books, Strapdown …strapdownassociates.com/Appendices F, G, H to Variation... · 2020-06-24 · CV CV() ×= ×C−1 ⎡⎣()⎤⎦ (F-2)