Ece Formula Sheet Final

8/11/2019 Ece Formula Sheet Final

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One stop series for

GATE/IES/PSU-2014

(Formula Sheet for ECE Dept.,)


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Formula Sheet(ECE Department)

Index Page. No.

1. Mathematics --- --- 1 to 9

2. EDC & Analog Electronics --- 10 to 18

3. Digital Electronics --- --- 19 to 21

4. Microprocessors --- --- 22 to 25

5. Signals & Systems --- --- 26 to 31

6. Control Systems --- --- 32 to 34

7. Communications --- --- 35 to 38

8. Electro-Magnetic Theory --- 39 to 45

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Institute of Engineering Studies (IES,Bangalore) Mathematics Formula SheetMatheMatics

Matrix :- If |A| = 0

Singular matrix ; |A|

0 Non singular matrix

Scalar Matrixis a Diagonal matrix with all diagonal elements are equal Unitary Matrixis a scalar matrix with Diagonal element as 1 (AQ= (A)T= A) If the product of 2 matrices are zero matrix then at least one of the matrix has det zero Orthogonal Matrix if AAT= AT.A = I AT= A A = AT Symmetric

A = - ATSkew symmetricProperties :- (if A & B are symmetrical )

A + B symmetric KA is symmetric AB + BA symmetric AB is symmetric iff AB = BA

For any A A + AT symmetric ; A - AT skew symmetric. Diagonal elements of skew symmetric matrix are zero If A skew symmetric Asymmetric matrix ; Askew symmetric If A is null matrix then Rank of A = 0.

Consistency of Equations :-

r(A, B) r(A) is consistent r(A, B) = r(A) consistent &

if r(A) = no. of unknowns then unique solutionr(A) < no. of unknowns then solutions .

Hermition , Skew Hermition , Unitary & Orthogonal Matrices :-

AT= A then Hermition AT= A then Hermition Diagonal elements of Skew Hermition Matrix must be purely imaginary or zero Diagonal elements of Hermition matrix always real . A real Hermition matrix is a symmetric matrix. |KA| = K|A|Eigen Values & Vectors :- Char. Equation |A I| = 0.Roots of characteristic equation are called eigen values . Each eigen value corresponds to non zero

solution X such that (A I)X = 0 . X is called Eigen vector . Sum of Eigen values is sum of Diagonal elements (trace) Product of Eigen values equal to Determinent of Matrix . Eigen values of AT& A are same is Eigen value of A then 1/ A & || is Eigen value of adj A. , are Eigen values of A then

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Institute of Engineering Studies (IES,Bangalore) Mathematics Formula Sheet

KA K , K ..K A

,

..

.

A + KI + k , + k , .. + k(A KI) ( k), ( k) Eigen values of orthogonal matrix have absolute value of 1 . Eigen values of symmetric matrix also purely real . Eigen values of skew symmetric matrix are purely imaginary or zero . , , distinct eigen values of A then corresponding eigen vectors X, X, .. Xfor

linearly independent set .

adj (adj A) = |A| ; | adj (adj A) | = |A|()Complex Algebra :-

Cauchy Rieman equations

=; =Neccessary & Sufficient Conditions for f(z) to be analytic

f(z)/(Z a)+ dz = ! [ f(a) ] if f(z) is analytic in region C & Z =a is single point f(z) = f(z0) + f(z0) ()! + f(z0) ()! + + f(z0) ()! + . Taylor Series

if z0= 0 then it is called Mclauren Series f(z) = a(z z0)0 ; when a= (

)

! If f(z) analytic in closed curve C except @ finite no. of poles then f(z)dz = 2i (sum of Residues @ singular points within C )

Res f(a) = lim( ()= (a) / (a)=

lim

()! ((Z

a)f(z) )

Calculus :-

Rolles theorem :-

If f(x) is

(a) Continuous in [a, b]




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Institute of Engineering Studies (IES,Bangalore) Mathematics Formula Sheet(b) Differentiable in (a, b)

(c) f(a) = f(b) then there exists at least one value C (a, b) such that f(c)= 0 .Langranges Mean Value Theorem :-

If f(x) is continuous in [a, b] and differentiable in (a, b) then there exists atleast one value C in (a, b)

such that f(c) = (b)()b Cauchys Mean value theorem :-

If f(x) & g(x) are two function such that

(a) f(x) & g(x) continuous in [a, b]

(b) f(x) & g(x) differentiable in (a, b)

(c) g(x) 0 x in (a, b)Then there exist atleast one value C in (a, b) such that

f(c)/ g(c) = (b)()(b)()Properties of Definite integrals :-

a < c < b

f(x)

. dxb =

f(x)

. dx +

f(x)

. dxb

f(x)dx0 = f(a x)dx0 f(x). dx = 2 f(x)dx0 f(x) is even

= 0 f(x) is odd

f(x). dx0 = 2 f(x)dx0 if f(x) = f(2a- x) = 0 if f(x) = - f(2a x)

f(x). dx0 = n f(x)dx0 if f(x) = f(x + a) f(x). dxb = f(a + b x). dxb




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(+) = t sin at

(

+)

=

[ sin at + at cos at]

(+) = [ sin at - at cos at]

= Cos hat

= Sin hat

Laplace Transform of periodic function : L { f(t) } = ()

Numerical Methods :-

Bisection Method :-

(1) Take two values of x& xsuch that f(x) is +ve & f(x) is ve then x3= + find f(x3) if f(x3)+ve then root lies between x3& xotherwise it lies between x& x3.Regular falsi method :-

Same as bisection except x= x0- ()() f(x0)Newton Raphson Method :-

x+= x ()()Pi cards Method :-

y+= y0+ f(x,y ) = f(x, y)Taylor Series method :-

= f(x, y) y = y

0+ (x- x

0) (y

)

0 +

(

)

!

(y)

0+ .

(

)

!

(y)

0

Eulers method :-

y= y0+ h f(x0, y0) = f(x, yy()= y0+ [f(x0, y0) + f(x0 + h, y)




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y()= y0+ [f(x0, y0) + f(x0+, y()) ]:

:

Calculate till two consecutive value of y agree

y= y+ h f(x0 + h, y)y()= y0+ [f(x0 + h, y) + f(x0 + 2h, y)

Runges Method :-k= h f(x0, y0)k= h f(x0 +, y0 +k ) finally compute K = 6(K+ 4K+ K3)k= h f(x0+h , y0+ k)k3= h ( f (x0+h , y0+ k))

Runge Kutta Method :-k= h f(x0, y0)k= h f(x0 +, y0 +k ) finally compute K = 6(K+ 2K+ 2K3 + K4)k3= h f(x0 +, y0 +k ) approximation vale y = y0+ K .k3= h f (x0+h , y0+ k3)

Trapezoidal Rule :-

f(x). dx+ = [ ( y0+ y) + 2 (y+ y+ . y)]f(x) takes values y0, y..@ x0, x, x..

Simpsons one third rule :-




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Institute of Engineering Studies (IES,Bangalore) Mathematics Formula Sheet f(x). dx+ = 3[ ( y0+ y) + 4 (y+ y3+ . y) + 2 (y + y4 + . + y)]Simpson three eighth rule :-

f(x). dx+ = 38 [ ( y0+ y) + 3 (y+ y+ y4+ y5+ . y)+ 2 (y3 + y6 + . + y3) ]Differential Equations :-

Variable & Seperable :-

General form is f(y) dy = (x) dxSol: f(y) dy = (x) dx + C .

Homo generous equations :-

General form = (,)(,) f(x, y) & (x, y)Homogenous of same degree

Sol : Put y = Vx = V + x & solveReducible to Homogeneous :-

General form= +b++b+

(i) bbSol : Put x = X + h y = Y + k

= +b+(+bk+)+b+(+bk+) Choose h, k such that becomes homogenous then solve by Y = VX(ii)

= bbSol : Let = bb= = +b+(+b)+Put ax + by = t = /b




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Institute of Engineering Studies (IES,Bangalore) Mathematics Formula SheetThen by variable & seperable solve the equation .

Libnetz Linear equation :-

General form +py = Q where P & Q are functions of xI.F = e p.

Sol : y(I.F) = Q. (I. F)dx + C .Exact Differential Equations :-

General form M dx + N dy = 0 M f (x, y)N f(x, y)

If My = Nx then

Sol : M. dx + (termsofN containing x ) dy = C( y constant )

Rules for finding Particular Integral :-

(

D)

e =

(

)

e

= x ()e if f (a) = 0= x ()e if f(a) = 0(b)sin (ax + b) = () sin (ax + b) f(- a) 0

= x() sin (ax + b) f(- a) = 0 Same applicable for cos (ax + b)

= x

(

) sin (ax + b)

(D)x= [f(D)]x(D) e f(x) = e (D+) f(x)Vector Calculus :-




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Institute of Engineering Studies (IES,Bangalore) Mathematics Formula SheetGreens Theorem :-

(dx + dy)C = x ydx dyThis theorem converts a line integral around a closed curve into Double integral which is special case ofStokes theorem .

Series expansion :-

Taylor Series :-

f(x) = f(a) +()! (x-a) + ()! (x a)+ + ()! (x a)

f(x) = f(0) +(0)! x + (0)! x+ + (0)! x + . (mc lower series )

(1 + x)= 1+ nx + () x+ | nx| < 1e= 1 + x + !+ ..Sin x = x -

3!+ 5! - ..Cos x = 1 -

!+ 4! - ..




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Institute of Engineering Studies (IES,Bangalore) EDC & Analog Electronics Formula SheetEDC & Analog

Energy gap/=1.213.6 10.T ev/=0.7852.23 10.T evEnergy gap depending on temperature

EF= E- KT ln= Ev+ KT ln No. of electrons n = Nce(EE)/T (KT in ev) No. of holes p = Nve(EE)/T Mass action law np= ni2= NcNveEG/KT Drift velocity d= E (for si d107cm/sec) Hall voltage H= B.Iw. Hall coefficient RH= 1/. charge density =qN0= ne Conductivity = ; = RH. Max value of electric field @ junction E0= -qNd. nn0= -qNA. np0. Charge storage @ junction Q+= - Q= qA xn0ND= qA xp0NA

EDC

Diffusion current densities Jp= - q Dpdpdx Jn= - q Dndndx Drift current Densities = q(p p+ nn)E p, ndecrease with increasing doping concentration .

D= D= KT/q 25 mv @ 300 K Carrier concentration in N-type silicon nn0= ND; pn0= ni2/ ND Carrier concentration in P-type silicon pp0= NA; np0= ni2/ NA Junction built in voltage V0= VTln

Width of Depletion region Wdep= xp+ xn= 2q 1N + 1N (V0 + V)* 2 = 12.93

xx= NN Charge stored in depletion region qJ= q.NNN+N . A . Wdep Depletion capacitance Cj= AW; Cj0= AW/ V=0

Cj= Cj0/1 + VVmC

j= 2C

j0 (for forward Bias)

Forward current I = Ip + In; Ip= Aq ni2 DLN/ 1In= Aq ni2 DLN/ 1

Saturation Current Is= Aq ni2 DLN + DLN Minority carrier life time p= Lp2 / Dp ; n= Ln2 / Dn




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Institute of Engineering Studies (IES,Bangalore) EDC & Analog Electronics Formula Sheet Minority carrier charge storage Qp= pIp, Qn= pIn

Q = Qp+ Qn= TI T= mean transist time Diffusion capacitance Cd=

I = .g CdI.

carrier life time , g = conductance = I / I02= 2(TT)/10I01 Junction Barrier Voltage Vj= VB= Vr(open condition)

= Vr- V (forward Bias)= Vr+ V (Reverse Bias)

Probability of filled states above E f(E) =11+e()/

Drift velocity of e d107cm/sec Poisson equation

dVdx= = nq dvdx= E = nqx Transistor :- I

E= I

DE+ I

nE

I= Io IE Active region I= IE+ Io(1- eV/V)Common Emitter :-

I= (1+ ) Io+ IB 1 IEO= I1 Collector current when baseopen IBOCollector current when IE= 0 IBO> Io. VBE,sat or VB,sat - 2.5 mv /0C ; VE,satV,10 = - 0.25 mv /0C Large signal Current gain = III+I

D.C current gain dc= II= hFE (dc= hFE) when IB> IBo Small signal current gain = ICIRV = hfe= h1(I+I)hFEIC Over drive factor =

undersaturation Isat= forcedIBsatConversion formula :-

CC CE

hic= hie; hrc= 1 ; hfc= - (1+ hfe) ; hoc= hoeCB CE

hib= h1+h; hib= hh1+h - hre; hfb= h1+h ; hob= h1+hCE parameters in terms of CB can be obtained by interchanging B & E .

Specifications of An amplifier :-




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Institute of Engineering Studies (IES,Bangalore) EDC & Analog Electronics Formula Sheet

AI= h1+hZ Zi= hi+ hrAIZL Avs= A.ZZ+= A.ZZ+= A .Z A

V=

A ZZ

Y

0= h

o-

hhh+

A

Is=

A.Z+

=A.

Z

Choice of Transistor Configuration :-

For intermediate stages CC cant be used as AV< 1 CE can be used as intermediate stage CC can be used as o/p stage as it has low o/p impedance CC/CB can be used as i/p stage because of i/p considerations.

Stability & Biasing :- ( Should be as min as possible)

For S =IIV, S= IVI, S= I V,

I= S. Io + SVBE+ S For fixed bias S =

1+1 = 1 + Collector to Base bias S =

1+1+ 0 < s < 1+ = 1+1+ Self bias S =

1+1+ 1+ RE > 10 R2

R1= V

V ; R2= V

VV For thermal stability [ Vcc- 2Ic(R+ RE)] [ 0.07 Ico. S] < 1/ ; VE< V2

Hybrid pi()- Model :-

gm= |I| / VTr

be = h

fe/ g

m

rbb= hie- rberbc= rbe/ hregce= hoe- (1+ hfe) gbcFor CE :-

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fT= hfef ; fH= 12r=g2 C = Ce+ Cc(1 + gmRL)

fT= S.C current gain Bandwidth productfH= Upper cutoff frequency

For CC :-

fH= 1+g2 g2= f = g+g2(+)For CB:-

f= 1+h2r(+) = (1 + hfe) f = (1 + ) f fT= 1+f f> fT > f

Ebress moll model :-

I= - NIE+ Io(1- eV/

V)IE= - II+ IEo(1- eV/V)

IIo= NIEo

Multistage Amplifiers :-

fH* = fH21/n 1 ; fL= f2/1 Rise time t

r=

0.35

f=

0.35

B.

W

tr= 1.1 tr12 + tr22 + fL= 1.1 fL2 + fL2 +

1f = 1.1 1f + 1f + Differential Amplifier :-

Zi= hie+ (1 + hfe) 2Re= 2 hfeRe

2

Re

gm= |I|4V = I4V= gmof BJT/4 0DC value of CMRR =

h+h ; Re, Zi , Ad & CMRR Darlington Pair :-




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Institute of Engineering Studies (IES,Bangalore) EDC & Analog Electronics Formula Sheet AI= (1 + 1) (1 + 2) ; Av1 ( < 1) Zi= (1+h)1+hh [ if Q1& Q2have same type ] = AIRe2 Ro= (1+h)+ 2h1+h gm= (1 + 2) gm1

Tuned Amplifiers : (Parallel Resonant ckts used ) :

f0= 12L Q Q factor of resonant ckt which is very high B.W = f0/Q

fL= f0- BW2 fH= f0+ BW2 For double tuned amplifier 2 tank circuits with same f0used . f0= fLfH.MOSFET (Enhancement) [ Channel will be induced by applying voltage]

NMOSFET formed in p-substrate If VGSVtchannel will be induced & iD(Drain source ) Vt+ve for NMOS iD(VGS- Vt) for small VDS

VDS channel width @ drain reduces .VDS= VGS- Vt channel width 0 pinch off further increase no effect

For every VGS> Vtthere will be VDS,sat iD= Kn [ (VGS- Vt) VDS- 12VDS2 ] triode region ( VDS< VGS- Vt)

Kn = nCox i

D=

12K

n

[ V

DS2]

saturation

rDS= 1K (VV) Drain to source resistance in triode region

PMOS :-




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Institute of Engineering Studies (IES,Bangalore) EDC & Analog Electronics Formula Sheet Device operates in similar manner except VGS, VDS, Vtare ve iDenters @ source terminal & leaves through Drain .

V

GS

V

t

induced channel V

DS

V

GS- V

t

Continuous channel

iD= Kp [(VGS Vt)2- 12VDS2 ] Kp = pCoxVDSVGS- VtPinched off channel .

NMOS Devices can be made smaller & thus operate faster . Require low power supply . Saturation region Amplifier For switching operation Cutoff & triode regions are used

NMOS PMOS

V

GS

V

t V

GS

V

t

induced channel

VGS- VDS> Vt VGS- VDS< Vt Continuous channel(Triode region)VDSVGS- Vt VDSVGS- Vt Pinchoff (Saturation)

Depletion Type MOSFET :-[ channel is physically implanted . i0flows with VGS= 0 ] For n-channel VGS +ve enhances channel . -ve depletes channel iD- VDScharacteristics are same except that Vtis ve for n-channel

Value of Drain current obtained in saturation when VGS= 0 IDSS.IDSS= 12 Kn Vt2.MOSFET as Amplifier :-

For saturation VD> VGS- Vt To reduce non linear distortion gs< < 2(VGS- Vt) id= Kn (VGS Vt)gs gm= Kn (VGS Vt)

= - g

mR

D

Unity gain frequency fT= g2(+)JFET :-

VGSVp iD= 0 Cut offLeading coaching center for GATE/IES/PSU in Bangalore & All over India for Online Tests/PracticeBranches: Jayanagar & Malleshwaram Ph: 0 99003 99699 Email:[email protected]



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Institute of Engineering Studies (IES,Bangalore) EDC & Analog Electronics Formula Sheet Vp VGS0, VDSVGS- Vp

iD = IDSS 2 1 2 Triode

Vp VGS0 , VDS VGS- Vpi=I1V=V1 g= || 1= || SaturationZener Regulators :-

For satisfactory operationVV IZ + IL

R

S=

VVI rI+

I

Load regulation = - (rz|| Rs) Line Regulation =

r+r . For finding min RL take Vsmin& Vzk, Izk(knee values (min)) calculate according to that .

Operational Amplifier:- (VCVS)

Fabricated with VLSI by using epitaxial method High i/p impedance , Low o/p impedance , High gain , Bandwidth , slew rate .

FET is having high i/p impedance compared to op-amp . Gain Bandwidth product is constant .

Closed loop voltage gain AL= A1 A feed back factor V0= 1 Vi dt LPF acts as integrator ; V0 = L i dt ; V0 = L dvdt (HPF)

For Op-amp integrator V0 = 1 i dt ; Differentiator V0= - dvdt Slew rate SR =

Vt = Vt . Vt = A. Vt Max operating frequency fmax= slewrate2. V = slewrate2 VA.




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Iflows for more than 1800& less than 3600 Q located in active region but near to cutoff ; = 60% Distortion & Noise interference less compared to class B but more in compared to class A

Eliminates cross over Distortion

Class C operation :- Iflows for < 180 ; Q located just below cutoff ; = 87.5% Very rich in Distortion ; noise interference is high .

Oscillators :-

For RC-phase shift oscillator f =126+4K hfe4k + 23 +29k where k = Rc/R

f =126 > 29

For op-amp RC oscillator f = 126 | Af| 29 Rf29 R1Wein Bridge Oscillator :-

f =12 hfe33

A 3 Rf2 R1Hartley Oscillator :-

f = 12(L+L) |hfe| LL| | LL|A| LL

Colpits Oscillator :-

f =1

2L |h

fe|

| | | A|




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Institute of Engineering Studies (IES,Bangalore) Digital Electronics Formula Sheet

Digital Electronics

Fan out of a logic gate =II

orII

Noise margin : VOH- VIH or VOL- VIL Power Dissipation PD= VccIcc= VccI+I2 I Icwhen o/p low

I Icwhen o/p high . TTL , ECL & CMOS are used for MSI or SSI

Logic swing : VOH - VOL RTL , DTL , TTL saturated logic ECL Un saturated logic Advantages of Active pullup ; increased speed of operation , less power consumption .

For TTL floating i/p considered as logic 1 & for ECL it is logic 0 .

MOS mainly used for LSI & VLSI . fan out is too high

ECL is fastest gate & consumes more power .

CMOS is slowest gate & less power consumption

NMOS is faster than CMOS .

Gates with open collector o/p can be used for wired AND operation (TTL)

Gates with open emitter o/p can be used for wired OR operation (ECL)

ROM is nothing but combination of encoder & decoder . This is non volatile memory .

SRAM : stores binary information interms of voltage uses FF.

DRAM : infor stored in terms of charge on capacitor . Used Transistors & Capacitors .

SRAM consumes more power & faster than DRAM .

CCD , RAM are volatile memories .

1024 8 memory can be obtained by using 1024 2 memories

No. of memory ICs of capacity 1k 4 required to construct memory of capacity 8k 8 are 16

DAC ADC FSV = VR1 12 * LSB = Voltage range / 2

n

Resolution =stepsizeFSV =

V/2V1

=1

21 100% * Resolution =FSV21

Accuracy = 12LSB =

12 * Quantisation error =

V2 %

Analog o/p = K. digital o/p

PROM , PLA & PAL :-

AND OR

Fixed Programmable

Programmable fixed

Programmable Programmable

PROM

PAL

PLA




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Flash Type ADC : 2n1 comparators2n resistors2nn Encoder

Fastest ADC :-

Successive approximation ADC : n clk pulses

Counter type ADC : 2n- 1 clk pulses Dual slope integrating type : 2n+1clock pulses .

Flip Flops :-

a(n+1) = S + RQ= D

= JQ+ KQ= TQ+ TQ

Excitation tables :-

For ring counter totalno.of states = n

For twisted Ring counter = 2n (Johnson counter / switch tail Ring counter ) . To eliminate race around condition tpdclock < < tpdFF. In Master slave master is level triggered & slave is edge triggered

Combinational Circuits :-

Multiplexer :-

0

0

00

0

1

x

1 1

0

1

R

1 0

1

S

x

0

1

0

00 0

1 0

1

0

1

D

0

1

0

1

1 0

1

01

1 0

1

0

1

0

J K

0

1

x 1

0x

x

x 1

0

1

T

0

1

1

0




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2ni/ps ; 1 o/p & n select lines. It can be used to implement Boolean function by selecting select lines as Boolean variables

For implementing n variable Boolean function 2n1 MUX is enough . For implementing n + 1 variable Boolean 2n1 MUX + NOT gate is required .

For implementing n + 2 variable Boolean function 2n1 MUX + Combinational Ckt isrequired

If you want to design 2m 1 MUX using 2n1 MUX . You need 2mn 2n1 MUXes

Decoder :-

n i/p & 2no/ps used to implement the Boolean function . It will generate required min terms @ o/p & those terms

should be OR ed to get the result .

Suppose it consists of more min terms then connect the max terms to NOR gate then it will give the

same o/p with less no. of gates .

If you want to Design m 2mDecoder using n 2nDecoder . Then no. of n 2nDecoder

required = 22 .

In Parallel (n bit ) total time delay = 2ntpd. For carry look ahead adder delay = 2 tpd.




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Institute of Engineering Studies (IES,Bangalore) Microprocessors Formula Sheet

Microprocessors

Clock frequency =1

2crystal frequency

Hardware interruptsTRAP (RST 4.5) 0024H both edge level

RST 7.5 Edge triggered 003CH

RST 6.5 0034 HRST 5.5 level triggered 002C

INTR Non vectored

Software interrupts RST 0 0000HRST 1 0008H

2 0010H Vectored: 0018H

:7 0038H

HOLD & HLDA used for Direct Memory Access . Which has highest priority over all interrupts .

Flag Registers :-

Sign flag :- After arthematic operation MSB is resolved for sign flag . S = 1 -ve result

If Z = 1 Result = 0

AC : Carry from one stage to other stage is there then AC = 1

P : P =1 even no. of ones in result .

CY : if arthematic operation Results in carry then CY = 1

For INX & DCX no flags effected In memory mapped I/O ; I/O Devices are treated as memory locations . You can connect max of

65536 devices in this technique .

In I/O mapped I/O , I/O devices are identified by separate 8-bit address . same address can be used

to identify i/p & o/p device .

Max of 256 i/p & 256 o/p devices can be connected .

1

Halt

write

Read

fetch

S 01 S

0

1

0

1

0

1

0

ACS X P CYXZ X




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Programmable Interfacing Devices :-

8155 programmable peripheral Interface with 256 bytes RAM & 16-bit counter 8255 Programmable Interface adaptor

8253 Programmable Interval timer

8251 programmable Communication interfacing Device (USART)

8257 Programmable DMA controller (4 channel)

8259 Programmable Interrupt controller

8272 Programmable floppy Disk controller

CRT controller

Key board & Display interfacing Device

RLC :- Each bit shifted to adjacent left position . D7becomes D0.

CY flag modified according to D7

RAL :- Each bit shifted to adjacent left position . D7becomes CY & CY becomes D0.

ROC :-CY flag modified according D0

RAR :- D0becomes CY & CY becomes D7

CALL & RET Vs PUSH & POP :-

CALL & RET PUSH & POP

When CALL executes , p automatically stores * Programmer use PUSH to save the contents16 bit address of instruction next to CALL on the rp on stackStack

CALL executed , SP decremented by 2 * PUSH executes SP decremented by 2 .

RET transfers contents of top 2 of SP to PC * same here but to specific rp .

RET executes SP incremented by 2 * same here

Some Instruction Set information :-

CALL Instruction

CALL 18T states SRRWW

CC Call on carry 9 18 states

CM Call on minus 9-18

CNC Call on no carry




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Institute of Engineering Studies (IES,Bangalore) Microprocessors Formula SheetCZ Call on Zero ; CNZ call on non zero

CP Call on +ve

CPE Call on even parity

CPO Call on odd parity

RET : - 10 T

RC : - 6/ 12 T states

Jump Instructions :-

JMP 10 T

JC Jump on Carry 7/10 T states

JNC Jump on no carry

JZ Jump on zero

JNZ Jump on non zero

JP Jump on Positive

JM Jump on Minus

JPE Jump on even parity

JPO Jump on odd parity.

PCHL : Move HL to PC 6T

PUSH : 12 T ; POP : 10 T

SHLD : address : store HL directly to address 16 T

SPHL : Move HL to SP 6T

STAX : Rp store A in memory 7T

STC : set carry 4T

XCHG : exchange DE with HL 4T

XTHL :- Exchange stack with HL 16 T

For AND operation AY flag will be set & CY Reset

For CMP if A < Reg/mem : CY 1 & Z 0 (Nothing but A-B)

A > Reg/mem : CY 0 & Z 0

A = Reg/mem : Z1 & CY 0 .




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DAD Add HL + RP (10T) fetching , busidle , busidle

DCX , INX wont effect any flags . (6T)

DCR, INR effects all flags except carry flag . Cy wont be modified

LHLD load HL pair directly

RST 12T states SPHL , RZ, RNZ ., PUSH, PCHL, INX , DCX, CALL fetching has 6T states

PUSH 12 T ; POP 10T




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Institute of Engineering Studies (IES,Bangalore) Signals & Systems Formula SheetLaplace Transform :-

x(t) =1j X(s)e+jj ds

X(s) = x(t)e dsInitial & Final value Theorems :x(t) = 0 for t < 0 ; x(t) doesnt contain any impulses /higher order singularities @ t =0 then

x( 0+) = lim ()x() = lim ()

Properties of ROC :-

1. X(s) ROC has strips parallel to jaxis2. For rational laplace transform ROC has no poles

3. x(t) finite duration & absolutely integrable then ROC entire s-plane4. x(t) Right sided then ROC right side of right most pole excluding pole s = 5. x(t) left sided ROC left side of left most pole excluding s= - 6. x(t) two sided ROC is a strip7. if x(t) causal ROC is right side of right most pole including s = 8. if x(t) stable ROC includes j-axisZ-transform :-

x[n] =1j x(z)z1dz

X(z) = x[n]= zInitial Value theorem :

If x[n] = 0 for n < 0 then x[0] = lim()Final Value theorem :-

lim[]= lim1( 1)X(z)Properties of ROC :-




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Institute of Engineering Studies (IES,Bangalore) Signals & Systems Formula Sheet

1.ROC is a ring or disc centered @ origin2. DTFT of x[n] converter if and only if ROC includes unit circle3. ROC cannot contain any poles

4. if x[n] is of finite duration then ROC is enter Z-plane except possibly 0 or 5. if x[n] right sided then ROC outside of outermost pole excluding z = 06. if x[n] left sided then ROC inside of innermost pole including z =07. if x[n] & sided then ROC is ring8. ROC must be connected region9.For causal LTI system ROC is outside of outer most pole including 10.For Anti Causal system ROC is inside of inner most pole including 011. System said to be stable if ROC includes unit circle .12. Stable & Causal if all poles inside unit circle13. Stable & Anti causal if all poles outside unit circle.

Phase Delay & Group Delay :-

When a modulated signal is fixed through a communication channel , there are two different delays to beconsidered.

(i) Phase delay:Signal fixed @ o/p lags the fixed signal by () phaseP= - () where () = K H(j)

Frequency response of channel

Group delay = d()d = for narrow Band signalSignal delay / Envelope delay

Probability & Random Process:-

P (A/B) = P(AB)P(B) Two events A & B said to be mutually exclusive /Disjoint if P(A B) =0Two events A & B said to be independent if P (A/B) = P(A) P(A B) = P(A) P(B)P(Ai / B) = P(AB)P(B) = P P(A) P P(A) CDF :-Cumulative Distribution function F

x(x) = P { X

x}

Properties of CDF :

Fx() = P { X } = 1 Fx(- ) = 0 Fx(x1X x) = Fx(x) - Fx(x1)




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Institute of Engineering Studies (IES,Bangalore) Signals & Systems Formula Sheet Its Non decreasing function P{ X > x} = 1 P { X x} = 1- Fx(x)PDF :-

Pdf = fx(x) = ddxFx(x)Pmf = fx(x) = P{X = x= } (x = x)

Properties:- fx(x) 0 Fx(x) = fx(x) * u(x) = fxx (x) dx Fx() = fx (x) dx =1 so, area under PDF = 1

P { x1< X x} = fx(x)dxxx Mean & Variance :-

Mean x= E {x} = x fx (x) dxVariance = E { (X x)} = E {x} - xE{g(x)} = g(x)fx (x)dxUniform Random Variables :

Random variable X ~u(a,b) if its pdf of form as shown below

fx(x) = 1 ; ;


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Institute of Engineering Studies (IES,Bangalore) Signals & Systems Formula Sheet

fx(x) = 1 e(x)/X ~N (1)Mean = x 1 e(x)/dx = Variance =

1 x e(x)/ dx = Exponential Distribution :-

fx(x) = exu(x)Fx(x) = ( 1- ex) u(x)Laplacian Distribution :-

fx(x) = e|x|Multiple Random Variables :-

FXY(x , y) = P { X x , Y y } FXY(x , ) = P { X x } = Fx(x) ; Fxy(, y) = P { Y < y } = FY(y) Fxy(-, y) = Fxy(x, - ) = Fxy(-, -) = 0 fx(x)= fxy (x, y)dy ; fY(y) = fxy (x, y) dx F

Y/

X

YX

x

=

P{Yy, Xx}P

{

X

x}

=F(x,y)

F(

x)

fY/X(y/x) = f(x,y)f(x)

Independence :- X & Y are said to be independent if FXY(x , y) = FX(x) FY(y)fXY(x, y) = fX(x) . fX(y) P { X x, Y y} = P { X x} . P{Y y}Correlation:

Corr{ XY} = E {XY} = fxy (x, y). xy. dx dyIf E { XY} = 0 then X & Y are orthogonal .

Uncorrelated :-Covariance = Cov {XY} = E { (X - x) (Y- y}

= E {xy} E {x} E{y}.If covariance = 0 E{xy} = E{x} E{y}




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Institute of Engineering Studies (IES,Bangalore) Signals & Systems Formula Sheet Independence uncorrelated but converse is not true.Random Process:-Take 2 random process X(t) & Y(t) and sampled @ t

1, t

X(t1), X(t) , Y(t1) , Y (t) random variablesAuto correlation Rx(t1, t) = E {X(t1) X(t) }Auto covariance Cx(t1, t) = E { X(t1) - x(t1)) (X(t) - x(t) } = Rx(t1, t) - x(t1) x(t)cross correlation Rxy(t1, t) = E { X(t1) Y(t) }cross covariance Cxy(t1, t) = E{ X(t1) - x(t1)) (Y(t) - y(t) } = Rxy(t1, t) - x(t1) y(t)CXY(t1, t) = 0 Rxy(t1, t) = x(t1) y(t) Un correlated RXY(t1, t) = 0 Orthogonal cross correlation 0 FXY(x, y ! t1, t) = Fx(x! t1) Fy(y ! t) independentProperties of Auto correlation :-

Rx(0) = E { x} Rx() = Rx(-) even | Rx() | Rx(0)

Cross Correlation

Rxy() = Ryx(-) Rxy () Rx(0) . Ry(0) 2 | R

xy(

)|

R

x(0) + R

y(0)

Power spectral Density :-

P.S.D Sx(j) = Rx ()ejdRx() = 1 (j)ejd

Sy(j) = Sx(j) |H(j)| Power = Rx(0) = 1 (j)d Rx() = k () white processProperties : Sx(j) even Sx(j) 0




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Institute of Engineering Studies (IES,Bangalore) Control Systems Formula SheetControl Systems

Time Response of 2ndorder system :-

Step i/P : C(t) = 1-

e (sin 1 2t tan

)

e(t) =e sin tan

ess= lime sin tan

Damping ratio ; Damping factor < 1(Under damped ) :

C(t) = 1- = e Sin tan = 0 (un damped) :

c(t) = 1- cos t= 1 ( ritically damped ) :

C(t) = 1 - e(1 + t)

> 1 (over damped) :

C(t) = 1 - e

2

T = Tuded> Tveded> Tudeded> Tld

Time Domain Specifications :

Rise time t= = tan

Peak time t= Max over shoot % M= e/100 Settling time ts= 3T 5% tolerance




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Institute of Engineering Studies (IES,Bangalore) Control Systems Formula Sheet= 4T 2% tolerance

Delay time td= +0.7 Dampingfactor2 2= (lM)

+

(lM)

Time period of oscillations T = 2 No of oscillations =

2/=

2

t1.5 td t= 2.2 T Resonant peak M= 2 ; = 1 22 >> < < b Bandwidth b= (1 22 + 44 42 + 2)/2

Static error coefficients :-

Step i/p : ess= lim

()= lim0

()= lim0

()

+

ess= +KP(positional error) K= lim0 ()() Ramp i/p (t) : ess = K Kv= lim0 ()() Parabolic i/p (t2/2) : ess= 1/ K K= lim0 s

2()()

Type < i/p ess= Type = i/p essfiniteType > i/p ess= 0

Sensitivity S =A/AK/Ksensitivity of A w.r.to K.

Sensitivity of over all T/F w.r.t forward path T/F G(s) :

Open loop: S =1

Closed loop : S =

+G(s)H(s)

Minimum S value preferable

Sensitivity of over all T/F w.r.t feedback T/F H(s) : S =G(s)H(s)

+G(s)H(s)

Stability

RH Criterion :-

Take characteristic equation 1+ G(s) H(s) = 0

All coefficients should have same sign

There should not be missing s term . Term missed means presence of at least one +ve real part root




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Institute of Engineering Studies (IES,Bangalore) Control Systems Formula Sheet

If char. Equation contains either only odd/even terms indicates roots have no real part & posses only

imag parts there fore sustained oscillations in response.

Row of all zeroes occur if(a) Equation has at least one pair of real roots with equal image but opposite sign

(b) has one or more pair of imaginary roots(c) has pair of complex conjugate roots forming symmetry about origin.




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Institute of Engineering Studies (IES,Bangalore) Communications Formula SheetR = m(t) cos 2f P= P/ 2 (SNR) = P/4PI/ = = P= P= 0= (SNR)b (SNR) = P/4P/ = = P = 0= (SNR)b.

A = P+P. P= = P+PNoise in Angle Modulation :-

= P b , PM

3 P b

, FM

PCM :- Min. no of samples required for reconstruction = 2= f; =Bandwidth of msg signal .Total bits required = v fbps . v bits / sampleBandwidth = Rb/2 = v f/ 2 = v. SNR = 1.76 + 6.02 vAs Number of bits increased SNR increased by 6 dB/bit . Band width also increases.Delta Modulation :-By increasing step size slope over load distortion eliminated [ Signal raised sharply ]By Reducing step size Grannualar distortion eliminated . [ Signal varies slowly ]

Digital Communication

Matched filter:impulse response a(t) = P( T t) . P(t) i/pMatched filter o/p will be max at multiples of T . So, sampling @ multiples of T will give max SNR(2ndpoint )matched filter is always causal a(t) = 0 for t < 0Spectrum of o/p signal of matched filter with the matched signal as i/p ie, except for a delay factor ;

proportional to energy spectral density of i/p.

0(f)= H(f) (f) = (f) *(f) eLeading coaching center for GATE/IES/PSU in Bangalore & All over India for Online Tests/PracticeBranches: Jayanagar & Malleshwaram Ph: 0 99003 99699 Email:[email protected]



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Institute of Engineering Studies (IES,Bangalore) Communications Formula Sheet

0(f)= |(f)|ej

o/p signal of matched filter is proportional to shifted version of auto correlation fine of i/p signal

0(t) = R(t T)At t = T 0(T) = R(0) which proves 2ndpointCauchy-Schwartz in equality :-

|g(t)g(t)dt| g(t) dt |g(t)| dtIf g(t) = c g(t) then equality holds otherwise

+

Bamdwidth of Raised cosine filter f

=

+

Bit rate

=

+

roll of factorT signal time periodFor Binary PSK P= Q = Q = erfc .4 PSK P= 2Q 1 FSK:-

For BPSK

P= Q = Q = erfc All signals have same energy (Const energy modulation )Energy & min distance both can be kept constant while increasing no. of points . But BandwidthCompramised.




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Institute of Engineering Studies (IES,Bangalore) Electro-Magnetic Theory Formula SheetElectromagnetic Fields

Vector Calculus:-

A. (B C) = C. (A B) = B. (C A)

A(BC) = B(A.C) C(A.B) Bac Cab ruleScalar component of A along B is A= A Cos = A . a= (.)|| Vector component of A along B is A= A Cos . a= (.)|| Laplacian of scalars :-

A.ds = (.) Divergence theorem L .I = ( ) Stokes theorem A = ( .A) - A .A 0 solenoidal / Divergence loss ; .A > 0 source ; .A < 0 sink

A 0 irrotational / conservative/potential.

A= 0 Harmonic .Electrostatics :- Force on charge Q located @ r F =

Q4 Q()||3N= ; F= QQ4R3. R E @ point r due to charge located @ = 4 ()|3NK= Q E due to line charge @ distance E = . a(depends on distance) E due to surface charge is E = a. aunit normal to surface (independent of distance) For parallel plate capacitor @ point P b/w 2 plates of 2 opposite charges is

E =a - ()

E due to volume charge E = Q4Ra.Electric flux density D = E D independent of mediumFlux = s .

Gauss Law :-Total flux coming out of any closed surface is equal to total charge enclosed by surface .= Q D. ds= Q= . dv . DElectric potential V= Q = - E.dI (independent of path)

V

= -

Q4

a

. dr a

= V

- V

(for point charge )

Potential @ any point (distance = r), where Q is located same where , whose position is vector @ rV =

Q4||V(r) = Q4 + C . [ if C taken as ref potential ]E = 0, E = - VLeading coaching center for GATE/IES/PSU in Bangalore & All over India for Online Tests/PracticeBranches: Jayanagar & Malleshwaram Ph: 0 99003 99699 Email:[email protected]



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Institute of Engineering Studies (IES,Bangalore) Electro-Magnetic Theory Formula SheetFor monopole E ; Dipole E 3.V ; V

Electric lines of force/ flux /direction of E always normal to equipotential lines .

Energy Density W= QVN= = D. E dv= Edv Continuity Equation .J = - . = e/T where T= Relaxation / regeneration time = /(less for good conductor )

Boundary Conditions :- E = E Tangential component of E are continuous across dielectric-dielectric Boundary . Tangential Components of D are dis continues across Boundary .

E= E; = / . Normal components are of D are continues , where as E are dis continues.

D-

D=

; E

=

E

;

=

=

H= H B= BtB= B H= H

Maxwells Equations :-faraday law V= E.dI= - B.dsTransformer emf = E.dI= - ds E = -

s

Motional emf = E= (B).H = J + Electromagnetic wave propagation :- H = J + D = E E= E = - B = H H= . D J E.B 0

= -

=

/ ; E.H = 0 E

H in UPW

For loss less medium E- E = 0 = j( + j) = + j.= 1 + 1




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Institute of Engineering Studies (IES,Bangalore) Electro-Magnetic Theory Formula Sheet

= 1 + + 1

E(z, t) = Eezcos(t z) ; H= E/ . = + || < || = /+ / tan 2= /. = + j attenuation constant Neper /m . | N|= 20 log =8.686 dB For loss less medium = 0; = 0. phase shift/length ; = / ; = 2/.

= = / = tan loss tanjent = 2 If tan

is very small (

< > ) good conductor Complex permittivity = 1 = - j . Tan = = .Plane wave in loss less dielectric :- ( 0)

= 0 ; = ; ; 2/ ; / 0. E & H are in phase in lossless dielectric

Free space :-(= 0, = , = )

= 0 ,

=

; u = 1/

,

2/ ; /

< 0 120 0

Here also E & H in phase .

Good Conductor :- > > / ; = = f; u = 2/ ; = 2/ ; = W 45 Skin depth 1/ 2e/4 + Skin resistance R

R= R.

R= .

Poynting Vector :-

(E H) ds = - [ E H] dv EdvS vLeading coaching center for GATE/IES/PSU in Bangalore & All over India for Online Tests/PracticeBranches: Jayanagar & Malleshwaram Ph: 0 99003 99699 Email:[email protected]



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(z) = ||ezcos az Total time avge power crossing given area P= P (s)ds

S

Direction of propagation :- ()aa= aaa= aBoth E & H are normal to direction of propagationMeans they form EM wave that has no E or H component along direction of propagation .

Reflection of plane wave :-(a) Normal incidence

Reflection coefficient = = +Tcoefficient = = +Medium-I Dielectric , Medium-2 Conductor :-> :->0 , there is a standing wave in medium & Twave in medium 2.Max values of | E| occursZ - n/ ; n 0, 1, 2.Z (+) (+)4

< :- Eoccurs @ Z= (+) Z= (+) = (+)4 Z= nZ= =

Hmin occurs when there is |t|maxS =

|||| =

|||| =

+||||; | | = +Since || < 1 1 Transmission Lines :- Supports only TEM mode LC = ; G/C = /.

V

z -

rV= 0 ; I

z -

rI= 0

= (R + jL)( G + jC)= + j V(z, t) = V+ezcos (t- z) + Vezcos (t + z) Z= VI = R+ = G+= R+G+Lossless Line : (R = 0 =G; = 0)




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Institute of Engineering Studies (IES,Bangalore) Electro-Magnetic Theory Formula Sheet= + j= jLC ; 0, w LC ; 1/ f LC , u 1/ LCZ L/C

Distortion less :(R/L = G/ )

RG ; LGR CRG LC Z RG ; 1/f LC ; u V; uz 1/C , u/z 1/Li/p impedance :-

Z= Z+Z +Z for lossless line = j tan hjl = j tan lZ= Z+Z +Z

VSWR =

=

ZZZ+Z

CSWR = - Transmission coefficient S = 1 + SWR =

VV =II =

+| ||| =ZZ =

ZZ(Z> Z) (Z< Z)

|Z| = VI = SZ |Z| = VI = Z/S

Shorted line :- = -1 , S = Z= Z= jZtan l = -1 , S = Z= Z= j Ztan l. Zmay be inductive or capacitive based on length 0

If l < / 4 inductive (Z+ve)4< l< /2 capacitive (Z-ve)Open circuited line :-

Z= Z= -jZcot l= 1 s = l< / 4 capacitive4< l + b Propegation mode j 0

k b f= + b u = phase velocity = is lossless dielectric medium = u/f= ( )+()

= 1 = / W = phase constant in dielectric medium. u= / = 2/ = u/f phase velocity & wave length in side wave guide T= = - = = 1

T= 1 impedance of UPW in mediumTE Modes :- (= 0)Hz= Hcos cos ez




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T= = / 1

T> TTEDominant modeAntennas :-

Hertzian Dipole :- H= I4 sin e E= HHalf wave Dipole :-

H= I ; E= H

Documents

Ece Formula Sheet Final