Upload
others
View
9
Download
0
Embed Size (px)
Citation preview
1
Chapter. 5 Impedance Matching and Tuning
Maximum power is delivered when the load is matched to the
line (assuming the generator is matched), and power loss in
the feed line is minimized.
Impedance matching sensitive receiver components (antenna,
low-noise amplifier, etc.) improves the signal-to-noise ratio of
the system.
Impedance matching in a power distribution network (such as
an antenna array feed network) will reduce amplitude and
phase errors.
FIGURE 5.1 A lossless network matching an arbitrary load impedance to a transmission line. - Factors that may be important in the selection of a particular
matching network:
1) Complexity: simple
2) Bandwidth: restricted frequency band
3) Implementation: easy
4) Adjustability: tunable
2
5.1 MATCHING WITH LUMPED ELEMENTS (L NETWORKS) - L-section
- Simplest type including two reactive elements
- zL = ZL / Z0
- X, B: lumped or distributed element depending on the operating
frequency.
(a) (b) FIGURE 5.2 L section matching networks. (a) Network for zL inside the 1 + jx circle. (b)
Network for zL outside the 1 + jx circle
Analytic Solutions
- ZL = RL + jXL (assumption: Z0 < RL, inside of 1 + jx circle)
- Impedance seen looking into the matching network:
)1.5()/(1
1
0
in
ZjXRjB
jXZLL
=++
+=
3
- Rearranging and separating into real and imaginary parts:
)2.5()( 00 aZRZXXRB LLL −=−
)2.5()1( 0 bXRBZBXX LLL −=−
- Solving Eq. (5.2a) for X and substituting into Eq. (5.2b)
)3.5(1
)2.5(
)(0)(where
)3.5(/
00
00
0022
220
220
bBRZ
RZX
B
aBR
ZRZBXX
ZRRZXR
aXR
RZXRZRXB
LL
L
L
LL
LLLL
LL
LLLLL
−+=
←−+
=
>>−+
+−+±
=
Two solutions: dual valued components (B, X)
- ZL = RL + jXL (assumption: Z0 > RL, outside of 1 + jx circle)
- Admittance seen looking into the matching network:
)4.5(1)(
1
0
in
Z
XXjRjBY
LL
=
+++=
4
- Rearranging and separating into real and imaginary parts:
)5.5()( 00 aRZXXBZ LL −=+
)5.5()( 0 bRBZXX LL =+
- Solving for X and B:
)6.5()( 0 aXRZRX LLL −−±=
)6.5(/)(
0
0 bZ
RRZB LL−
±=
Two solutions: dual valued components (B, X)
Smith Chart Solutions
[Ex. 5.1] Design an L-section matching network to match a series
RC load with an impedance ZL = 200 - j100 Ω to a 100 Ω line, at a
frequency of 500 MHz.
Sol. 1) zL = 2 - j1
yL = 0.4 + j0.2
y = yL + jb = 0.4 + j0.5 ← jb = j0.3
Interconnection with admittance circle of 1 - jx
z = 1 - j1.2
x = 1.2
5
(a)
Using Eq.(5.3a, b), b = 0.29 and x = 1.22
000
0
,pF92.02 Z
bCCZbbYB
fZbC
ωω
π====←==
0 0038.8 nH ,
2xZ xZL X xZ ωL Lπf ω
= = ← = = =
6
Sol. 2) yL = 0.4 + j0.2
y = yL + jb = 0.4 - j0.5 ← jb = -j0.7
Interconnection with admittance circle of 1 + jx
z = 1 + j1.2
x = -1.2
Using Eq.(5.3a, b), b = -0.69 and x = -1.22
00 0
1 1 12.61 pF ,2
C jX jxZ j CfxZ C xZπ ω ω
= = ← = = − = −
0 00
0
146.1 nH ,2
Z ZjbL jB jbY j Lfb Z L bπ ω ω
= = ← = = = − = −
(b)
(c)
FIGURE5.3 Continued. (b) The two possible L section matching circuits. (c) Reflection.
7
coefficient magnitudes versus frequency for the matching circuits of (b).
5.2 SINGLE-STUB TUNING
- Single open-circuited or short circuited length of transmission line
(a “stub”), connected either in parallel or in series with the
transmission feed line at a certain distance from the load
The shunt tuning stub is especially easy to fabricate in microstrip
or stripline form.
- Shunt-stub case
YL → Y0 + jB using trasmission line having length d
→ Y0 using trasmission line having length l
8
- Series-stub case
ZL → Z0 + jX using trasmission line having length d
→ Z0 using trasmission line having length l
(a)
(b)
FIGURE 5.4 Single-stub tuning circuits. (a) Shunt stub. (b) Series stub.
- λ/4 transmission line with a shunt stub
- Open-circuited stubs are easier to fabricate since a via hole through
the substrate to the ground plane is not needed.
9
Shunt Stubs
[Ex. 5.2] Single-Stub Shunt Tuning
- ZL = 15 + j10 Ω, f = 2 GHz
- Design two single-stub shunt tuning networks to match this load to
a 50 Ω line.
- Plot the reflection coefficient magnitude from 1 GHz to 3 GHz for
each solution.
Solution
- zL = 0.3 + j0.2
- SWR circle
- Convert to the load admittance, yL.
- SWR circle intersects the 1 + jb circle at two points, denoted as y1
and y2 in Figure 5.5a. → d1 or d2
- d1 = (0.328 – 0.284)λ = 0.044 λ
d2 = (0.5 – 0.284) + 0.171λ = 0.387 λ
- The matching stub is kept as close as possible to the load, to
improve the bandwidth of the match and to reduce losses caused by
a possibly large standing wave ratio on the line between the stub
and the load.
- y1 = 1 – j1.33, y2 = 1 + j1.33
10
- l1 = 0.147 λ for jb = j1.33
l2 = 0.353 λ for jb = -j1.33
(a)
(b)
0.328
0.147
0.171
0.284
0.353
11
(c)
FIGURE 5.5 Solution to Example 5.2. (a) Smith chart for the shunt-stub tuners. (b) The two shunt-stub tuning solutions. (c) Reflection coefficient magnitudes versus frequency for the tuning circuits of (b).
- Derivation of formulas for d and l
ZL = 1/YL = RL + jXL
Impedance Z down a length, d, of line from the load:
dt
tjXRjZtjZjXRZZ
LL
LL
βtanwhere
)7.5()(
)(
0
00
=++++
=
ZjBGY 1
=+=
12
02
02
2 1)8.5()(
)1(whereZ
atZXR
tRGLL
L =+++
=
)8.5(])([
))((2
02
0
002
btZXRZ
tZXtXZtRBLL
LLL
+++−−
=
d (which implies t) is chosen so that G = Y0 = 1 / Z0
0)(2)( 2200
200 =−−+−− LLLLL XRZRtZXtZRZ
)9.5(for,/])[(
00
022
0 ZRZR
ZXRZRXt L
L
LLLL ≠−
+−±=
If RL = Z0, then t = -XL / 2Z0.
If RL ≠ Z0,
<+
≥=
==
−
−
.0for),tan(21
0for,tan21
2tantan
1
1
tt
ttd
ddt
ππ
πλ
λπβ
(5.10)
To find the required stub lengths, BS = -B.
1) For an open-circuited stub,
jBS = jY0tanβlo = jY0tan(2πlo/λ) =- jB
)11.5(tan2
1tan21
0
1
0
1 aYB
YBl so
−=
= −−
ππλ
13
2) For a short-circuited stub,
jBS = -jY0cotβls = - jY0cot(2πls/λ) = -jB
)11.5(tan21tan
21 0101 b
BY
BYl
s
s
=
−= −−
ππλ
If the lengths (lo, ls) are negative, λ/2 can be added to give a
positive result.
Shunt Stubs
[Ex. 5.2] Single-Stub Series Tuning
- ZL = 100 + j80 Ω, f = 2 GHz
- Match a load to a 50 Ω using a single series open-circuited stub.
- Plot the reflection coefficient magnitude from 1 GHz to 3 GHz for
each solution.
Solution
- zL = 2 + j1.6
- SWR circle
- The SWR circle intersects the 1 + jx circle at two points, denoted
as z1 and z2.
- The shortest distance, d1, from the load to the stub:
d1 = (0.328 – 0.208) λ = 0.120 λ
The second distance:
d2 = (0.5 – 0.208) + 0.172 λ = 0.463 λ
14
- z1 = 1 – j1.33, z2 = 1 + j1.33
- l1 = 0.397 λ for jx = j1.33 (open stub)
l2 = 0.103 λ for jx = -j1.33 (open stub)
(a)
(b)
0.328
0.208
0.172
0.353 → 0.353 – 0.25 = 0.103
0.284
0.147 ← 0.25 + 0.147 = 0.397
15
(c)
FIGURE 5.6 Solution to Example 5.3. (a) Smith chart for the series-stub tuners. (b) The two series-stub tuning solutions. (c) Reflection coefficient magnitudes versus frequency for the tuning circuits of (b).
- Derivation of formulas for d and l for series-stub tuner
YL = 1 / ZL = GL + jBL
Impedance Y down a length, d, of line from the load:
00
0
00
/1,tanwhere
)12.5()(
)(
ZYdtjBGjtYjtYjBGYY
LL
LL
==++++
=
β
YjXRZ 1
=+=
02
02
2 1)13.5()(
)1(whereY
atYBG
tGRLL
L =+++
=
16
)13.5(])([
))((2
02
0
002
btYBGY
tYBtBYtGXLL
LLL
+++−−
=
Now d (which implies t) is chosen so that R = Z0 = 1 / Y0.
0
2200
200
/1)13.5(.Eq0)(2)(
YaBGYGtYBtYGY LLLLL
=←=−−+−−
00
022
0 for,/])[(
YGYG
YBGYGBt L
L
LLLL ≠−
+−±=
If GL = Y0, then t = -BL / 2Y0.
If GL ≠ Y0,
<+
≥=
==
−
−
.0for),tan(21
0for,tan21
2tantan
1
1
tt
ttd
ddt
ππ
πλ
λπβ
(5.15)
To find the required stub lengths, XS = -X.
1) For a short-circuited stub,
jXS = jZ0tanβls = jZ0tan(2πls/λ) = -jX
)16.5(tan2
1tan21
0
1
0
1 aZX
ZXl ss
−=
= −−
ππλ
17
2) For a open-circuited stub,
jXs = -jZ0cotβlo = -jZ0cot(2πlo/λ) = - jX
)16.5(tan21tan
21 0101 b
XZ
XZl
s
o
=
−= −
ππλ
If the lengths (lo, ls) are negative, λ/2 can be added to give a
positive result.
5.3 DOUBLE-STUB TUNING - Disadvantage of single-stub tuning: requiring a variable length of
line between the load and the stub.
Difficult if an adjustable tuner was desired
(a) (b)
FIGURE 5.7 Double-stub tuning. (a) Original circuit with the load an arbitrary distance from the first stub. (b) Equivalent circuit with load at the first stub.
18
Smith Chart Solution
- The susceptance of the first stub, b1 (or b1’, for the second solution),
moves the load admittance (yL = g0 + jb) to y1 (or y1’).
- The amount of rotation is d wavelengths toward the load
→ Transforming y1 (or y1’) to y2 (or y2’): 1 + jb circle
- The second stub then adds a susceptance b2 (or b2’), which brings
us to the center of the chart, and completes the match.
FIGURE 5.8 Smith chart diagram for the operation of a double-stub tuner.
19
[Ex. 5.4] Double-Stub Tuning
- Design a double-stub shunt tuner to match a load impedance ZL =
60 - j80 Ω to a 50 Ω line at 2 GHz.
- The stubs are to be open-circuited stubs, and are spaced λ/8 apart.
- Plot the reflection coefficient magnitude versus frequency from 1 ~
3 GHz.
yL = 0.3 + j0.4
By moving every point on the g = 1 circle λ/8 toward the load, we
then find the susceptance of the first stub, which can be one of
two possible values: b1 = 1.314 or b1’ = -0.114
→ y2 = 1 – j3.38 or y2‘ = 1 + j1.38
Susceptance of the second stub: b2 = 3.38 or b2’ = -1.38
Lengths of the short-circuited stubs are then found as,
l1 = 0.146 λ, l2 = 0.482 λ,
or l1’ = 0.204 λ, l2’ = 0.350 λ
20
(a)
(b)
90o rotation Of unit circle
21
(c)
FIGURE 5.9 Solution to Example 5.4. (a) Smith chart for the double-stub tuners. (b) The two double-stub tuning solutions. (c) Reflection coefficient magnitudes versus frequency for the tuning circuits of (b).
Analytic Solution
- Load admittance with the first stub:
)17.5()( 111 BBjGjBYY LLL ++=+=
- Admittance just to the right of the second stub:
00
10
0102
/1,tan where
)18.5()(
)(
ZYdtjBjBGjtYtYBBjGYY
LL
LL
==++++++
=
β
- Re(Y2) = Y0:
22
)19.5(0)(12
210
2
2
02 =
−−+
+−
ttBtBY
ttYGG L
LL
)20.5()1(
)(4112
1222
0
210
2
2
2
0
+−−
−±+
=tY
tBtBYtttYG L
L
- Since GL is real,
1)1(
)(40 2220
210
2
≤+
−−≤
tYtBtBYt L
)21.5(sin
10 20
2
2
0 dY
ttYGL β=
+≤≤⇒
- After d has been fixed, the first stub susceptance can be determined
from (5.19) as:
)22.5()1( 22
02
01 t
tGYGtYBB LL
L−+±
+−=
- jB2 = -jIm(Y2):
)23.5()1( 0
22200
2 tGYGtGtGYY
BL
LLL +−+±=
- Open-circuited stub length: )24.5(tan21
0
1 aYBlo
= −
πλ
23
- Short-circuited stub length: )24.5(tan2
1 01 bBYls
−
= −
πλ
Where B = B1, B2
5.4 THE QUARTER-WAVE TRANSFROMER
- A single-section λ/4 transformer: narrow band impedance match
A multi-section λ/4 transformer: broad band impedance match
- One drawback of the quarter-wave transformer: match a real load
impedance.
FIGURE 5.10 A single-section quarter-wave matching transformer. l=λ0/4 at the design
frequency f0.
)25.5(01 LZZZ =
where l = λ0/4 : electrical length at operating frequency, f0.
24
- Input impedance seen looking into the matching section:
2/ and tan where
)26.5(1
11in
πθββ ===++
=
llttjZZtjZZZZ
L
L
- Reflection coefficient:
L
LL
LL
ZZZZZZjtZZZZZZjtZZZ
ZZZZ
02
1
02
101
02
101
0in
0in )27.5()()()()(
=←
+++−+−
=+−
=Γ
)28.5(2 00
0
LL
L
ZZtjZZZZ
++−
=Γ
- Reflection coefficient magnitude:
2/10
220
0
]4)[( LL
L
ZZtZZZZ
++−
=Γ
2/1200
2200 ])/(4[)]/()[(
1ZZZZtZZZZ LLLL −+−+
=
2/120
20
200 ])/(4[])/(4[1
1ZZtZZZZZZ LLLL −+−+
=
)29.5(sec])/(4[1
12/122
00 θZZZZ LL −+=
)sectan11( 222 θθ =+=+ t
25
- If the frequency is near the design frequency, f0, then l ≈ λ0/4 and θ
≈ π/2. Then sec2θ >> 1.
)30.5(2/nearfor,cos2 0
0 πθθL
L
ZZZZ −
≅Γ
FIGURE 5.11 Approximate behavior of the reflection coefficient magnitude for a single-
section quarter-wave transformer operation near its design frequency.
- If we set a maximum value, Γm, of the reflection coefficient
magnitude that can be tolerated, then we can define the bandwidth
of the matching transformer as
)31.5(2
2
−=∆ mθπθ
2
0
02 sec
211
−+=
Γ mL
L
m ZZZZ
θ
26
)32.5(2
1cos
0
0
2 ZZZZ
L
L
m
mm −Γ−
Γ=θ
- If we assume TEM line,
00 242
ff
fv
vfl p
p
ππβθ ===
πθ 02 ff m
m =
- Fractional bandwidth:
πθmmm
ff
fff
ff 4222)(2
00
0
0
−=−=−
=∆
)33.5(2
1cos42
0
0
2
1
−Γ−
Γ−= −
ZZZZ
L
L
m
m
π
FIGURE 5.12 Reflection coefficient magnitude versus frequency for a single-section
quarter-wave matching transformer with various load mismatches.
27
- When non-TEM lines (such as waveguides) are used, the
propagation constant is no longer a linear function of frequency,
and the wave impedance will be frequency dependent.
- The effect of reactances associated with discontinuities of
transmission line must be considered.
[Ex. 5.5] Quarter-Wave Transformer Bandwidth
Design a single-section quarter-wave matching transformer to match
a 10 Ω load to a 50 Ω line, at f0 = 3 GHz. Determine the percent
bandwidth for which the SWR ≤ 1.5.
Ω=== 36.22)10)(50(01 LZZZ
The length of the matching section is λ/4 at 3 GHz.
2.015.115.1
1SWR1SWR
=+−
=+−
=Γm
Fractional bandwidth:
−Γ−
Γ−=
∆ −
0
0
2
1
0
2
1cos42
ZZZZ
ff
L
L
m
m
π
−−−= −
5010)10)(50(2
)2.0(12.0cos42
2
1
π
%.29or29.0=
28
5.5 THE THEORY OF SMALL REFLECTIONS
- Theory of small reflections: total reflection coefficient caused by
the partial reflections from several small discontinuities
Single-Section Transformer
- Derivation an approximate expression for the overall reflection
coefficient Γ
FIGURE 5.13 Partial reflections and transmissions on a single-section matching
transformer.
)34.5(12
121 ZZ
ZZ+−
=Γ
29
)35.5(12 Γ−=Γ
)36.5(2
23 ZZ
ZZ
L
L
+−
=Γ
)37.5(2121
2121 ZZ
ZT+
=Γ+=
)38.5(2121
1212 ZZ
ZT+
=Γ+=
- Expression of the total reflection as an infinite sum of partial
reflections and transmissions:
...42
232112
2321121 +ΓΓ+Γ+Γ=Γ −− θθ jj eTTeTT
)39.5(0
232
2321121 ∑
∞
=
−− ΓΓΓ+Γ=n
jnnnj eeTT θθ
- Using the geometric series and small reflection condition,
1for,1
10
<−
=∑∞
=n
n xx
x
)41.5(1
1)1)(1(
11,1,
)40.5(1
231
231
231
2311
23211
121212112
232
232112
1
θ
θ
θ
θθ
θ
θ
j
j
j
jj
j
j
ee
eee
TTeeTT
−
−
−
−−
−
−
ΓΓ+Γ+Γ
≅
ΓΓ+ΓΓ+Γ−+ΓΓΓ−Γ
=
Γ−=Γ+=Γ+=Γ−=Γ←ΓΓ−Γ
+Γ=Γ
30
)42.5(231
θje−Γ+Γ≅Γ
- Intuitive ideas:
1) The total reflection is dominated by the reflection from the
initial discontinuity between Z1 and Z2 (Γ1), and the first
reflection from the discontinuity between Z2 and ZL (Γ3e-2jθ).
2) The e-2jθ term accounts for the phase delay when the incident
wave travels up and down the line.
Multi-Section Transformer
- N equal-length (commensurate) sections of transmission lines.
FIGURE 5.14 Partial reflection coefficients for a multisection matching transformer.
- Derivation an approximate expression for the total reflection coefficient Γ.
31
)43.5(,01
010 a
ZZZZ
+−
=Γ
)43.5(,1
1 bZZZZ
nn
nnn +
−=Γ
+
+
)43.5(. cZZZZ
NL
NLN +
−=Γ
- Assumptions:
1) All Zn increase or decrease monotonically across the transformer.
2) ZL is real.
0
0
if0if0
ZZZZ
Ln
Ln
<<Γ>>Γ⇒
- Overall reflection coefficient:
)42.5()44.5()( 242
210 ←Γ++Γ+Γ+Γ=Γ −−− θθθθ jN
Njj eee
- If the transformer be made symmetrical, so that Γ0 = ΓN, Γ1 = ΓN-1,
Γ2 = ΓN-2, etc. Then (5.44) can be written as;
)45.5(][][)( )2()2(10 ++Γ++Γ=Γ −−−−− θθθθθθ NjNjjNjNjN eeeee
Where if N is odd, the last term is Γ(N-1)/2(ejθ+e-jθ), while if N is
even the last term is ΓN/2.
32
)46.5(evenfor ]21
)2cos(...)2cos(cos[2)(
2/
10
aN
nNNNe
N
njN
Γ++
−Γ++−Γ+Γ=Γ −
θθθθ θ
)46.5( odd for ]cos)2cos(...)2cos(cos[2)(
2/)1(
10
bNnNNNe
N
njN
θθθθθ θ
−
−
Γ++−Γ++−Γ+Γ=Γ