This document provides analytical design equations for simple impedance matching networks like L-match and inverted L-match networks. It introduces basic concepts like impedance, admittance, ABCD parameters and input impedance/admittance. Design procedures for L-match network using analytical equations and Smith chart are presented. The document contains an example problem demonstrating the design of an L-match network to match a given load impedance to a source impedance. Analytical equations to design inverted L-match network are also provided.

1. National Institute of Technology
Rourkela
Analytical Design Equations of Simple
Matching Network
EE3004 : Electromagnetic Field Theory
Dr. Rakesh Sinha
(Assistant Professor)
Circuit and Electromagnetic Co-Design Lab at NITR
Department of Electrical Engineering
National Institute of Technology (NIT) Rourkela
April 11, 2022

2. Circuit-EM Co-Design Lab
Outline
1 Introduction
2 Preliminary Concept
3 L-match Network
4 L-match Network using Smith Chart
5 Inverted L-match Network
6 inv L-match Network using Smith Chart
7 Single Series Stub Matching Network
8 Single Shunt Stub Matching Network
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3. Circuit-EM Co-Design Lab
Introduction
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4. Circuit-EM Co-Design Lab
Introduction
q The impedance matching network (IMN) is one of the fundamental and
crucial component spanning over the entire spectrum of RF, microwave
and millimetre-wave.
q The IMNs are in general passive, reciprocal,lossless, two-port networks
q Application: antenna feed network, amplifier input and output matching,
power combiners and dividers, RFID, impedance compensation network
for energy harvesting and WPT etc.
q When delivering ac power, maximum real power is delivered to the
complex load, when the load impedance as seen through an IMN, by the
complex source is equal to the complex conjugate of the source
impedance
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5. Circuit-EM Co-Design Lab
Introduction
(a) (b)
Figure 1: (a) A two-port lossless passive reciprocal network transforming a load
impedance ZL = RL + jXL into complex conjugate of the source impedance
¯
ZS = RS − jXS; (b) The same network transforming the source impedance
ZS = RS + jXS into complex conjugate of the load impedance ¯
ZL = RL − jXL
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6. Circuit-EM Co-Design Lab
Introduction
q Depending on the frequency of operation, the IMNs can be implemented
using lumped inductor and capacitor or using distributed transmission line
(TL) or combination of lumped and distributed elements.
q Lumped elements topology: L-match, Inverted L-match, Π-match and
T-match.
q Distributed elements topology: Single section TL, Stepped impedance,
L-type or single stub (series and shunt), Π-type or double stub and T-type.
q Most of the book follow Smith Chart method to design a Matching
Network. So one have to carry Smith Chart and geometry box to design a
matching network.
q In this presentation four analytical techniques are provided to design
simple matching network like L-match, inverted L-match and single stub
matching network (series and shunt).
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7. Circuit-EM Co-Design Lab
Preliminary Concept
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8. Circuit-EM Co-Design Lab
Preliminary Concept
q Impedance of a load in general define as Z = R + jX, where the real
part R is resistive part responsible for loss and the imaginary part X is
the reactive part responsible for stored energy.
q Similarly, Admittance of a load in general define as Y = G + jB, where G
is conductance responsible for loss and B is the susceptance responsible
for stored energy.
q Y = 1
Z ; G = R
|Z|2 ; B = − X
|Z|2 ; |Z|2
= R2
+ X2
q Similarly, Z = 1
Y ; R = G
|Y |2 ; X = − B
|Y |2 ; |Y |2
= G2
+ B2
q Impedance of an inductor of inductance L is Z = jωL, similarly
admittance is Y = − j
ωL .
q Admittance of an capacitor of capacitance C is Y = jωC, similarly
impedance is Z = − j
ωC .
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9. Circuit-EM Co-Design Lab
Preliminary Concept
q If the imaginary part of a load impedance is positive, then the load is
inductive in nature.
q Else if the imaginary part of a load impedance is negative, then the load
is capacitive in nature.
q If the imaginary part of a load admittance is positive, then the load is
capacitive in nature.
q Else if the imaginary part of a load admittance is negative, then the load
is inductive in nature.
q If the real part of load impedance or admittance is zero, then load is
purely reactive in nature.
q If the imaginary part of load impedance or admittance is zero, then load is
purely resistive in nature.
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10. Circuit-EM Co-Design Lab
Preliminary Concept-ABCD parameter
Figure 2: Two-port Network
q The ABCD parameter of a two-port network shown in Fig. 2 can be
defined as
V1
I1
=
A B
C D
V2
−I2
. (1)
q When port-2 as input port the ABCD matrix define as
V2
I2
=
1
AD − BC
D B
C A
V1
−I1
. (2)
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11. Circuit-EM Co-Design Lab
Preliminary Concept-ABCD parameter
q For a reciprocal two-port network
AD − BC = 1. (3)
q For lossless network the ABCD parameters define as
A B
C D
=
a jb
jc d
(4)
q where a, b, c and d are pure real number.
q For a lossless reciprocal two-port network
ad + bc = 1. (5)
q For a symmetric network
A = D. (6)
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12. Circuit-EM Co-Design Lab
Preliminary Concept-ABCD parameter
Circuit ABCD parameters of a
Series Impedance Z is:
A B
C D
=
1 Z
0 1
Shunt Admittance Y is:
A B
C D
=
1 0
Y 1
Transmission Line of characteristic
impedance Z0 or admittance Y0 = 1
Z0
and electrical length θ = βl is:
A B
C D
=
cos θ jZ0 sin θ
jY0 sin θ cos θ
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13. Circuit-EM Co-Design Lab
Preliminary Concept-Input Impedance
Figure 3: Port-2 of a two-port network terminated by load impedance ZL
q The input impedance seen at port-1, when port-2 is terminated by load
impedance ZL or admittance YL, is given as
Zin1 =
V1
I1
=
AV2 − BI2
CV2 − DI2
=
AZL + B
CZL + D
=
A + BYL
C + DYL
(7)
q The input impedance seen at port-2, when port-1 is terminated by load
impedance ZL admittance YL, is given as
Zin2 =
DZL + B
CZL + A
=
D + BYL
C + AYL
(8)
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14. Circuit-EM Co-Design Lab
Preliminary Concept-Input Admittance
q The input admittance seen at port-1, when port-2 is terminated by load
admittance YL or impedance ZL, is given as
Yin1 =
DYL + C
BYL + A
=
D + CZL
B + AZL
(9)
q The input admittance seen at port-2, when port-1 is terminated by load
admittance YL or impedance ZL, is given as
Yin2 =
AYL + C
BYL + D
=
A + CZL
B + DZL
(10)
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15. Circuit-EM Co-Design Lab
Input Impedance/Admittance of TL
q The input impedance of a transmission line of characteristic impedance
Z0 and electrical length θ is terminated by load impedance ZL or
admittance YL, is given as
Zin = Z0
ZL cos θ + jZ0 sin θ
jZL sin θ + Z0 cos θ
= Z0
Y0 cos θ + jYL sin θ
jY0 sin θ + YL cos θ
(11)
q The input admittance of a transmission line of characteristic impedance
Z0 and electrical length θ is terminated by load impedance ZL or
admittance YL, is given as
Yin = Y0
YL cos θ + jY0 sin θ
jYL sin θ + Y0 cos θ
= Y0
Z0 cos θ + jZL sin θ
jZ0 sin θ + ZL cos θ
(12)
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16. Circuit-EM Co-Design Lab
Input Impedance/Admittance of short/open
circuited TL
q The input impedance/admittance of a short circuited (i.e., ZL = 0 or
YL = ∞) transmission line of characteristic impedance Z0 and electrical
length θ , is given as
Zin = jZ0 tan θ; Yin = −jY0 cot θ (13)
q The input impedance/admittance of a open circuited (i.e., ZL = ∞ or
YL = 0) transmission line of characteristic impedance Z0 and electrical
length θ , is given as
Zin = −jZ0 cot θ; Yin = jY0 tan θ (14)
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17. Circuit-EM Co-Design Lab
L-match Network
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18. Circuit-EM Co-Design Lab
L-match Network
Figure 4: L-match network consist of shunt susceptance B1 and series reactance X2,
matches the complex load impedance ZL = RL + jXL to a real source impedance
Z0 = 1/Y0 or admittance Y0 = 1/Z0.
q The system can be normalized with respect to real source impedance
and the normalized parameters are given as
b1 = B1Z0; x2 = X2
Z0
; zL = rL + jxL = ZL
Z0
= RL+jXL
Z0
(15)
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19. Circuit-EM Co-Design Lab
L-match Network
Figure 5: Normalized L-match network consist of shunt susceptance b1 and series
reactance x2, matches the complex load impedance zL = rL + jxL to a real source
impedance z0 = 1 or admittance y0 = 1.
q The input admittance yin2 can be calculated as
yin2 = 1 − jb1 =
1
zL + jx2
=
1
rL + j(xL + x2)
(16)
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20. Circuit-EM Co-Design Lab
L-match Network
q The above equation (16) can be rewritten as
(1 − jb1) (rL + j(xL + x2)) = 1
rL + b1(xL + x2) + j ((xL + x2) − rLb1) = 1
(17)
q By separating the real and imaginary parts of (17), one can obtain
(xL + x2) =
1 − rL
b1
(18)
(xL + x2) = b1rL (19)
q By comparing the above equations one can obtain the solution b1 and x2 of
L-match network as
b1 = ±
r
1 − rL
rL
(20)
x2 = −xL ±
p
(1 − rL)rL (21)
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21. Circuit-EM Co-Design Lab
L-match Network
q The solution of renormalized L-match network for capacitive shunt
element (B1 0) can be given as
B1 = Y0
r
1 − rL
rL
(22)
X2 = −Z0
xL −
p
(1 − rL)rL
(23)
q The solution of renormalized L-match network for inductive shunt element
(B1 0) can be given as
B1 = −Y0
r
1 − rL
rL
(24)
X2 = −Z0
xL +
p
(1 − rL)rL
(25)
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22. Circuit-EM Co-Design Lab
L-match network
Problem-1
Design a lumped L-match Network which matches load impedance of ZL = 40 − j30 Ω
to a source having internal impedance of ZS = 50 Ω at 1.2 GHz frequency.
Solution-1
q From the problem statement one can write
Z0 = 50 Ω,Y0 = 0.02 f, ZL = 40 − j30 Ω, zL = 0.8 − j0.6
rL = 0.8, xL = −0.6, f0 = 1.2 GHz
q The first solution set with capacitive shunt element can be written as
B1 = Y0
r
1 − rL
rL
= 0.01 f (26)
X2 = −Z0
xL −
p
(1 − rL)rL
= 50 Ω. (27)
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23. Circuit-EM Co-Design Lab
L-match network
Solution-1
q As shunt element B1 is positive can be implemented using capacitor C1
and the series element can be implemented using inductor L2 due to
positive value of X2.
C1 =
B1
2πf0
= 1.3263 pF L2 =
X2
2πf0
= 6.6315 nH
q The second solution set with inductive shunt element can be written as
B1 = −Y0
r
1 − rL
rL
= −0.01 f (28)
X2 = −Z0
xL +
p
(1 − rL)rL
= 10 Ω. (29)
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24. Circuit-EM Co-Design Lab
L-match network
Solution-1
q As shunt element B1 is negative can be implemented using inductor L1 and the
series element can also be implemented using inductor L2 due to positive value
of X2.
L1 = −
1
2πf0B1
= 13.263 nH L2 =
X2
2πf0
= 1.3263 nH
(a) (b)
Figure 6: L-match network: a. first solution b. second solution
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25. Circuit-EM Co-Design Lab
L-match Network using Smith Chart
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26. Circuit-EM Co-Design Lab
L-match Network using Smith Chart
q Following steps to be followed to design L-match network using smith
chart.
1. Normalized the load impedance ZL to zL = ZL/Z0, where Z0 is the source
impedance.
2. Please ensure that zL is outside z = 1 + jx circle. If inside you need to go
for inv-L match network.
3. Draw constant SWR circle |z| = |zL|.
4. Draw y = 1 + jb circle.
5. Move to the point z1 = 1/y1 through constant resistance circle z = rL + jx
such that constant rL circle cut y = 1 + jb circle.
6. The desired series reactance for the L-match is jx1 = z1 − zL.
7. Convert z1 into y1 = 1 + jb1 via reflection.
8. Add susceptance such that y1 = 1 + jb1 move to matched point y = 1.
9. The desired shunt susceptance is −jb1 = 1 − y1.
10. Follow step 5-9 for second solution.
11. Scaled the reactance and susceptance to X1 = Z0x1 and
B1 = b1/Z0 = Y0b1.
12. Get the L or C value at the design frequency.
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27. Circuit-EM Co-Design Lab
L-match Network using Smith Chart
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28. Circuit-EM Co-Design Lab
Inverted L-match Network
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29. Circuit-EM Co-Design Lab
Inverted-L-match Network
Figure 7: Inverted L-match network consist of series reactance X1 and shunt
susceptance B2, matches the complex load impedance ZL = RL + jXL or admittance
YL = GL + jBL to a real source impedance Z0 = 1/Y0 or admittance Y0 = 1/Z0.
q The system can be normalized with respect to real source admittance Y0 and the
normalized parameters are given as
x1 = x1Y0; b2 = B2
Y0
; yL = gL + jbL = YL
Y0
= GL+jBL
Y0
(30)
gL + jbL =
1
rL + jxL
=
rL − jxL
|zL|2
(31)
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30. Circuit-EM Co-Design Lab
Inverted L-match Network
Figure 8: Normalized inverted L-match network consist of series reactance x1 and
shunt susceptance b2, matches the complex load admittance yL = gL + jbL to a real
source impedance z0 = 1 or admittance y0 = 1.
q The input impedance zin2 can be calculated as
zin2 = 1 − jx1 =
1
yL + jb2
=
1
gL + j(bL + b2)
(32)
q Note that L-match (16) and inverted L match (32) are dual problem and
the solutions are similar to each other.
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31. Circuit-EM Co-Design Lab
Inverted L-match Network
q The above equation (32) can be rewritten as
(1 − jx1) (gL + j(bL + b2)) = 1
gL + x1(bL + b2) + j ((bL + b2) − gLx1) = 1
(33)
q By separating the real and imaginary parts of (33), one can obtain
(bL + b2) =
1 − gL
x1
(34)
(bL + b2) = x1gL (35)
q By comparing the above equations one can obtain the solution x1 and b2 of
L-match network as
x1 = ±
r
1 − gL
gL
(36)
b2 = −bL ±
p
(1 − gL)gL (37)
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32. Circuit-EM Co-Design Lab
Inverted L-match Network
q The solution of renormalized inverted L-match network for inductive
series element (X1 0) can be given as
X1 = Z0
r
1 − gL
gL
(38)
B2 = −Y0
bL −
p
(1 − gL)gL
(39)
q The solution of renormalized inverted L-match network for capacitive
series element (X1 0) can be given as
X1 = −Z0
r
1 − gL
gL
(40)
B2 = −Y0
bL +
p
(1 − gL)gL
(41)
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33. Circuit-EM Co-Design Lab
Inverted L-match network
Problem-2
Design a lumped inverted L-match network which matches load impedance of
ZL = 40 − j30 Ω to a source having internal impedance of ZS = 50 Ω at 1.2 GHz
frequency.
Solution-2
q From the problem statement one can write
Z0 = 50 Ω,Y0 = 0.02 f, ZL = 40 − j30 Ω, yL = 0.8 + j0.6
gL = 0.8, bL = 0.6, f0 = 1.2 GHz
q The first solution set with inductive series element can be written as
X1 = Z0
r
1 − gL
gL
= 25 Ω (42)
B2 = −Y0
bL −
p
(1 − gL)gL
= −0.004 f (43)
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34. Circuit-EM Co-Design Lab
Inverted L-match network
Solution-2
q As series element X1 is positive can be implemented using inductor L1
and the shunt element can also be implemented using inductor L2 due to
negative value of B2.
L1 =
X1
2πf0
= 3.3157 nH L2 = −
1
2πf0B2
= 33.157 nH
q The second solution set with capacitive series element can be written as
X1 = −Z0
r
1 − gL
gL
= −25 Ω (44)
B2 = −Y0
bL +
p
(1 − gL)gL
= −0.02 f (45)
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35. Circuit-EM Co-Design Lab
Inverted L-match network
Solution-2
q As series element X1 is negative can be implemented using capacitor C1 and the
shunt element can be implemented using inductor L2 due to negative value of B2.
C1 = −
1
2πf0X1
= 5.3052 pF L2 = −
1
2πf0B2
= 6.6315 nH
(a) (b)
Figure 9: Inverted L-match: a. First solution b. Second solution
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36. Circuit-EM Co-Design Lab
inv L-match Network using Smith Chart
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37. Circuit-EM Co-Design Lab
inv-L-match Network using Smith Chart
q Following steps to be followed to design inv-L-match network using smith
chart.
1. Normalized the load impedance ZL to zL = ZL/Z0, where Z0 is the source
impedance.
2. Please ensure that zL is inside z = 1 + jx circle. If outside you need to go
for L match network.
3. Draw constant SWR circle |z| = |zL|.
4. Draw y = 1 + jb circle in Z-smith chart, which can be used as z = 1 + jx
circle in Y-smith chart.
5. Convert zL into yL via reflection in constant SWR circle.
6. Move to the point y1 = 1/z1 through constant conductance circle y = gL + jb
such that constant gL circle cut x = 1 + jx circle.
7. The desired shunt susceptance for the inv-L-match is jb1 = y1 − yL.
8. Convert y1 into z1 = 1 + jx1 via reflection.
9. Add reactance such that z1 = 1 + jx1 move to matched point z = 1.
10. The desired series reactance is −jx1 = 1 − z1.
11. Follow step 6-10 for second solution.
12. Scaled the reactance and susceptance to X1 = Z0x1 and
B1 = b1/Z0 = Y0b1.
13. Get the L or C value at the design frequency.
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38. Circuit-EM Co-Design Lab
inv-L-match Network using Smith Chart
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39. Circuit-EM Co-Design Lab
Single Series Stub Matching Network
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40. Circuit-EM Co-Design Lab
Single Series Stub Matching Network
Figure 10: Single series stub matching network consist of series stub (characteristic
impedance Z0 and electrical length θs = βls) having input impedance of jX and main
line (characteristic impedance Z0 and electrical length θm = βlm), matches the
complex load impedance ZL = RL + jXL to a real source impedance Z0 = 1/Y0 or
admittance Y0 = 1/Z0.
q The system can be normalized with respect to real source impedance Z0
and the normalized parameters are given as
x = X
Z0
; zL = rL + jxL = ZL
Z0
= RL+jXL
Z0
(46)
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41. Circuit-EM Co-Design Lab
Single Series Stub Matching Network
Figure 11: Normalized single series stub matching network, matches the complex load
impedance ZL = rL + jxL to a real source impedance z0 = 1 or admittance y0 = 1.
q The input impedance zin2 can be calculated as
zin2 = 1 − jx =
zL cos θm + j sin θm
jzL sin θm + cos θm
=
(rL + jxL) cos θm + j sin θm
j(rL + jxL) sin θm + cos θm
(47)
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42. Circuit-EM Co-Design Lab
Single Series Stub Matching Network
q The above equation (47) can be rewritten as
(1 − jx) (j(rL + jxL) sin θm + cos θm)
= (rL + jxL) cos θm + j sin θm
cos θm − (xL − xrL) sin θm + j ((rL + xxL) sin θm − x cos θm)
= rL cos θm + j (xL cos θm + sin θm)
(48)
q By separating the real and imaginary parts of (48), one can obtain
tan θm =
1 − rL
xL − rLx
(49)
tan θm =
xL + x
xxL − (1 − rL)
(50)
q By comparing the above equations one can write
1 − rL
xL − rLx
=
xL + x
xxL − (1 − rL)
(51)
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43. Circuit-EM Co-Design Lab
Single Series Stub Matching Network
q The the equation (51) can be rewritten as
(1 − rL)xxL − (1 − rL)2
= xL
2
+ xxL − rLxxL − rLx2
(52)
x2
=
(1 − rL)2
+ xL
2
rL
(53)
q The above equation has two solutions: one is inductive series stub
(x 0) and other one is capacitive series stub (x 0). The solution are
given as
x = ±
s
(1 − rL)2 + xL
2
rL
(54)
q The parameters related to main line electrical length can be obtain as
t = tan θm =
1 − rL
xL − rLx
=
1 − rL
xL ∓
p
rL(1 − rL)2 + rLxL
2
(55)
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44. Circuit-EM Co-Design Lab
Single Series Stub Matching Network
q The electrical length of the main line and shunt line can be obtained as
θm = tan−1
(t) (56)
θs = tan−1
(x) for short stub
= cot−1
(−x) for open stub.
(57)
q If θm,s is negative then add 180◦
to it, when θm,s is in degree. The
physical lengths of the main line and shunt line can be obtained as
lm,s =
θm,s
360◦
λg (58)
q Here λg is the guided wavelength of the transmission lines.
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45. Circuit-EM Co-Design Lab
Single Series Stub Matching Network
Problem-3
Design a single series stub matching network which matches load impedance
of ZL = 40 − j30 Ω to a source having internal impedance of ZS = 50 Ω at 1.2
GHz frequency.
Solution-3
q From the problem statement one can write
Z0 = 50 Ω,Y0 = 0.02 f, ZL = 40 − j30 Ω, zL = 0.8 − j0.6
rL = 0.8, xL = −0.6, f0 = 1.2 GHz
q The first solution set with inductive series stub can be written as
x =
s
(1 − rL)2 + xL
2
rL
= 0.70711 Ω (59)
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46. Circuit-EM Co-Design Lab
Single Series Stub Matching Network
Solution-3
q The parameters related to main line electrical length can be obtain as
t = tan θm =
1 − rL
xL − xrL
= −0.17157 (60)
q The electrical length of the main line and the series stub can calculated as
θm = tan−1
(t) = 170.26◦
θs(short) = tan−1
(x) = 35.264◦
θs(open) = cot−1
(−x) = 125.26◦
q Corresponding physical length are given as
lm =
θm
360
λg = 0.47296λg ls(short) =
θs
360
λg = 0.097957λg
ls(open) =
θs
360
λg = 0.34796λg
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47. Circuit-EM Co-Design Lab
Single Series Stub Matching Network
Solution-3
q The second solution set with capacitive series stub can be written as
x = −
s
(1 − rL)2 + xL
2
rL
= −0.70711 Ω (61)
q The parameters related to main line electrical length can be obtain as
t = tan θm =
1 − rL
xL − xrL
= −5.8284 (62)
q The electrical length of the main line and the series stub can calculated
as
θm = tan−1
(t) = 99.736◦
θs(short) = tan−1
(x) = 144.74◦
θs(open) = cot−1
(−x) = 54.736◦
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48. Circuit-EM Co-Design Lab
Single Series Stub using Smith Chart
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49. Circuit-EM Co-Design Lab
Single Shunt Stub Matching Network
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50. Circuit-EM Co-Design Lab
Single Shunt Stub Matching Network
Figure 12: Single shunt stub matching network consist of shunt stub ( electrical length
θs = βls) having input admittance of jB and main line ( electrical length θm = βlm),
matches the complex load admittance YL = GL + jBL to a real source admittance
Y0 = 1/Z0.
q The system can be normalized with respect to real source admittance Y0
and the normalized parameters are given as
b = BZ0; yL = gL + jbL = YLZ0 = (GL + jBL)Z0 (63)
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51. Circuit-EM Co-Design Lab
Single Shunt Stub Matching Network
Figure 13: Normalized single shunt stub matching network, matches the complex load
admittance yL = gL + jbL to a real source admittance y0 = 1.
q The input admittance zin2 can be calculated as
yin2 = 1 − jb =
yL cos θm + j sin θm
jyL sin θm + cos θm
=
(gL + jbL) cos θm + j sin θm
j(gL + jbL) sin θm + cos θm
(64)
q Note that series stub matching (47) and shunt stub matching (64) are dual
problem and the solutions are similar to each other.
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52. Circuit-EM Co-Design Lab
Single Shunt Stub Matching Network
q The above equation (64) can be rewritten as
(1 − jb) (j(gL + jbL) sin θm + cos θm)
= (gL + jbL) cos θm + j sin θm
cos θm − (bL − bgL) sin θm + j ((gL + bbL) sin θm − b cos θm)
= gL cos θm + j (bL cos θm + sin θm)
(65)
q By separating the real and imaginary parts of (65), one can obtain
tan θm =
1 − gL
bL − gLb
(66)
tan θm =
bL + b
bbL − (1 − gL)
(67)
q By comparing the above equations one can write
1 − gL
bL − gLb
=
bL + b
bbL − (1 − gL)
(68)
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53. Circuit-EM Co-Design Lab
Single Shunt Stub Matching Network
q The the equation (68) can be rewritten as
(1 − gL)bbL − (1 − gL)2
= bL
2
+ bbL − gLbbL − gLb2
(69)
b2
=
(1 − gL)2
+ bL
2
gL
(70)
q The above equation has two solutions: one is capacitive shunt stub (b 0) and
other one is inductive shunt stub (b 0). The solution are given as
b = ±
s
(1 − gL)2 + bL
2
gL
(71)
q The parameters related to main line electrical length can be obtain as
t = tan θm =
1 − gL
bL − gLb
=
1 − gL
bL ∓
p
gL(1 − gL)2 + gLbL
2
(72)
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54. Circuit-EM Co-Design Lab
Single Shunt Stub Matching Network
q The electrical length of the main line and shunt line can be obtained as
θm = tan−1
(t) (73)
θs = tan−1
(b) for open stub
= cot−1
(−b) for short stub.
(74)
q If θm,s is negative then add 180◦
to it, when θm,s is in degree. The
physical lengths of the main line and shunt line can be obtained as
lm,s =
θm,s
360◦
λg. (75)
q Here λg is the guided wavelength of the transmission lines.
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55. Circuit-EM Co-Design Lab
Single Shunt Stub Matching Network
Problem-4
Design a single shunt stub matching network which matches load impedance
of ZL = 40 − j30 Ω to a source having internal impedance of ZS = 50 Ω at 1.2
GHz frequency.
Solution-4
q From the problem statement one can write
Z0 = 50 Ω,Y0 = 0.02 f, ZL = 40 − j30 Ω, yL = 0.8 + j0.6
gL = 0.8, bL = 0.6, f0 = 1.2 GHz
q The first solution set with capacitive shunt stub can be written as
b =
s
(1 − gL)2 + bL
2
gL
= 0.70711 Ω (76)
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56. Circuit-EM Co-Design Lab
Single Shunt Stub Matching Network
Solution-4
q The parameters related to main line electrical length can be obtain as
t = tan θm =
1 − gL
bL − bgL
= 5.8284 (77)
q The electrical length of the main line and the series stub can calculated as
θm = tan−1
(t) = 80.264◦
θs(open) = tan−1
(b) = 35.264◦
θs(short) = cot−1
(−b) = 125.26◦
q Corresponding physical length are given as
lm =
θm
360
λg = 0.22296λg ls(open) =
θs
360
λg = 0.097957λg
ls(short) =
θs
360
λg = 0.34796λg
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57. Circuit-EM Co-Design Lab
Single Shunt Stub Matching Network
Solution-4
q The second solution set with inductive shunt stub can be written as
b = −
s
(1 − gL)2 + bL
2
gL
= −0.70711 Ω (78)
q The parameters related to main line electrical length can be obtain as
t = tan θm =
1 − gL
bL − bgL
= 0.17157 (79)
q The electrical length of the main line and the series stub can calculated
as
θm = tan−1
(t) = 9.7356◦
θs(open) = tan−1
(x) = 144.74◦
θs(short) = cot−1
(−x) = 54.736◦
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58. Circuit-EM Co-Design Lab
Single Shunt Stub using Smith Chart
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59. National Institute of Technology
Rourkela
Thanks.