Indraprastha Institute of

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Lecture – 14 Date: 18.09.2014

• L – Type Matching Network

• Examples
• Nodal Quality Factor
• T- and Pi- Matching Networks
• Microstrip Matching Networks
• Series- and Shunt-stub Matching
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L – Type Matching Network (contd.)
Various configurations

Their usefulness is regulated by the specified source and load

impedances and the associated matching requirements
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L – Type Matching Network (contd.)
Design procedure for two element L – Type matching Network
1. Find the normalized source and load impedances.
2. In the Smith chart, plot circles of constant resistance and conductance
that pass through the point denoting the source impedance.
3. Plot circles of constant resistance and conductance that pass through the
point of the complex conjugate of the load impedance (zM = zL*).
4. Identify the intersection points between the circles in steps 2 and 3. The
number of intersection points determine the number of possible L-type
matching networks.
5. Find the values of normalized reactances and susceptances of the
inductors and capacitors by tracing a path along the circles from the
source impedance to the intersection point and then to the complex
conjugate of the load impedance.
6. Determine the actual values of inductors and capacitors for a given
frequency.
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Example – 1
Using the Smith chart, design all possible configurations of discrete two-
element matching networks that match the source impedance ZS = (50+j25)Ω
to the load ZL = (25-j50)Ω. Assume a characteristic impedance of Z0 = 50Ω and
an operating frequency of f = 2 GHz

Solution
1. The normalized source and load impedances are:

zS  Z S / Z 0  1  j 0.5 yS  0.8  j 0.4

zS  Z L / Z 0  0.5  j1 yS  3  j 0.8
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Example – 1 (contd.)
2. Mark zS and then plot (zL)*=0.5+j1
circles of constant
resistance and
conductance that passes
through zS. A D

zS=1+j0.5

B
3. Mark (zL)* and then plot C
circles of constant
resistance and
conductance that passes
through (zL)*.
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Example – 1 (contd.)
4. The intersection points of these circles are A, B, C and D with the
normalized impedances and inductances as:
z A  0.5  j 0.6 y A  0.8  j1 z B  0.5  j 0.6 yB  0.8  j1

zC  1  j1.2 yC  3  j 0.5 z D  1  j1.2 yD  3  j 0.5

5. There are four intersection points and therefore four L-type matching
circuit configurations are possible.
zS → zA → (zL)* Shunt L, Series L
zS → zB → (zL)* Shunt C, Series L

zS → zC → (zL)* Series C, Shunt L

zS → zD → (zL)* Series L, Shunt L
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Example – 1 (contd.)
6. Find the actual values of the components
In the first case:
zS → zA : the normalized admittance is changed by
Z0
jbL2  y A  yS   j 0.6  L2    6.63nH
bL2 
zA → zL : the normalized impedance is changed by
jxL1   z L   z A  j 0.4
* xL1 Z 0
 L1   1.59nH


Therefore the circuit is:

zS → zA → (zL)*
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Example – 1 (contd.)
Similarly:

zS → zB → (zL)* zS → zC → (zL)*

zS → zD → (zL)*
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Forbidden Region, Frequency Response, and Quality Factor
Self Study - Section 8.1.2 in the Text Book

• The L-type matching networks can be considered as resonance circuits

with 𝑓0 being the resonance frequency.
• These networks can be described by a loaded quality factor, 𝑄𝐿 , given by:
f0 Similarity to
QL 
BW bandpass filter

• However, analysis of matching circuit based on bandpass filter concept is

complex → In addition, it only allows approximate estimation of the
bandwidth.
• More simpler and accurate method is design and analysis through the use
of nodal quality factor, 𝑄𝑛 .
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Nodal Quality Factor
• During L-type matching network analysis it was apparent that at each
node the impedance can be expressed in terms of equivalent series
impedance 𝑍𝑠 = 𝑅𝑠 + 𝑗𝑋𝑠 or admittance 𝑌𝑃 = 𝐺𝑃 + 𝑗𝐵𝑃 .
• Therefore, at each node we can define 𝑄𝑛 as • Similarly,
the ratio of the absolute value of reactance
𝑋𝑠 to the corresponding resistance 𝑅𝑠 . BP
Qn 
XS GP
Qn 
RS

• The “nodal quality factor” and loaded quality factor are related as:
True for any L-type matching network
Q
QL  n
2 For more complicated networks, QL = Qn

• BW of the matching network can be easily estimated once the “nodal

quality factor” is known.
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Nodal Quality Factor (contd.)

• To simplify the matching network design process even further, we can
draw constant 𝑄𝑛 contours in the Smith chart.
• We know the Smith chart expression:
1   2r  i2 2i
z  r  jx   j
(1   r ) 2  i2 (1   r ) 2   i2

• The expression for 𝑄𝑛 :

x 2 i
Qn  z
r 1   2r  i2

2
 1  1
i2    r    1 
𝑸𝒏 circles!  Qn  Qn2

𝑄𝑛 circles in the Smith chart help in direct

determination of 𝑄𝐿 .
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Nodal Quality Factor (contd.)

• Once you go through section 8.1.2, it will be apparent that quality factor of
matching network is extremely important.
• For example, broadband amplifier requires matching circuit with low-Q.
Whereas oscillators require high-Q networks to eliminate undesired
harmonics in the output signal.
• It will also be apparent that L-type matching networks have no control over
the values of 𝑄𝑛 → Limitation!!!
• To gain more freedom in choosing the values of Q or Qn, another element
in the matching network is incorporated → results in T- or Pi-network
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T- and Pi- Matching Networks

• The knowledge of nodal quality factor (𝑄𝑛 ) of a network enables
estimation of loaded quality factor → hence the Band Width (BW).
• The addition of third element into the matching network allows control of
𝑄𝐿 by choosing an appropriate intermediate impedance.

Example – 2
• Design a T-type matching network that transforms a load impedance ZL =
(60 – j30)Ω into a Zin = (10 + j20)Ω input impedance and that has a
maximum Qn of 3. Compute the values for the matching network
components, assuming that matching is required at f = 1GHz.
Solution
• Several possible configurations! Let us focus on just one!
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Example – 2 (contd.)
General Topology of a T-matching Network Gives the
name T

• First element in series (Z1) is purely reactive, therefore the combined

impedance of (ZL and Z1) will reside on the constant resistance circle of r = rL
• Similarly, Z3 (that is purely reactive!) is connected in series with the input,
therefore the combined impedance ZB (consisting of ZL, Z1, and Z2) lies on
the constant resistance circle r = rin
• Network needs to have a Qn of 3 → we should choose impedance in such a
way that ZB is located on the intersection of constant resistance circle r = rin
and Qn = 3 circle → helps in the determination of Z3
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Example – 2 (contd.)
• The constant resistance circle of
zin intersects the 𝑄𝑛 = 3 circle at
point B. This gives value of 𝑍3 . zin
• The constant resistance circle
𝑟 = 𝑟𝐿 and a constant
conductance circle that passes
through B helps in the zL
determination of and allows
determination 𝑍2 and 𝑍1 .

Final solution at 1 GHz

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Example – 3
• For a broadband amplifier, it is required to develop a Pi-type matching
network that transforms a load impedance ZL = (10 – j10)Ω into an input
impedance of Zin = (20 + j40)Ω. The design should involve the lowest
possible Qn. Compute the values for the matching network components,
assuming that matching is required at f = 2.4GHz.
Solution
• Several Configurations possible (including the forbidden!). One such is below:

Z2
Z3 Z1

• Since the load and source impedances are fixed, we can’t develop a
matching network that has Qn lower than the values at locations ZL and Zin
• Therefore in this example, the minimum value of Qn is determined at the
input impedance location as Qn = |Xin|/Rin = 40/20 = 2
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Example – 3 (contd.)
• In the design, we first plot
constant conductance circle g = zin
gin and find its intersection with
Qn=2 circle (point B) →
determines the value of Z3
• Next find the intersection point
(labeled as A) of the g=gL circle
and constant-resistance circle zL
that passes through B →
determines value of Z2 and Z1
Final solution at 2.4 GHz
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Example – 3 (contd.)
• It is important to note that the relative positions of Zin and ZL allows only
one optimal Pi-type network for a given specification.
• All other realizations will result in higher Qn → essentially smaller BW!
• Furthermore, for smaller ZL the Pi-matching isn’t possible!

It is thus apparent that BW can’t be enhanced arbitrarily by reducing

the 𝑄𝑛 . The limits are set by the desired complex 𝑍𝑖𝑛 and 𝑍𝐿 .

With increasing frequency and correspondingly reduced wavelength

the influence of parasitics in the discrete elements are noticeable →
distributed matching networks overcome most of the limitations (of
discrete components) at high frequency
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Microstrip Line Matching Networks
• In the lower RF region, its often a standard practice to use a hybrid
approach that combines lumped and distributed elements.
• These types of matching circuits usually contain TL segments in series and
capacitors in shunt.

• Inductors are avoided in these designs as they tend to have higher resistive
losses as compared to capacitors.
• In principle, only one shunt capacitor with two TL segments connected in
series on both sides is sufficient to transform any given load impedance to
any input impedance.
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Microstrip Line Matching Networks (contd.)
• Similar to the L-type matching network, these configurations may also
involve the additional requirement of a fixed Qn, necessitating additional
components to control the bandwidth of the circuit.
• In practice, these configurations are extremely useful as they permit
tuning of the circuits even after manufacturing → changing the values of
capacitors as well as placing them at different locations along the TL
offers a wide range of flexibility → In general, all the TL segments have
the same width to simplify the actual tuning →the tuning ability makes
these circuits very appropriate for prototyping.
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Example – 4
Design a hybrid matching network that transforms the load ZL = (30 + j10) Ω to
an input impedance Zin = (60 + j80) Ω. The matching network should contain
only two series TL segments and one shunt capacitor. Both TLs have a 50Ω
characteristic impedance, and the frequency at which the matching is required
is f = 1.5 GHz

Solution
• Mark the normalized load impedance (0.6 + j0.2) on the Smith chart.
• Draw the corresponding SWR circle.
• Mark the normalized input impedance (1.2 + j1.6) on the Smith chart.
• Draw the corresponding SWR circle.
• The choice of the point from which we transition from the load SWR circle
to the input SWR circle can be made arbitrarily.
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Example – 4 (contd.)
Normalized load
to the point A
gives length of
point B to
the first segment
normalized input
of TL
impedance gives
length of the
second segment
of TL

A to B provides
the necessary
susceptance
value for the
shunt capacitor