EC-542: Microwave Integrated Circuits

Lecture Module#2: Analysis of Some Common 3-port

& 4-Port Microwave Networks
Karun Rawat
Karun.rawat.in@ieee.org

IIT Roorkee
RF Group, Department of Electronics and Communications Engineering
Indian Institute of Technology Roorkee, India

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 Important
 To all members of Radio Amplifier and Power Transceiver Lab, RF Group,
I.I.T Roorkee.

 These slides have been prepared from the following text books and are
being used for class lecture/demonstration only:

 Gonzalez, Microwave Transistor Amplifiers: Analysis and Design

 D.M. Pozar, Microwave Engineering.

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 Design 1: T-Junction Power Divider (1/2)
 Lossless,
reciprocal 3-port
junction can not
be matched
simultaneously at
each port.

 The Input admittance at the junction is given by:

where, jB represents lumped susceptance corresponding to the stored

energy associated with fringing fields and higher order modes.
 If T.Ls are lossless and B=0 (by some reactive tuning):

Courtesy: D.M. Pozar, Microwave Engineering, Wiley. 3/40

 Design 1: T-Junction Power Divider (2/2)

 The Characteristic impedances are:

 The I/P impedance of junction:

 Looking into 150 Ω line: impedance is 50||75=30Ω; Looking into 75

Ω line: impedance is 50||150=37.5 Ω.
Courtesy: D.M. Pozar, Microwave Engineering, Wiley. 4/40
 Design 2: Resistive Divider (1/2)
 If a three-port divider
contains lossy components, it
can be made to be matched at
all ports, although the two
output ports may not be
isolated.

 If all ports are terminated in the characteristic impedance Z0, the

impedance Z, seen looking into the Z0/3 resistor followed by a
terminated output line, is:

 The input Impedance can be given by:

 The network is symmetric from all three ports (see the

geometry/topology), thus all O/P ports are matched. S11=S22=S33=05/40
Courtesy: D.M. Pozar, Microwave Engineering, Wiley.
 Design 2: Resistive Divider (1/2)
 If the voltage at port 1 is V1, then
by voltage div., the voltage V at
the centre of junction is:

 The O/P voltages are:

 Thus, S21=S31=S23=1/2, so the O/P powers are 6 dB below the I/P.

 The s-parameter matrix is given as:

 Note Isolation is poor, however

all ports are matched.
Courtesy: D.M. Pozar, Microwave Engineering, Wiley. 6/40
 Design 3: Wilkinson Power
Divider

If we consider all impedances are normalized w.r.t Z0

Port 2
V2

1 2V0
4
λ/
2,

Port 1
2
V1
1
2
,
λ/
4

V3
Courtesy: D.M. Pozar, Microwave Engineering, Wiley.
Port 3 7/40
 Design 3: Wilkinson Power
Divider : Even & Odd Mode Excitation
V2
Port 2
 The circuit can be divided into two
1 V0 V0 structure around graphical/physical
4
λ/
2, symmetry.
Port 1
2  Defining two separate modes of
V1
2
,
operations: Even Mode & Odd Mode
λ/
4
 Superposition of two modes
1
produces overall voltages at each
V3
-V0 V0
Port 3 nodes.
Port 2 Port 2
V2 V2

1 V0 1 V0
λ /4 1 4
λ/ 1
2 Port 1 2, Port 1 2,
2

V1 V1
2 2
, 1
λ/ 2
4 2
, 1
λ/
1 4

1
V3
-V0
Odd Mode Port 3
Even Mode V3
V0
Port 3
Courtesy: D.M. Pozar, Microwave Engineering, Wiley. 8/40
 Design 3: Wilkinson Power
Divider : Odd Mode Excitation
Port 2
V2
Port 2 V2o

1 V0 1 V0
4
λ/
λ/
4 1 2, 1
2 Port 1 2,

V1 2
2 2
, 1
λ/
4
2
1 2 1
,
λ/
4
V3 1
Port 3 -V0

V3o
Zin = Open Port 3 -V0

V2o
V3o  V2o
1 V0
4
λ/
2, 1

V2o
Open
1 V0 V
1 V2o  0
2
Courtesy: D.M. Pozar, Microwave Engineering, Wiley. 9/40
 Design 3: Wilkinson Power
Divider : Even Mode Excitation (1/2) Port 2
V2

1 V0
4
λ/ 1
Port 1 2,
2

V1

2 2
, 1
λ/
4

V3
Port 3 V0

V2e  Since Characteristic Impedance

4
1 V0 of λ/4 line is √2 , therefore, Zin=1.
λ/

V1e
2, Zin =1  This results in:
V2e
V0
Open V2e  V3e  V2e
1 V0 2
2
Zin =1

Courtesy: D.M. Pozar, Microwave Engineering, Wiley. 10/40

 Design 3: Wilkinson Power
Divider : Even Mode Excitation (2/2)
 
V2e

Γ
V  x   V  e jβx  Γe jβx
1 V0
4
, λ/

V1e
2
 Let x=0 at port 1 and x=-λ/4 at
port 2.
 λ V0
V 
V0 jV0
V2e  V     jV 1  Γ  
 
2
 4 2 j 2 1  Γ  2  Γ  1
jV0  Γ  1
 Then: V1e  V  0  V 
1  Γ  
2  Γ  1

 The reflection coefficient Γ is:

 jV0
 Therefore, V1
e

2

Courtesy: D.M. Pozar, Microwave Engineering, Wiley. 11/40

 Design 3: Wilkinson Power
Divider: After Superposition
The overall Node voltages at each port are calculated by superposition:
Port 2
V2
 jV0 jV0
1 2V0 V1  V1e  V1o  0  
2,
λ /4 2 2
Port 1
V0 V0
V1
2
V2  V2e  V2o    V0
1
2
,
2 2
λ/
4
V0 V0
1
V3  V3e  V3o   0
V3 2 2
Port 3

Courtesy: D.M. Pozar, Microwave Engineering, Wiley. 12/40

 Design 3: Wilkinson Power
Divider: Calculating S-parameters (1/3)
The overall Node voltages at each port are calculated by superposition:
Port 2
V2
 jV0 jV0
1 2V0 V1  V1e  V1o  0  
2,
λ /4 2 2
Port 1
V0 V0
V1
2
V2  V2e  V2o    V0
1
2
,
2 2
λ/
4
V0 V0
1
V3  V3e  V3o   0
V3 2 2
Port 3

Since ports 1 and 3 are terminated in matched loads, we know that the
incident wave on those ports are zero. As a result, the total voltage is
equal to the value of the exiting waves at those ports:
 jV0
V1  0; V1  V3  0; V3  0
2

Courtesy: D.M. Pozar, Microwave Engineering, Wiley. 13/40

 Design 3: Wilkinson Power
Divider: Calculating S-parameters (2/3)
Port 2
V2
Since port 2 is matched for both even
1
4
2V0
and odd mode. Therefore, this port is
λ/
2,
matched.
V2  V0 V2  0
Port 1
2
V1
1
2
,
λ/
Recalling from previous slides:
4
 jV0
V1  0; V1  V3  0; V3  0
1

V3 2
Port 3

Note: V1 ,V1 , V2 , V2 ,V3 , V3 are calculated at plane defining port 1, 2
,3 respectively. The S-parameters can be calculated as:
V1 j V3
S12   S32   0 S22  0 The Only
V2 2 V2 parameter left is
Using Bilateral Symmetry:1 1, 2 3, 3 2 S11.
j
S13   S23  0 S33  0
2 14/40
Courtesy: D.M. Pozar, Microwave Engineering, Wiley.
 Design 3: Wilkinson Power
Divider : Calculating S-Parameters (3/3)
Port 2
Port 2 V2
V2
1
1
4
4 λ/ 1
λ/ 2,
2,
2V0 2
Port 1 V1
2 Port 1
2V0 V1
1 2
,
λ/ 2
4 2V0
2
,
1 λ/ 1
4

V3 1
Port 3  Note: No even-odd V3
mode required here. Port 3

Port 2
V2

1  Since Source is
Zin=2
2,
λ/
4
matched at Port 1.
2V0 2 V1 2
V1  V0 V1  V0 V1  0
V1 2
2V0 S11  0
Courtesy: D.M. Pozar, Microwave Engineering, Wiley. 15/40
 Design 3: Wilkinson Power
Divider: S-Parameters
 From Previous slides
V1 j V3
S12   S32   0 S22  0
V2 2 V2
j
S13   S23  0 S33  0 S11  0
2
 From Reciprocity:
j j
S31  S13   S21  S12  
2 2
 j j 
 0   
 2 2
 j 
  
S  0 0 
 2 
 j 
 0 0 
Courtesy: D.M. Pozar, Microwave Engineering, Wiley.
 2  16/40
 Quadrature Hybrid: 4-Port N/W
 Quadrature Hybrid (Symmetric Coupler): θ=Ф=π/2; α=β=1/√2
0 α jβ 0 0 1 j 0
α jβ   
0 0 1  1 0 0 j 
 S    jβ 0 0 α
S    j 0 0 1
2
   
0 jβe α 0 0 j 1 0 

 Similar realization in Microstrip is branch-line coupler.

Courtesy: D.M. Pozar, Microwave Engineering, Wiley. 17/40

 Design of Branch-Line Coupler:
Even Mode

A1 is a wave of
unit amplitude

1 1
B1e  B4e  Γe B2e  B3e  Te
2 2 Courtesy: D.M. Pozar, Microwave Engineering, Wiley. 18/40
 Design of Branch-Line Coupler:
Odd Mode

1 1 1 1
B1o  Γo B2o  To B4o   Γo B3o   To
2 2 2 2
Courtesy: D.M. Pozar, Microwave Engineering, Wiley. 19/40
 Design of Branch-Line Coupler:
Overall Voltage Waves at Each Port

Courtesy: D.M. Pozar, Microwave Engineering, Wiley. 20/40

 Design of Branch-Line Coupler:
Overall Voltage Waves at Each Port

A  D 1
B  0; C  Y
 Also for odd-mode one can get

A  D  cos βl
Courtesy: D.M. Pozar, Microwave Engineering, Wiley. B  jZ0 sin βl; C  jY0 sin21/40
βl;
 Design of Branch-Line Coupler:
Overall Voltage Waves at Each Port
 Converting ABCD parameter to s-parameters (s-parameter is
equivalent to reflection and transmission coefficient)

22/40
 Design of Branch-Line Coupler:
S-Parameters
 Considering the voltage waves at each ports, we can obtain certain
sets of s-parameters depending on port of excitation.

w.r.t Input A1  10 at Port 1

Therefore, we retrieve S11,

S21, S31, S41.

Similarly by exciting port 4, we can retrieve S14, S24, S34, S44. The
analysis will be same as the circuit topology is quite similar in this case.
The other s-paramaters can also be obtained in a same manner and also
considering reciprocity.

23/40
 Application of Branch-Line Coupler:
Γ ΓTL

Ground
Port 1 Port 2
θ/2 Δϕ/2
ΓS1 Transmission
Input 0 j 1 0 Line
  S1

Ground
1  j 0 0 1 
S   
2 1 0 0 j
Output   S2
0 1 j 0
ΓS2

Ground
Port 4
θ/2 Δϕ/2
Port 3 Transmission
Γ ΓTL Line

When both switches changes

 e j  θ  π 
from OFF to ON state, the  Both S1& S 2 are ON
Γ
relative phase between Output & e j  θ  ΔΦ  Both S1& S 2 are OFF
Input changes by Δϕ.

24/40
 180o Hybrid: 4-Port N/W

0 1 1 0
 0 1
 j 1 0
 S   1 0 0 1
2
 
0 1 1 0
 Microstrip realization is also called rat-race coupler.
 Four port with 180O phase shift between
two output port.
 Signal applied to port 1 (Σ-port) split into
two in-phase component at ports 2 and
3.
 Signal applied to port 4 (Δ-port) split
into two out of phase component at
ports 2 and 3.
 As a combiner: if signals applied to ports
2 and 3, sum appears at Σ-port and
Courtesy: D.M. Pozar, Microwave Engineering, Wiley.
difference appears at Δ-port. 25/40
 180o Hybrid: Even Mode Analysis
Excitation port: Port 1

w.r.t Input
A1  10
at Port 1 (Σ port).

Therefore, we can retrieve S11, S21, S31, S41.

Courtesy: D.M. Pozar, Microwave Engineering, Wiley. 26/40

 180o Hybrid: Even Mode Analysis
Excitation port: Port 1

w.r.t Input
A1  10
at Port 1 (Σ port).

Courtesy: D.M. Pozar, Microwave Engineering, Wiley. 27/40

 180o Hybrid: Odd Mode Analysis
Excitation port: Port 1

w.r.t Input
A1  10
at Port 1 (Σ port).

Courtesy: D.M. Pozar, Microwave Engineering, Wiley. 28/40

 180o Hybrid: Superposition

Considering reciprocity, These results

These gives form the first row and column of the
S11, S21, S31, scattering matrix in
S41.
0 1 1 0 
 
 j 1
 S   1 

2
 
Courtesy: D.M. Pozar, Microwave Engineering, Wiley.  0  29/40
 180o Hybrid: Even Mode
Analysis (Excitation port: Port 4)

 Since Port 1 and 4 have

different role : Σ and Δ

Courtesy: D.M. Pozar, Microwave Engineering, Wiley. 30/40

 180o Hybrid: Even Mode
Analysis (Excitation port: Port 4)

w.r.t Input
A4  10
at Port 1 (Σ port).

Considering reciprocity,

These gives 0 1 1 0 
 1
S14, S24, S34,  S    j 1
S44. 2 1 1
 
 0 1 1 0 

Port 2 and 1 can be seen

Courtesy: D.M. Pozar, Microwave Engineering, Wiley. as reciprocal. 31/40
 180o Hybrid: Analysis
(Excitation port: Port 2 and 3)

Excitation at Port 3 will be handled with same analysis as in the case

where excitation was at port 1. This will give S33 and S23.Considering
reciprocity we can get S32 =S23 also.
0 1 1 0 
 
 j 1 0 0 1
 S   1 0 1
2
 
 0 1 1 0 
 Excitation at port 2, will be handled with same analysis as in the case
where excitation was at port 4. This will give S22.
0 1 1 0
 0 1
 j 1 0
 S   1 0 0 1
2
 
0 1 1 0
Courtesy: D.M. Pozar, Microwave Engineering, Wiley. 32/40
 Coupled Line Coupler

Even Mode Odd Mode

Input Impedance at port 1, can be expressed by superposition as:

Courtesy: D.M. Pozar, Microwave Engineering, Wiley. 33/40

 Coupled Line Coupler

Even Mode Odd Mode

From Symmetry we can say:

For Even Mode: I1e  I3e ; I 4e  I 2e ; V1e  V3e ; V4e  V2e

For Odd Mode: I1o   I3o ; I 4o   I 2o ; V1o  V3o ; V4o  V2o

Input Impedance at port 1, can be expressed by superposition as:

Courtesy: D.M. Pozar, Microwave Engineering, Wiley. 34/40

 Coupled Line Coupler

Even Mode Odd Mode

Courtesy: D.M. Pozar, Microwave Engineering, Wiley.

The Input Impedances for even and odd mode are:

The Voltages and currents for each mode are:

The Input Impedance at port 1:


35/40
 Coupled Line Coupler

Even Mode Odd Mode

The Input Impedance at port 1:



To match port 1: Zin=Z0 , This is possible if:

Z 0  Z 0e Z 0o

If port 1 is matched i.e. Zin=Z0 , Then V1=V0 and Voltage at port 3 is:

Courtesy: D.M. Pozar, Microwave Engineering, Wiley. 36/40

 Coupled Line Coupler

Even Mode Odd Mode

If port 1 is matched i.e. Zin=Z0 , Then V1=V0 and Voltage at port 3 is:

Using Following three conditions on above equations:

Z 0  Z 0e Z 0o

One can get:

Courtesy: D.M. Pozar, Microwave Engineering, Wiley. 37/40

 Coupled Line Coupler

Even Mode Odd Mode

If port 1 is matched i.e. Zin=Z0 , Then V1=V0 and Voltage at port 3 is:

Using Following equation in above expression

One can get:

Courtesy: D.M. Pozar, Microwave Engineering, Wiley. 38/40

 Coupled Line Coupler

Even Mode Odd Mode

Courtesy: D.M. Pozar, Microwave Engineering, Wiley.

Rearranging the above expression:

Similarly one can get:

If θ=90o, C =V3/V0 mid-band Coupling Coefficient and V2 / V0   j 1  C 2

39/40
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