Two Port Network Parameters

2-Port Network Parameters

Mir Muhammad Lodro

Lecturer
Department of Electrical Engineering
Sukkur IBA

Contents
 Introduction
 Two Port Networks
 Z Parameters
 Y Parameters
 S Parameters
 Return Loss
 Insertion Loss
 Transmission (ABCD) Matrix
 VNA Calibration
 SOLT Structure
Microwave and Radar Engineering

Two Port Networks
 Linear networks can be completely characterized by parameters
measured at the network ports without knowing the content of the
networks.
 Networks can have any number of ports.
 Analysis of a 2-port network is sufficient to explain the theory and applies
to isolated signals (no crosstalk).

I1 I2
+ +

Port 2
Port 1

2 Port
V1 V2
-
Network -

 The ports can be characterized with many parameters (Z, Y, S,
ABDC). Each has a specific advantage.
 Each parameter set is related to 4 variables:
 2 independent variables for excitation
 2 dependent variables for response


Z Parameters
Impedance Matrix: Z Parameters for 2-port Network

V1   Z11 Z12   I1 
V    Z  I  V   Z I 
 2   21 Z 22   2 
V1  Z11I1  Z12 I 2
V2  Z 21I1  Z 22 I 2
 Advantage: Z parameters are intuitive.
 Relates all ports to an impedance & is easy to calculate.
 Disadvantage: Requires open circuit voltage measurements, which are difficult
to make.
 Open circuit reflections inject noise into measurements.
 Open circuit capacitance is non-trivial at high frequencies.


Y Parameters
Admittance Matrix: Y Parameters for 2-port Networks

 I1  Y11 Y12  V1  I   Y V 
 I   Y Y  V 
 2   21 22   2 
I1  Y11V1  Y12V2
I 2  Y21V1  Y22V2
 Advantage: Y parameters are also somewhat intuitive.
 Disadvantage: Requires short circuit voltage measurements, which are
difficult to make.
 Short circuit reflections inject noise into measurements.
 Short circuit inductance is non-trivial at high frequencies.


Example- Z-parameters

ZA ZB
+ + V1  Z11I1  Z12 I 2
I1 I2
V2  Z 21I1  Z 22 I 2
Port 1

Port 2
V1 ZC V2

- -

V1 V1 V1 I 2 ZC
Z11 
I1

V1
 Z A  ZC Z12    ZC
I 20
Z A  ZC
I2 I10
I2

V2 I1 Z C V2 V2
Z 21    ZC Z 22    Z B  ZC
I2 V2
I1 I 2 0
I1 I10
Z B  ZC

Frequency Domain: Vector Network Analyzer
(VNA)

 VNA offers a means to characterize circuit elements as a function of
frequency.
 VNA is a microwave based instrument that provides the ability to
understand frequency dependent effects.
 The input signal is a frequency swept sinusoid.
 Characterizes the network by observing transmitted and reflected power
waves.
 Voltage and current are difficult to measure directly.
 It is also difficult to implement open & short circuit loads at high frequency.
 Matched load is a unique, repeatable termination, and is insensitive to length,
making measurement easier.
 Incident and reflected waves the key measures.
 We characterize the device under test using S parameters.

S Parameters
a1 a2
+ +

Port 2
Port 1
2 Port
V1 b1 b2 V2
-
Network -

 We wish to characterize the network by observing transmitted and
reflected power waves.
 ai represents the square root of the power wave injected into port i.

V1
ai  P  P V2
R R
 bi represents the square root of the power wave injected into port j.

V j
bj 
R

S Parameters
a1 a2
 We can use a set of linear equations
+ +
to describe the behavior of the

Port 2
Port 1
2 Port
network in terms of the injected and V1 b1 b2 V2
reflected power waves. -
Network -

b1  S11a1  S12a2
bj
power measured at port j
b2  S 21a1  S 22a2 Sij  
ai power measured at port i

 For the 2 port case:

 b1   S11 S12   a1 
b   S S 22  a2 
 2   21  


Scattering Matrix – Return Loss
 S11, the return loss, is a measure of
the power returned to the source.

 When there is no reflection from the
load, or the line length is zero, S11 is
equal to the reflection coefficient.

V1
b Z0 V1 Vreflected Z  50
S11  1      0  0
a1 a 2 0
V1 V1 Vincident Z 0  50
Z0

In general:

bi
Sii   0
ai a j 0 , j 0


Scattering Matrix – Return Loss
 When there is a reflection from the load, S11 will be composed of
multiple reflections due to standing waves.
 Use input impedance to calculate S11 when the line is not
perfectly terminated.

1   ( z  0)
Z in  Z ( z  0)  Z o
1   ( z  0)
RS = 50 S11 for a transmission line will exhibit
periodic effects due to the standing waves.

Zin  If the network is driven with a 50 source, S11
is calculated as follows:

Z in  50
S11   v 
Z in  50
In this case S11 will be maximum when Zin is real. An imaginary component implies a
phase difference between Vinc and Vref. No phase difference means they are perfectly
aligned and will constructively add.

Scattering Matrix – Insertion Loss
a1 a2
+ +

Port 2
Port 1
Z0
2 Port
V1 b1 b2 Z0 V2
-
Network -

 When power is injected into Port 1 and measured at Port 2, the power
ratio reduces to a voltage ratio:

V2
b2 Z o V2  Vtransmitted
S 21      
a1 a 20 V1 V1 Vincident
Zo
 S21, the insertion loss, is a measure of the power
transmitted from port 1 to port 2.

S Parameters
Sij = Gij is the reflection coefficient of the ith
port if i=j with all other ports matched

Sij = Tij is the forward transmission coefficient
of the ith port if I>j with all other ports
matched

Sij = Tij is the reverse transmission coefficient b  S a
of the ith port if I<j with all other ports
matched

Vi 
bi
Sij  Sij 
bi

Z 0i
aj aj V j
a k 0 , k  j a k 0 , k  j
Z0 j
Vk 0 ,k  j



Comments on Losses
 True losses come from physical energy losses.
 Ohmic (i.e. skin effect)
 Field dampening effects (loss tangent)
 Radiation (EMI)

 Insertion and return losses include other effects, such as
impedance discontinuities and resonance, which are not true
losses.

 Loss free networks can still exhibit significant insertion and
return losses due to impedance discontinuities.


Reflection Coefficients
 Reflection coefficient at the load:

Z L  Z0
L 
Z L  Z0
 Reflection coefficient at the source:
Z S  Z0
S 
Z S  Z0
 Input reflection coefficient:

S12 S 21 L S12  L
2
in  S11   S11  Assuming S12 = S21 and S11 = S22.
1  S 22  L 1  S11 L
 Output reflection coefficient:
S12 S 21 S
 out  S 22 
1 S11 S

Transmission Line Z0 Measurements
 Impedance vs. frequency
 Recall
1  e 2 jl
Z in  Z 0
1  e 2 jl
 Zin vs f will be a function of delay () and ZL.
 We can use Zin equations for open and short circuited lossy
transmission.

Z in,open  Z 0 tanh l 
Z 0  Z in, shortZ in,open
Z in,short  Z 0 coth l 
Using the equation for Zin, rin, and Z0,
we can find the impedance.


Advantages/Disadvantages of S Parameters
Advantages:
 Ease of measurement: It is much easier to measure
power at high frequencies than open/short current and
voltage.

Disadvantages:
 They are more difficult to understand and it is more
difficult to interpret measurements.


Transmission (ABCD) Matrix
 The transmission matrix describes the network in terms of both voltage
and current waves (analagous to a Thévinin Equivalent).
V1  AV2  BI 2 I1 I2

I1  CV2  DI 2 + +

Port 1

Port 2
V1
2 Port V2
Network
V1 A B V2 - -

I1 C D I2
 The coefficients can be defined using superposition:

V1 V1
A B
V2 I 2 0 I2 V2  0

I1
C D
I1
V2 I 2 0 I2 V2  0

Transmission (ABCD) Matrix
 Since the ABCD matrix represents the ports in terms of currents and
voltages, it is well suited for cascading elements.
I1 I2 I3
+ A B A B +
V1 V2 V3
C D1 C D2
- -
I1

 The matrices can be mathematically cascaded by multiplication:
V1 A B V2
 
I1 C D 1 I2 V1 A B A B V3
  
V2 A B V3 I1 C D 1 C D 2 I 3
 
I2 C D 2 I3

 This is the best way to cascade elements in the frequency domain.
 It is accurate, intuitive and simple to use.

Converting to and from the S-Matrix

 The S-parameters can be measured with a VNA, and
converted back and forth into ABCD, the Matrix
 Allows conversion into a more intuitive matrix
 Allows conversion to ABCD for cascading
 ABCD matrix can be directly related to several useful
circuit topologies


Advantages/Disadvantages of ABCD Matrix

Advantages:
 The ABCD matrix is intuitive: it describes all ports with voltages and
currents.
 Allows easy cascading of networks.
 Easy conversion to and from S-parameters.
 Easy to relate to common circuit topologies.

Disadvantages:
 Difficult to directly measure: Must convert from measured
scattering matrix.


VNA Calibration

 Proper calibration is critical!!!
 There are two basic calibration methods
 Short, Open, Load and Thru (SOLT)
 Calibrated to known standard( Ex: 50)
 Measurement plane at probe tip
 Thru, Reflect, Line(TRL)
 Calibrated to line Z0
 Helps create matched port condition.


SOLT Calibration Structures

OPEN SHORT

S S

Signal
G G

Ground

LOAD THRU

S S S

G G G

Calibration Substrate


Thanks


Two Port Network Parameters

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