Chapter 18
Two-Port Circuits
18.1
18.2
18.3
The Terminal Equations
The Two-Port Parameters
Analysis of the Terminated Two-Port
Circuit
18.4
Interconnected Two-Port Circuits
1
Motivation

Thévenin and Norton equivalent circuits are
used in representing the contribution of a circuit
to one specific pair of terminals.

Usually, a signal is fed into one pair of
terminals (input port), processed by the system,
then extracted at a second pair of terminals
(output port). It would be convenient to relate
the v/i at one port to the v/i at the other port
without knowing the element values and how
they are connected inside the “black box”.
2
How to model the “black box”?
Source
(e.g.
CD
player)

Load
(e.g.
speaker)
We will see that a two-port circuit can be
modeled by a 22 matrix to relate the v/i
variables, where the four matrix elements can
be obtained by performing 2 experiments.
3
Restrictions of the model

No energy stored within the circuit.

No independent source.

Each port is not a current source or sink, i.e.
i1  i1, i2  i2 .

No inter-port connection, i.e. between ac, ad,
bc, bd.
4
Key points

How to calculate the 6 possible 22 matrices of
a two-port circuit?

How to find the 4 simultaneous equations in
solving a terminated two-port circuit?

How to find the total 22 matrix of a circuit
consisting of interconnected two-port circuits?
5
Section 18.1
The Terminal Equations
6
s-domain model

The most general description of a two-port
circuit is carried out in the s-domain.

Any 2 out of the 4 variables {V1, I1, V2, I2} can be
determined by the other 2 variables and 2
simultaneous equations.
7
Six possible sets of terminal equations (1)
 V1   z11 z12   I1 
  ; Z  is the impedance matrix;
   

 V2   z21 z22   I 2 

  I1    y11 y12   V1 ; Y   Z -1 is the admittance matrix;
  I 2   y21 y22  V2 

 V1   a11  a12  V2 
  ; A is a transmission matrix;
   

  I1  a21  a22   I 2 

 V2    b11  b12   V1 ; B   A-1 is a transmission matrix;
  I 2  b21  b22   I1 

8
Six possible sets of terminal equations (2)
 V1   h11 h12   I1 
  ; H  is a hybrid matrix;
   

  I 2  h21 h22  V2 

  I1    g11 g12   V1 ; G   H -1 is a hybrid matrix;
 V2   g 21 g 22   I 2 


Which set is chosen depends on which
variables are given. E.g. If the source voltage
and current {V1, I1} are given, choosing
transmission matrix [B] in the analysis.
9
Section 18.2
The Two-Port Parameters
1.
2.
Calculation of matrix [Z]
Relations among 6 matrixes
10
Example 18.1: Finding [Z] (1)

Q: Find the impedance matrix [Z] for a given
resistive circuit (not a “black box”):
V1   z11
V    z
 2   21

z12   I1 
 

z22   I 2 
By definition, z11 = (V1/I1) when I2 = 0, i.e. the
input impedance when port 2 is open.  z11=
(20 )//(20 ) = 10 .
11
Example 18.1: (2)

By definition, z21= (V2/I1) when I2 = 0, i.e. the
transfer impedance when port 2 is open.

When port 2 is open:
15 

V2  5   15  V1  0.75V1 ,
V
V
 1  z11  10 ,  I1  1 ,
 I1
10 
V2
0.75V1
 z21 

 7.5 .
I1 V1 (10 )
12
Example 18.1: (3)

By definition, z22= (V2/I2) when I1 = 0, i.e. the
output impedance when port 1 is open.  z22 =
(15 )//(25 ) = 9.375 .

z12 = (V1/I2) when I1 = 0, 
20 

V1  20   5  V2  0.8V2 ,
V
V2
 2  z22  9.375 ,  I 2 
,
 I 2
9.375 
V1
0.8V2
 z12  
 7.5 .
I 2 V2 (9.375 )
13
Comments

When the circuit is well known, calculation of
[Z] by circuit analysis methods shows the
physical meaning of each matrix element.

When the circuit is a “black box”, we can
perform 2 test experiments to get [Z]: (1) Open
port 2, apply a current I1 to port 1, measure the
input voltage V1 and output voltage V2. (2) Open
port 1, apply a current I2 to port 2, measure the
terminal voltages V1 and V2.
14
Relations among the 6 matrixes

If we know one matrix, we can derive all the
others analytically (Table 18.1).

[Y]=[Z]-1, [B]=[A]-1, [G]=[H]-1, elements between
mutually inverse matrixes can be easily related.

E.g.
 z11
z
 21
z12   y11


z22   y21
1
y12 
1  y22  y12 

,



y22 
y  y21 y11 
where y  detY   y11 y22  y12 y21.
15
Represent [Z] by elements of [A] (1)

[Z] and [A] are not mutually inverse, relation
between their elements are less explicit.

By definitions of [Z] and [A],
V1   z11
V    z
 2   21
z12   I1 
  ,

z22   I 2 
V1   a11  a12  V2 
  ,
 I   a

 1   21  a22   I 2 
the independent variables of [Z] and [A] are {I1,
I2} and {V2, I2}, respectively.

Key of matrix transformation: Representing the
distinct independent variable V2 by {I1, I2}.
16
Represent [Z] by elements of [A] (2)

By definitions of [A] and [Z],
V1  a11V2  a12 I 2  (1)

 I1  a21V2  a22 I 2  ( 2)
1
a22
(2)  V2 
I 2  z21 I1  z22 I 2  (3),
I1 
a21
a21
 1
a22 
(1), (3)  V1  a11 
I1 
I 2   a12 I 2
a21 
 a21

 a11a22
a11
I1  

 a12  I 2  z11 I1  z12 I 2  ( 4)
a21

 a21
 z11

 z21
z12  1 a11 a 
, where a  detA.




z22  a21  1 a22 
17
Section 18.3
Analysis of the Terminated
Two-Port Circuit
1.
2.
Analysis in terms of [Z]
Analysis in terms of [T][Z]
18
Model of the terminated two-port circuit

A two-port circuit is typically driven at port 1 and
loaded at port 2, which can be modeled as:

The goal is to solve {V1, I1, V2, I2} as functions
of given parameters Vg, Zg, ZL, and matrix
elements of the two-port circuit.
19
Analysis in terms of [Z]

Four equations are needed to solve the four
unknowns {V1, I1, V2, I2}.
 V1  z11 I1  z12 I 2  (1)
 two - port equations

V2  z21 I1  z22 I 2  ( 2)
V1  Vg  I1Z g  (3)
 constraint equations due to terminations

V2   I 2 Z L  ( 4)
 1 0 z11
 0 1 z
21

0 Zg
1

1
0
0
z12  V1   0 
{V1, I1, V2, I2} are
z22  V2   0 
      ,
0   I1  Vg  derived by inverse
    
Z L   I 2   0  matrix method.
20
Thévenin equivalent circuit with respect to port 2

Once {V1, I1, V2, I2} are solved, {VTh, ZTh} can
be determined by ZL and {V2, I2}:
ZL

V2  Z  Z VTh  (1)

Th
L

 I 2  V2  VTh  ( 2)

ZTh
Z L

1
 V2  VTh  V2 Z L  VTh   Z L
;  
   


I 2   ZTh   V2   ZTh   1
1
 V2 
V2 Z L 
.



I2 
 V2 
21
Terminal behavior (1)

The terminal behavior of the circuit can be
described by manipulations of {V1, I1, V2, I2}:
V1
z12 z21
;
 z11 
 Input impedance: Z in 
I1
z22  Z L
 z21Vg
;
 Output current: I 2 
( z11  Z g )( z22  Z L )  z12 z21
I2
z21
;

 Current gain:
I1
z22  Z L

V
Voltage gains:  2 
z21Z L
;
z11Z L  z
 V1

z21Z L
V2 
;
Vg ( z11  Z g )( z22  Z L )  z12 z21
22
Terminal behavior (2)
z21
Vg ;
 Thévenin voltage: VTh 
z11  Z g
z12 z21
;
 Thévenin impedance: Z Th  z22 
z11  Z g
23
Analysis in term of a two-port matrix [T][Z]

If the two-port circuit is modeled by [T][Z],
T={Y, A, B, H, G}, the terminal behavior can be
determined by two methods:

Use the 2 two-port equations of [T] to get a new
44 matrix in solving {V1, I1, V2, I2} (Table 18.2);

Transform [T] into [Z] by Table 18.1, borrow the
formulas derived by analysis in terms of [Z].
24
Example 18.4: Analysis in terms of [B] (1)

Q: Find (1) output voltage V2, (2,3) average
powers delivered to the load P2 and input port
P1, for a terminated two-port circuit with known
Rg
[B].
Vg
RL
  20 3 k -b12
B  
 -b

2
mS
0
.
2
 22

25
Example 18.4 (2)

Use the voltage gain formula of Table 18.2:
bZ L
V2

;
Vg b12  b11Z g  b22 Z L  b21Z g Z L
b  b11b22  b12b21  ( 20)( 0.2)  ( 3 k)( 2 mS)  4  6  2,

V2
( 2)(5 k)
10
 ,

Vg ( 3 k)  ( 20)(0.5 k)  ( 0.2)(5 k)  ... 19
10
 V2  5000  263.160 V.
19
26
Example 18.4 (3)

The average power of the load is formulated by
2
1 V2
1 263.160 V

 6.93 W.
P2 
5 k 
2 RL
2
2


The average power delivered to port 1 is
1 2
formulated by P1  I1 ReZ in .
2
( 0.2)(5 k)  (3 k)
V b Z b
 133.33 ;
Z in  1  22 L 12 
( 2 mS)(5 k)  20
I1 b21Z L  b11
5000 V

 0.7890 A,
I1 
Z g  Z in (500 )  (133.33 )
Vg
1
 P1  (0.789) 2 (133.33)  41.55 W.
2
27
Section 18.4
Interconnected Two-Port
Circuits
28
Why interconnected?

Design of a large system is simplified by first
designing subsections (usually modeled by
two-port circuits), then interconnecting these
units to complete the system.
29
Five types of interconnections of two-port circuits
a. Cascade: Better
use [A].
b. Series: [Z]
c. Parallel: [Y]
d. Series-parallel: [H].
e. Parallel-series: [G].
30
Analysis of cascade connection (1)

Goal: Derive the overall matrix [A] of two
cascaded two-port circuits with known
transmission matrixes [A'] and [A"].
A
A
Overall two-port
circuit [A]=?
31
Analysis of cascade connection (2)
 V1 
V2
V1 
   A    A    (1)
 I1
 I 2 
 I1 
  a12
  V2 
V1
V2   a11
   A    
   ,
  a22
   I 2 
 I1 
 I 2  a21

  V2 
 a12
 V1   a11
V2 
 
     A     (2)

   I 2 
 I1   a21
 I 2 
a22
V2 
V2 
V1 
By (1), (2),     A  A      A  ,
 I 2 
 I 2 
 I1 
 a11
  a12
 a21

 a11  a12   a11
 A   A A , 

 a11
  a22
 a21

a21  a22  a21
 a12
  a12
 a22
 ) 
 (a11
.
 a12
  a22
 a22
 ) 32
 (a21
Key points

How to calculate the 6 possible 22 matrices of
a two-port circuit?

How to find the 4 simultaneous equations in
solving a terminated two-port circuit?

How to find the total 22 matrix of a circuit
consisting of interconnected two-port circuits?
33
Practical Perspective
Audio Amplifier
34
Application of two-port circuits

Q: Whether it would be safe to use a given audio
amplifier to connect a music player modeled by
{Vg = 2 V (rms), Zg = 100 } to a speaker modeled
by a load resistor ZL = 32  with a power rating of
100 W?
35
Find the [H] by 2 test experiments (1)
V1   h11
 Definition of hybrid matrix [H]:    
 I 2  h21

h12   I1 
   ;
h22  V2 
Test 1:
I1= 2.5 mA (rms)
V2 = 0 (short)
V1= 1.25 V (rms)
I2= 3.75 A
(rms)
1.25 V
V1
Input

 500 .
V1  h11 I1 ,  h11 
impedance
I1 V 0 2.5 mA
2
3.75 A
I2
Current

 1500. gain
I 2  h21 I1 ,  h21 
I1 V 0 2.5 mA
2
36
Find the [H] by 2 test experiments (2)
V1   h11
 Definition of hybrid matrix [H]:    
 I 2  h21

h12   I1 
   ;
h22  V2 
Test 2:
I1 = 0 (open)
V2 = 50 V (rms)
V1 = 50 mV
(rms)
I2 = 2.5 A
(rms)
V1
V1  h12V2 ,  h12 
V2
I2
I 2  h22V2 ,  h22 
V2
I1  0
I1  0
50 mV

 103.
50 V
Voltage
gain
2.5 A
-1 Output

 ( 20 ) . admittance
50 V
37
Find the power dissipation on the load

For a terminated two-port circuit:
the power dissipated on ZL is
PL  Re V I   Re (  I 2 Z L ) I   I 2 ReZ L ,
*
2 2
*
2
2
where I2 is the rms output current phasor.
38
Method 1: Use terminated 2-port eqs for [H]

By looking at Table 18.2:
I2 
h21Vg
(1  h22 Z L )( h11  Z g )  h12h21Z L
 1.98 A (rms),
103 
 h11 h12  500 
where 
;


-1 
h21 h22   1500 ( 20 ) 
Vg  2 V (rms), Z g  100 , Z L  32 .
Not safe!
 PL  I 2 ReZ L   (1.98) 2 (32)  126 W  100 W.
2
39
Method 2: Use system of terminated eqs of [Z]

Transform [H] to [Z] (Table 18.1):
 z11
z
 21

0.02
z12  1  h h12   470
.






z22  h22   h21 1   30,000 20 
By system of terminated equations:
V1   1 0 z11
V   0  1 z
21
 2  
0 Zg
 I1   1
  
1
0
 I2   0
1
z12 
 0   1.66 V 
 0    63.5 V
z22 
    
.
0
Vg   3.4 mA 


  
ZL 
0
1
.
98
A

  
 PL  I 2 ReZ L   (1.98)2 (32)  126 W  100 W.
2
40