Unit  5
Two Port Networks
Open circuit impedance parameters
The Z parameters of a twoport for the positive directions of voltages and currents may be defined by expressing the port voltages V1 and V2 in terms of the currents I1 and I2. Here V1 and V2 are dependent variables, and I1, I2 are independent variables. The voltage at port 11’ is the response produced by the two currents I1 and I2. Thus,
Zparameter:
v1 = Z11I1 + Z12I2
v2 = Z11I1 + Z22I2
=
Z11 = I2=0
Z12 = I1=0
Z21 = I2=0
Z22 = I1=0
I1 and I2 are excitations at port 1 & 2 respectively.
V1 and V2 are the responses at port 1 and 2 respectively.
Equivalent circuit:
Symmetry:
I2=0 =I1 = 0
Z11 = Z22
Reciprocal two port N/W:
I2=0= I1=0
Z12 = Z21
I1& I2 should be independent
Que 1.
Find Zparameter
Solution: I1 = I2
Current dependent so Zparameter doesn’t exist
Que 2.
Find zparameter
Solution: V1 =R (I1 + I2)
V2 = R (I1 + I2)
Z11 = Z12 = Z21 = Z22 = R
Que 3.
Solution: V1 = I1Za + I1Zc + I2Zc
= (Za + Zc)I1 + ZcI2
V2 = I2Zb + I2Zc + I1Zc
= (Zb + Zc)I1 + ZcI1
Z11 = (Za + Zc)
Z12 = Zc = Z21
Z22 = (Zb + Zc)
Que 4.
Solution: V1 = Za(I1  I)
(I  I1)Za+ IZc+ Zb(I + I2) = 0
I(Za + Zb + Zc) – I1Za + I2Zb = 0
I =
V1 = ZaI1  Za
= I1 + I2
V2 = Zb(I2 + I)
= ZbI2 + Zb
= I2 + I2
Z11 =
Z12 = Z21 =
Z22 =
Can be solved by YA conversion
Que 1.
Solution:
Z11 = I2=0
V1  (Za + Zb) = 0
= Z11 =
Z21 = I2=0
V2  Zb +Za = 0
=
Z12 =
Z22 =
Que 2.
Find Z21?
Solution: Z21 = I2=0
I1/2 =
= I1
V2 = I1/2
= × I1
= I1
Z21 = I2 = 0 = I1 Ω
Que 3.
Solution: + + 2V1 = 0
2Vx – 2V1 + Vx – V2 + 2V1 = 0
3Vx = V2
Vx =
I1 = V1 +
= 3V1 – 2Vx
= 3V1 – 2
V1 =
+ 2V2 = I2
3V2  = I2
V2 = I2
V2 = I2
V1 =
=
Z11 =
Z12 =
Z21 = 0
Z22 =
Short circuit Admittance parameters
Yparameter:
I1 = Y11V1 + Y12V2
I2 = Y21V1 + Y22V2
Y11 = V2=0
Y12 = V1=0
Y21 = V2=0
Y22 = V1=0
V1& V2 should be independent
Equivalent circuit: 
Symmetrical two port N/W: 
I2 = 0 = I1=0
Y12 = Y22
Reciprocal two port N/W: 
I1=0 =I2 = 0
Y12 = Y21
Que 1.
Solution: V1 – I1R – V2 = 0
V1 – V2 = I1R
I1 = V1  V2
V2 = I2R + V1
I2 =  V1 + V2
Y11 =
Y12 = Y21 =
Y22 =
Que 2.
Solution: Yparameter does not exist as V1 = V2
Que 3.
Solution: I1 = V1Ya + (V1 – V2)Yc
I1 = (Ya + Yc)V1  YcV2
I2 = V2Yb + (V2 – V1)Yc
I1 = (Yb + Yc)V2  YcV1
Y11 = Yb + Yc
Y12 = Y21 =  Yc
Y22 = Yb + Yc
Que 4.
Solution:
Que 5:
Find Yparameter?
Solution:
I1 + – 2V2 + + 2V1 = 0
I1= V1 + V1  3V2 + 2V1
I1= 4V1  3V2V
V2 + 2V2 2V1 = 2(I2 + 2V1)
 2V1 + 3V2 = 2(I2 + 2V1)
3V2  2V1 – 4V1 = 2I2
I2 = 3V1 + V2
Y11 = 4
Y12 = 3
Y21 = 3
Y22 =
Transmission parameter [ABCD]
V1 = AV2  BI2
I1 = CV2  DI2
A = I2=0
B = V2=0
C = I2=0
D = V2=0
Symmetrical two port N/W :
I2 = 0 = I1=0
A = D
Reciprocal two port N/W :
I1=0 =I2 = 0
AD – BC = 1
Que 1.
Solution:
3V1 – I1 + + = 0
+ = I1
V1 – V2 = I1
V1 = V2 + I1
V1= V2 I1(1)
I2 = 3V1 + V2 + V2 – V1
I2 = 2V1 + 2V2
2V1 = I2  2V2
2V1 =  2V2 + I2
V1 = V2 + I2 (2)
A = 1
B =
From (1) & (2)
V2 + I2 = I1  V2
V2  V2 + I2 = I1
I1 = V2 + V2  I2
I1 = V2  I2
C =
D =
Que 2.
Solution: V1 = RI1 + V2 (1)
I2R = V2 – V1
I2 = V2  V1
I2R = V2 – V1
V1 = I2R  V2 (2)
A = 1
B = R
From (2) in (1)
V2  I2R = V2 + RI1
I2 = I1
C = 0
D = 1
Que 3.
Solution: V1 = R(I1 + I2)
V2 = R(I1 + I2)
V1 = V2 + 0I2
A = 1
B = 0
V2 = RI1 + RI2
RI1= V2  RI2
I1 = V2 – I2
C = , D = 1
Hparameter:
V1 = h11I1 + h12V2
I2 = h21I1 + h22V2
Equivalent circuit for Hparameter:
Fig: Equivalent circuit for Hparameter
h11 = V2=0
h12 = I1=0
h21 = V2=0
h22 = I1=0
Symmetrical two port N/W: 
I2 = 0 = I1=0
∆h = 0
Reciprocal two port N/W: 
I1=0 =I2 = 0
h12 = h21
Que. Find all hparameter?
Solution: + = I1
V1  = I1
V1 = + I1
V1 = + I1
 + 3I1 = I2
3I1 + V2 = I2
From (1)
I2 = 3I1 + V2 – [ I1 + V2]
I2 = I1 + V2
h11 =
h12 =
h21 =
h22 =
Key takeaway
Relationship of twoport variables
Fig: 2 Port network
Fig: 1 port network
Fig: 3 port network
zparameter open circuit impedance
yparameter short circuit admittance
hparameter hybrid parameter
gparameter inverse hybrid parameter
ABCDparameter transmission parameter
A’B’C’D’ parameter inverse transmission
sparameter scattering
Used for very high frequency application
Relationship of Two Part Variables
Condition for Reciprocity and Symmetry
Sr No.  Parameter  Reciprocity  Symmetry 
1.  Z Parameter  Z12 = Z21  Z11= Z22 
2.  Y Parameter  Y12 = Y21  Y12 = Y21 
3.  h Parameter  h12 = h21  Δ = 1 
4.  ABCD Parameter  ADBC = 1  A = D 
5.  Inverse h Parameter  g12 = g21  Δ = 1 
6.  Inverse Transmission  A1D1 – B1 C1= 1  A1 = D1 
Series connection
V1 = V1a +V1b
I1 = I1a = I1b
V2 = V2a +V2b
I2 = I2a = I2b
=
=
=
Parallel connection
V1 = V1a = V1b
=
=
=
Series parallel connection
I1 = I1a = I1b
V1 = V1a +V1b
If two 2ports are connected in series parallel then overall hparameter is sum of individual hparameter
=
Parallel series connection
=
Cascade connection
V1 = V1a , V2a = V1b , V2 = V2b
I1 = I1a , I2a = I1b , I2 = I2b
Transmission parameter for N/W (A)
Transmission parameter for N/W (B)
Overall transmission parameter
Que 1.
Find out overall transmission parameter?
Solution:
Que 2.
Find overall Yparameter?
Solution:
Key takeaway
Relation between Two Port Parameters:
Δ = X11 X22 – X12 X21 , ΔT = AD – BC
 [z]  [y]  [h]  [T] 
[z]  Z11 Z12  y22 – y12 Δy Δy
 Δh – h12 h22 h22  A – ΔT C C 
Z21 Z22  y21 – y11 Δy Δy
 h21 h h22 h22
 1 D C C  





[y]  Z22 Z12 Δz Δz
 y11 y12  1 h12 h11 h11
 D – ΔT B B 
Z21 Z11 Δz Δz  y21 y22  h21 Δn h11 h11  1 A B B
 





[h]  Δz Z12 Z22 Z22
 1 y12 y11 y11
 h11 h12  B ΔT D D 
Z21 1 Z22 Z22  y21 Δy y11 y11  H21 h22  1 C D D
 
[T]  Z21 ΔZ Z21 Z21  y22 1 y21 y21  Δh h11 h21 h21
 A B 
1 Z22 Z21 Z21
 Δy y11 y21 y21
 h22 1 h21 h21  C D 
One of the common fourterminal twoport network is the lattice, or bridge network shown in Figure. Lattice networks are used in filter sections and are also used as attenuaters. Lattice structures are sometimes used in preference to ladder structures in some special applications. Za and Zd are called series arms, Zb and Zc are called the diagonal arms. It can be observed that, if Zd is zero, the lattice structure becomes a psection. The lattice network is redrawn as a bridge network as shown in Figure.
Z11 = I2=0
When
If the network is symmetric, then and
When is the voltage across
Substituting the value of from Eq, we have
If the network is symmetric,
When the input port is open,
The network can be redrawn as shown in Fig.
Substituting the value of in Eq, we get
If the network is symmetric,
We have
If the network is symmetric,
From the above equations,
And
Q) Obtain the lattice equivalent of a symmetrical T network figure?
A) A twoport network can be realised as a symmetric lattice if it is reciprocal and symmetric. The z parameter of the network.
Z11= 3
Z12=Z21= 2
Z22=3
Since, Z11=Z22
Z12=Z21
The given network is symmetrical and reciprocal. The parameters of the lattice network are
Za=Z11Z12=1
Zb=Z11+Z12=5
The lattice network is shown below
Q) Obtain the lattice equivalent of a symmetric pnetwork shown in Figure?
A) The Z parameters of the given network are
Z11=6 =Z22
Z12=Z21=4
Hence, the parameters of the lattice network are
Za=Z11Z12=2
Zb=Z11+Z12= 10
For this type of network series are represented as impedance and about cum represent admittance.
Now we can find the transfer function of above network KCL and KVL and then substituting the value in each equation the equation is reached which relative the output to input
Z=
Here, first Y6 is converted as Z6(=1 / Y6) Then combined with Z5 and this is called continued function method.
TNetworks
The circuit for T network is shown below
The z and y parameters of such networks are
Z1+Z3 = z11= I2=0
Z3=z12 = I1=0
Z3=z21 = I2=0
Z2+Z3=z22 = I1=0
As we see z12=z21 T network is always a reciprocal network
The yparameters can be found as
z= z11z22z12z21
y11= z22/z = Y1(Y2+Y3)/Y1+Y2+Y3
y22= z11/z = Y2(Y1+Y3)/Y1+Y2+Y3
y12=y21= z12/z = Y1Y2/Y1+Y2+Y3
Delta or π Network
A general π network is shown below. The y and z parameters can be found by short circuiting output and input ports respectively.
Ya+ Yb =y11 = V2=0
Yb=y12 = V1=0
Yb=y21 = V2=0
Yb+ Yc=y22 = V1=0
Since, y12=y21 the z parameters for the π network can be found by taking
Za=1/Ya
Zb=1/Yb
Zc=1/Yc
y= y11y22y12y21
z11=y22/y= Za (Zb +Zc)/Za +Zb +Zc
z22=y11/y= Zc (Zb +Za)/Za +Zb +Zc
z12=z21=y12/y= Za Zc/Za +Zb +Zc
References:
 Engineering Circuit Analysis”, by W H Hayt, TMH Eighth Edition
 “Network analysis and synthesis”, by F F Kuo, John Weily and Sons, 2nd Edition.
 “Circuit Theory”, by S Salivahanan, Vikas Publishing House 1st Edition, 2014
 “Network analysis”, by M. E. Van Valkenburg, PHI, 2000
 “Networks and Systems”, by D. R. Choudhary, New Age International, 1999
 Electric Circuit”, Bell Oxford Publications, 7th Edition.