open circuit impedance parameters or z parameters of two port network definition meaning

z parameters of two port network definition meaning or what are open circuit impedance parameters ?
Two-port Network
Two-port Network  A two-port network consists of two pairs of terminals in which one pair of terminals is designated as input and other pair being output.
In these network, the driving source is connected at input port and output is taken at the other port of the network.
Z-Parameters
Z-parameters are also known as open circuit impedance parameters. here, input and output voltages are expressed in terms of input and output currents. the equations are
V1 = Z11I1 + Z12I2
V2 = Z21 I1 + Z22 I2
In matrix form it is represented as
[V] =  [Z] [I], where [Z] is the impedance matrix
[V1] = [Z11  Z12] [I1]
[V2] = [Z21  Z22] [I2]
Where, Z11, Z12, Z21 and Z22 are called the Z-parameters.
These parameters are defined below
(i) Z11 is the input impedance when output is open circuited,
Z11 = V1/I1|I2 = 0
(ii) Z12 is the reverse transfer impedance when input is open-circuited.
Z12 = V1/I1|I2 = 0
(iii) Z21 is the forward transfer impedance when output is open-circuit
Z21 = V2/I1|I2 = 0
(iv) Z22 is the output impedance when input is open-circuited.
Z22 = V2/I2|I1 = 0
Condition for reciprocal network
Z12 = Z21
Condition for symmetry
Z11 = Z22
Example 1. Determine the Z-parameters for network shown below.
Sol.  Applying KVL in the network, the loop equations are written as
V1 = I1 R1 + (I1 + I2) R3 = (R1 + R3) I1 + R3I2 …………..(i)
V2 = I2R2 + (I1 + I2) R3 = R3 I1 + (R2 + R3) I2 …………(ii)
Comparing eqs. (i) and (ii) with standard equation of Z-parameters, we get
Z11 = R1 + R3; Z12 = R3
Z21 = R3; Z22 = R2 + R3
Y-parameters
These are also called admittance parameters. these are obtained by expressing currents at two ports in terms of voltages at two ports. thus voltages V1 and V2 are independent variables, while I1 and I2 are dependent variables.
The equations are
I1 = Y11V1 + Y12V2
I2 = Y21V1 + Y22V2
In matrix form it is represented as
[I] = [Y][V], where [Y] is the admittance matrix.
[I1] = [Y11  Y12][V1]
[I2] = [Y21 Y22] [V2]
Where, Y11, Y12, Y21, Y22 are called the y-parameters these are defined below.
(i) short-circuit driving point input admittance
Y11 = I1/V1|V2 = 0
(ii) short-circuit forward transfer admittance
Y21 = I2/V1|V2 = 0
(iii) short-circuit reverse transfer admittance
Y12 = I2/V2|V1 = 0
(iv) short-circuit driving point output admittance
V22 = I2/V2|V1 = 0
h-parameters
These are also called hybrid parameters. these parameters are obtained by expressing voltage at input port and the voltage at the output port.
In equation form,
V1 = h11 I1 + h12 V2
I2 = h21 I1 + h22 V2
where, h11, h12, h21, h22 are called hybrid parameters.
In matrix form,
[V1] = [h11   h12][I1]
[I2] = [h21  h22] [V2]
h-parameters are defined below
(i) short-circuit input impedance
h11 = V2/I1|V2 = 0
(ii) Forward short-circuit current gain
h21 = I2/I1|v2 = 0
(iii) reverse open-circuit voltage gain
h12 = V1/V2|I1 = 0
(iv) Open-circuit output admittance
h22 = I2/V2|I1 = 0
ABCD-Parameters or Transmission Parameters or Chain Parameters
These are generlly used in the analysis of power transmission in which the input port is referred as the sending end while the output port is referred as receiving end
in equation form,
V1 = AV2 + B(-I2)………….(1)
I1 = CV2 + D (-I2) …………..(2)
In matrix form,
[V1] = [A  B][V2]
[I1] = [C  D] [-I2]
(i) open-circuit reverse voltage gain
A = V1/V2|I2 = 0
(ii) open-circuit transfer admittance
C = I1/V2|I2 = 0
(iii) short-circuit transfer impedance
B = V1/-I2|V2 = 0
(iv) short-circuit reverse current gain
D = I1/-I2 |V2 = 0
Intro Exercise – 8

  1. The open- circuit impedance matrix of the two-port network shown in figure is

(a) [-2  1]/[-8  3]
(b) [-2  -8]/[1  3]
(c) [0  1]/[1  0]
(d) [-2  -1]/[-1  3]

  1. Two two-port networks are connected in cascade. the combination is to represent as a single two-port networks. the parameters of the network are obtained by multiplying the individual

(a) Z-parameter matrix
(b) h-parameter matrix
(c) y-parameter matrix
(d) ABCD-paramter matrix

  1. For a two-port network to be reciprocal

(a) Z11 = Z22
(b) Y21 = Y12
(c) h21 = – h12
(d) AD – BC = 0

  1. The condition that a two-port network is reciprocal, can be expressed in terms of its ABCD-parameters as

(a) AD – BC = 1
(b) AD – BC = 0
(c) AD – BC > 1
(d) AD – BC < 1

  1. The short-circuit admittance matrix of a two-port network is

[0   -1/2]  [1/2    0]
the two-port network is
(a) non-reciprocal and passive
(b) non-reciprocal and active
(c) neciprocal and passive
(d) reciprocal and active

  1. What is the value of [Z] in the given figure?

(a) [21/16  1/8 / 1/8  7/12]
(b) [7/9    1/6 / 1/6     7/4]
(c) [21/16    – 1/8 / -1/8  7/12]
(d) 7/9   1/3   1/3   7/4]

  1. The circuit is given in figure

What is the value of [Z] ?
(a) [-1/2  -3/2   -17/6   1/2]
(b) [1/2   3/2   17/6  1/2]
(c) [-17/6    -1/2  -1/6   3/2]
(d) [17/6  1/2   1/6   3/2]

  1. For the network shown below z-parameters will be

(a) [1/sc +sL1   sM /sM  R + sL2]
(b) [/sc + sL  R1 – sM  -sM  R + sL2]
(c) [1/sc + sL1 + sM /1   R + sL2 + 1/SC]
(d) sL1 + 1/SC – 1   -1   sL2 + R]

  1. For lattice type attenuator shown in the given figure, the characteristic impedance RO is

(a) R1 + R2/2
(b) 2R1 + R2/R1 + R2
(c) R1R2
(d) R1 + R2/2

  1. Which one of the following parameters does not exist for the two-port network shown in the given figure?

(a) h
(b) y
(c) Z
(d) none of these
Answers with Solutions

  1. (a)

V1 = Z11I1 + Z12I2
V2 = Z21I1 + Z22I2
V1 = – 2I1 + I2
V2 = 2I2 – 6I1 – V1
V2 = – 8I1 + 3I2
[V1/V2] = [-2  1/-8  3] [I1/I2]
[Z] = [-2  1/-8   3]

  1. (d)

Transmission parameters (ABCD-parameters) will be multiplied.

  1. (b,c)

Y21 = Y12, H21 = – H12

  1. (a)

AD – BC = 1, is the condition of reciprocal for a two-port network.

  1. (b)

[Y] = [0  -1/2  1/2   0] (Y12 = Y21]
So, network is non-reciprocal because Y12 = Y21 and Y12 is negative that mean either energy storing or providing device availlable so network is acitive also.

  1. (b)

let VA be the node.
voltage of center node.
VA = (V1/2) + (V2/3)/(1/2 + 1/2 + 1/3) = 3V1/8 + V2/4)
I1 = V/1 + V1-VA/2
I1 = 3/2V1 – 1/2 (3/8V1 + V2/4)
I1 = 21/16 V1 – V2/8……………….(i)
I2 = V2/3 + V2 – VA/3 = 2V2/3 – 1/3 (3V1/8 + V2/4)
I2 = – V1/8 + 7/12 V2 ………………(ii)
[Y] = [21/16  -1/8/-1/8  7/12]
[Z] = [Y]1 = [21/16   -1/8    -1/8   7/12]-1
[Z]  = [7/9  1/6  1/6  7/4]

  1. (d)

RA = 2 x 1/6 = 1/3
RB = 6/6 = 1
RE = 3/6 = 1/2
-V1 + 7/3 I1 + 1/2 (I1 + I2) = 0
V1 = (7/3 + 1/2) I1 + I2/2
V1 = 17/6 I1 + I2/2 ………….(1)
-V2 + 1.I2 + 1/2 (I1 + I2) = 0
V2 = I1/2 + I2 (1 + 1/2)
V2 = I1/2 + 3/2 I2 ……………(2)
[Z] = [17/6  1/2  1/2  3/2]

  1. (a)

using KVL,
V1(s)  = (1/sc + sL1) I1 (s) + sM I2 (s)
V2(s) = (R + sL2) I2 (s) + sM I1(s)
Z0 (s) = VI(s)/II(s)|Ik (s) = 0
Z (s) = [1/sc + sL1  sM    sM    R + sL2]

  1. (c)

R0 = RSC ROC
Since,   RSC = 2R1R2/R1 + R2 and ROC = R1R2/2
R0 = R1R2

  1. (c)

y-parameter = 1/Z [1  -1  -1  1]
y = 0
then z-parameters cannot exist.