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|I_{2 = 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, Y_21, 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]

2. 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

3. For a two-port network to be reciprocal

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

4. 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

5. 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

6. 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]

7. 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]

8. 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]

9. For lattice type attenuator shown in the given figure, the characteristic impedance R_O is

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

10. 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]

2. (d)

Transmission parameters (ABCD-parameters) will be multiplied.

3. (b,c)

Y₂₁ = Y₁₂, H₂₁ = – H₁₂

4. (a)

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

5. (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.

6. (b)

let V_A be the node.
voltage of center node.
V_A = (V₁/2) + (V₂/3)/(1/2 + 1/2 + 1/3) = 3V₁/8 + V₂/4)
I₁ = V/1 + V₁-V_A/2
I₁ = 3/2V₁ – 1/2 (3/8V₁ + V₂/4)
I₁ = 21/16 V₁ – V₂/8……………….(i)
I₂ = V₂/3 + V₂ – V_A/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]

7. (d)

R_A = 2 x 1/6 = 1/3
R_B = 6/6 = 1
R_E = 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]

8. (a)

using KVL,
V₁(s)  = (1/sc + sL₁) I₁ (s) + sM I₂ (s)
V₂(s) = (R + sL₂) I₂ (s) + sM I₁(s)
Z₀ (s) = V_I(s)/I_I(s)|_{Ik (s) = 0}
Z (s) = [1/sc + sL₁  sM    sM    R + sL₂]

9. (c)

R₀ = R_SC R_OC
Since,   R_SC = 2R₁R₂/R₁ + R₂ and R_OC = R₁R₂/2
R₀ = R₁R₂

10. (c)

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