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

V_{1} = Z_{11}I_{1} + Z_{12}I_{2}

V_{2} = Z_{21} I_{1} + Z_{22} I_{2}

In matrix form it is represented as

[V] = [Z] [I], where [Z] is the impedance matrix

[V_{1}] = [Z_{11} Z_{12}] [I_{1}]

[V_{2}] = [Z_{21} Z_{22}] [I_{2}]

Where, Z_{11}, Z_{12}, Z_{21} and Z_{22} are called the Z-parameters.

These parameters are defined below

(i) Z_{11 }is the input impedance when output is open circuited,

Z_{11} = V_{1}/I_{1}|I_{2 = 0}

(ii) Z_{12} is the reverse transfer impedance when input is open-circuited.

Z_{12} = V_{1}/I_{1}|_{I2 = 0}

(iii) Z_{21} is the forward transfer impedance when output is open-circuit

Z_{21} = V_{2}/I_{1}|_{I2 = 0}

(iv) Z_{22} is the output impedance when input is open-circuited.

Z_{22} = V_{2}/I_{2}|_{I1 = 0}

Condition for reciprocal network

Z_{12} = Z_{21}

Condition for symmetry

Z_{11} = Z_{22}

**Example 1**. Determine the Z-parameters for network shown below.

**Sol.** Applying KVL in the network, the loop equations are written as

V_{1} = I_{1} R_{1} + (I_{1} + I_{2}) R_{3} = (R_{1} + R_{3}) I_{1} + R_{3}I_{2} …………..(i)

V_{2} = I_{2}R_{2} + (I_{1} + I_{2}) R_{3} = R_{3} I_{1} + (R_{2} + R_{3}) I_{2} …………(ii)

Comparing eqs. (i) and (ii) with standard equation of Z-parameters, we get

Z_{11} = R_{1} + R_{3}; Z_{12} = R_{3}

Z_{21} = R_{3}; Z_{22} = R_{2} + R_{3}

**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 V_{1} and V_{2} are independent variables, while I_{1} and I_{2} are dependent variables.

The equations are

I_{1} = Y_{11}V_{1} + Y_{12}V_{2}

I_{2} = Y_{21}V_{1} + Y_{22}V_{2}

In matrix form it is represented as

[I] = [Y][V], where [Y] is the admittance matrix.

[I_{1}] = [Y_{11} Y_{12}][V_{1}]

[I_{2}] = [Y_{21} Y_{22}] [V_{2}]

Where, Y_{11}, Y_{12}, Y_{21,} Y_{22} are called the y-parameters these are defined below.

(i) short-circuit driving point input admittance

Y_{11} = I_{1}/V_{1}|_{V2 = 0}

(ii) short-circuit forward transfer admittance

Y_{21} = I_{2}/V_{1}|_{V2 = 0}

(iii) short-circuit reverse transfer admittance

Y_{12} = I_{2}/V_{2}|_{V1 = 0}

(iv) short-circuit driving point output admittance

V_{22} = I_{2}/V_{2}|_{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,

V_{1} = h_{11} I_{1} + h_{12} V_{2}

I_{2} = h_{21} I_{1} + h_{22} V_{2}

where, h_{11}, h_{12}, h_{21}, h_{22} are called hybrid parameters.

In matrix form,

[V_{1}] = [h_{11} h_{12}][I_{1}]

[I_{2}] = [h_{21} h_{22}] [V_{2}]

h-parameters are defined below

(i) short-circuit input impedance

h_{11 }= V_{2}/I_{1}|_{V2 = 0}

(ii) Forward short-circuit current gain

h_{21} = I_{2}/I_{1}|_{v2 = 0}

(iii) reverse open-circuit voltage gain

h_{12} = V_{1}/V_{2}|_{I1 = 0}

(iv) Open-circuit output admittance

h_{22} = I_{2}/V_{2}|_{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,

V_{1} = AV_{2} + B(-I_{2})………….(1)

I_{1} = CV_{2} + D (-I_{2}) …………..(2)

In matrix form,

[V_{1}] = [A B][V_{2}]

[I_{1}] = [C D] [-I_{2}]

(i) open-circuit reverse voltage gain

A = V_{1}/V_{2}|_{I2 = 0}

(ii) open-circuit transfer admittance

C = I_{1}/V_{2}|_{I2 = 0}

(iii) short-circuit transfer impedance

B = V_{1}/-I_{2}|_{V2 = 0}

(iv) short-circuit reverse current gain

D = I_{1}/-I_{2} |_{V2 = 0}

**Intro Exercise – 8**

- 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]

- 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

- For a two-port network to be reciprocal

(a) Z_{11} = Z_{22}

(b) Y_{21} = Y_{12}

(c) h_{21} = – h_{12}

(d) AD – BC = 0

- 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

- 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

- 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]

- 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]

- For the network shown below z-parameters will be

(a) [1/sc +sL_{1} sM /sM R + sL_{2}]

(b) [/sc + sL R_{1} – sM -sM R + sL_{2}]

(c) [1/sc + sL_{1} + sM /1 R + sL_{2} + 1/SC]

(d) sL_{1} + 1/SC – 1 -1 sL_{2} + R]

- For lattice type attenuator shown in the given figure, the characteristic impedance R
_{O }is

(a) R_{1} + R_{2}/2

(b) 2R_{1} + R_{2}/R_{1} + R_{2}

(c) R_{1}R_{2}

(d) R_{1} + R_{2}/2

- 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**

- (a)

V_{1} = Z_{11}I_{1} + Z_{12}I_{2}

V_{2} = Z_{21}I_{1} + Z_{22}I_{2}

V_{1} = – 2I_{1} + I_{2}

V_{2} = 2I_{2} – 6I_{1} – V_{1}

V_{2} = – 8I_{1} + 3I_{2}

[V_{1}/V_{2}] = [-2 1/-8 3] [I_{1}/I_{2}]

[Z] = [-2 1/-8 3]

- (d)

Transmission parameters (ABCD-parameters) will be multiplied.

- (b,c)

Y_{21} = Y_{12}, H_{21} = – H_{12}

- (a)

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

- (b)

[Y] = [0 -1/2 1/2 0] (Y_{12} = Y_{21}]

So, network is non-reciprocal because Y_{12} = Y_{21} and Y_{12} is negative that mean either energy storing or providing device availlable so network is acitive also.

- (b)

let V_{A} be the node.

voltage of center node.

V_{A} = (V_{1}/2) + (V_{2}/3)/(1/2 + 1/2 + 1/3) = 3V_{1}/8 + V_{2}/4)

I_{1} = V/1 + V_{1}-V_{A}/2

I_{1} = 3/2V_{1} – 1/2 (3/8V_{1} + V_{2}/4)

I_{1} = 21/16 V_{1} – V_{2}/8……………….(i)

I_{2} = V_{2}/3 + V_{2} – V_{A}/3 = 2V_{2}/3 – 1/3 (3V_{1}/8 + V_{2}/4)

I_{2} = – V_{1}/8 + 7/12 V_{2} ………………(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]

- (d)

R_{A} = 2 x 1/6 = 1/3

R_{B} = 6/6 = 1

R_{E} = 3/6 = 1/2

-V_{1} + 7/3 I_{1} + 1/2 (I_{1} + I_{2}) = 0

V_{1} = (7/3 + 1/2) I_{1} + I_{2}/2

V_{1} = 17/6 I_{1} + I_{2}/2 ………….(1)

-V_{2} + 1.I_{2} + 1/2 (I_{1} + I_{2}) = 0

V_{2} = I_{1}/2 + I_{2} (1 + 1/2)

V_{2} = I_{1}/2 + 3/2 I_{2} ……………(2)

[Z] = [17/6 1/2 1/2 3/2]

- (a)

using KVL,

V_{1}(s) = (1/sc + sL_{1}) I_{1 }(s) + sM I_{2 }(s)

V_{2}(s) = (R + sL_{2}) I_{2} (s) + sM I_{1}(s)

Z_{0} (s) = V_{I}(s)/I_{I}(s)|_{Ik (s) = 0}

Z (s) = [1/sc + sL_{1} sM sM R + sL_{2}]

- (c)

R_{0} = R_{SC} R_{OC}

Since, R_{SC} = 2R_{1}R_{2}/R_{1} + R_{2} and R_{OC} = R_{1}R_{2}/2

R_{0} = R_{1}R_{2}

- (c)

y-parameter = 1/Z [1 -1 -1 1]

y = 0

then z-parameters cannot exist.