Calculate Z parameter for the circuit given below

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Calculate Z parameter for the circuit given below—

1. 14 8
  13 13
  216 314
  117 117
2. 13 13
  14 8
  117 117
  216 314
3. 12 9
  13 13
  116 114
  117 117
4. None of these

Correct Option: B

The given circuit can be redrawn as shown below

Applying KVL in loop 1.
V₁ = 2 (I₁ – I) ................(i)
KVL in loop 2
– 3I + 2V₁ – (I₂ + I) 4 + 2 (I₁ – I) = 0
or 9I = 2V₁ – 4 I₂ + 2I₁

or I =	2V₁ + 2I₁ – 4I₂	...................(ii)
	9

From equations (i) and (ii)

V₁ = 2I₁ – 2	(2V₁ + 2I₁ – 4I₂)
	9

or 9 V₁		1 +	4		= 18I₁ – 4I₁ + 8I₂
			9

or 13V₁ = 14 I₁ + 8I₂

or V₁ =	14	I₁ +	8	I₂ ............(iii)
	13		13

We know that
V₁ = Z₁₁ I₁ + Z₁₂ I₂ ..........(iv)
(for Z-parameter)
Comparing equations (iii) and (iv), we get

Z₁₁ =	14	, Z₁₂ =	8
	13		13

Applying KVL in loop 3.
V₂ = 4 (I₂ + I)

or V₂ = 4 I₂ + 4		2V₁ + 2I₁ – 4I₂
		9

[from equation (ii)]

or V₂ = 4 I₂ +	8	V₁ +	8	I₁	–16	I₂
	9		9		9

or V₂ = 4 I₂ +	8		14	I₁ +	8	I₂		+	8	I₁	–16	I₂
	9		13		13				9		9

or V₂ = I₂		4 +	64	–	16		+ I₁		112	+	8
			117		9				117		9

or V₂ = I₂	(466 + 64 – 208)	+ I₁		112 + 104
	117			117

or V₂ =	314	I₂ +	216	I₁ .......(v)
	117		117

We know that
V₂ = Z₂₁ I₁ + Z₂₂ I₂ (for Z-parameters) ..............(vi)
Comparing equations (v) and (vi), we get

Z₂₁ =	216	, Z₂₂ =	314
	117		117

Hence alternative (A) is the correct choice.

12	9
13	13
116	114
117	117