Two Networks

1. Two Two-port networks are connected in cascade. The combination is to the represented as a single two – port network, by multiplying the individual (a) z-parameter matrices (c) y-parameter matrices (b) h-parameter matrices (d) ABCD parameter [GATE 1991: 2 Marks]

Soln. ABCD parameters relate the voltage and current at one port to voltage and current at the other port

Option (d)

2. For a 2-port network to be reciprocal, (a) z11 = z22 (c) h21 = -h12 (b) y21 = y12 (d) AD – BC = 0 [GATE 1992: 2 Marks]

Soln. 풚ퟐퟏ = 풚ퟏퟐ

풉ퟐ = −풉ퟏퟐ

Option (b) & (c)

3. The condition, that a 2-port network is reciprocal, can be expressed in terms of its ABCD parameters as……. [GATE 1994: 1 Mark]

Soln. AD – BC = 1

4. In the circuit of figure, the equivalent impedance seen across terminals A, B is

A

2 4 j 2

Zeq a b -j 2

2 4 B

(a) (16/3) Ω (c) (8/3 + 12j) Ω (b) (8/3) Ω (d) None of the above [GATE 1997: 2 Marks]

Soln. The product of the opposite arms are equal, so the bridge is balanced.

The point a and b are at the same potential

풁풆풒 = (ퟐ‖ퟒ) + (ퟐ‖ퟒ)

ퟒ ퟒ ퟖ = + = ퟑ ퟑ ퟑ Option (b)

5. The short-circuit admittance of a two-port network is 0 −1⁄ [ 2] 1 ⁄2 0 The two – port network is (a) non – reciprocal and passive (c) reciprocal and passive (b) non – reciprocal and active (d) reciprocal and active [GATE 1998: 1 Marks]

Soln. 풚ퟏퟐ ≠ 풚ퟐퟏ so the network is nonreciprocal. And active networks are nonreciprocal

Option (b)

6. A 2-port network is shown in the figure. The parameter h21 for this network can be given by

I1 I2 + + R R

V1 R V2

- - (a) -1/2 (c) -3/2 (b) +1/2 (d) +3/2 [GATE 1999: 1 Mark]

Soln. 푽ퟏ = 풉ퟏퟏ푰ퟏ + 풉ퟏퟐ푽ퟐ

푰ퟐ = 풉ퟐퟏ푰ퟏ + 풉ퟐퟐ푽ퟐ

푰ퟏ 풉ퟐퟏ = |푽ퟐ=ퟎ 푰ퟐ

푽ퟐ = 푹푰ퟐ + 푹(푰ퟏ + 푰ퟐ)

Or ퟐ푹푰ퟐ + 푹푰ퟏ = 푽ퟐ

When 푽ퟐ = ퟎ, ퟐ푹푰ퟐ + 푹푰ퟏ = ퟎ 푰 −푹 ퟏ So, ퟐ = = − 푰ퟏ ퟐ푹 ퟐ Option (a)

7. The admittance parameter Y12 in the 2-port network in figure is

I1 20 I2

E1 5 5 E2

(a) -0.2 nho (c) -0.05 mho (b) 0.1 mho (d) 0.05 mho [GATE 2001: 1 Mark]

ퟏ Soln. 풚 = − = −ퟎ. ퟎퟓ 풎풉풐풔 ퟏퟐ ퟐퟎ

Option (c)

8. The Z parameters Z11 and Z21 for the 2-port network in the figure are

I1 2 I2 4

E1 E2

10E1

−6 16 6 −16 (a) 푍 = Ω, 푍 = Ω (c) 푍 = Ω, 푍 = Ω 11 11 21 11 11 11 21 11 6 4 4 4 (b) 푍 = Ω, 푍 = Ω (d) 푍 = Ω, 푍 = Ω 11 11 21 11 11 11 21 11 [GATE 2001: 2 Marks]

Soln. For z – parameters

푬ퟏ = 풁ퟏퟏ푰ퟏ + 풁ퟏퟐ푰ퟐ

푬ퟐ = 풁ퟐퟏ푰ퟏ + 풁ퟐퟐ푰ퟐ

Writing KVL in LHS loop

푬ퟏ = ퟐ푰ퟏ + ퟒ푰ퟏ + ퟒ푰ퟐ − ퟏퟎ푬ퟏ

Or ퟏퟏ푬ퟏ = ퟔ푰ퟏ + ퟒ푰ퟐ − − − −(푰)

푬ퟏ ퟔ 풁ퟏퟏ = |푰ퟐ=ퟎ = Ω 푰ퟏ ퟏퟏ

푬ퟏ ퟒ 풁ퟏퟐ = |푰ퟏ=ퟎ = Ω 푰ퟐ ퟏퟏ

Writing KVL in RHS Loop

푬ퟐ = ퟒ(푰ퟏ + 푰ퟐ) − ퟏퟎ푬ퟏ − − − − − (푰푰) ퟔ푰 +ퟒ푰 Substituting 푬 = ퟏ ퟐ 풊풏 풆풒풖풂풕풊풐풏 − −(푰푰푰) ퟏ ퟏퟏ

ퟏퟎ(ퟔ푰 +ퟒ푰 ) 푬 = ퟒ(푰 + 푰 ) − ퟏ ퟐ ퟐ ퟏ ퟐ ퟏퟏ

푬ퟐ −ퟏퟔ 풁ퟐퟏ = |푰ퟐ=ퟎ = Ω 푰ퟏ ퟏퟏ

Option (c)

9. The Z11 and Z12 of the two-port network in the figure are

2 2 2 1 2

1 1

1 2

(a) 푍11 = 2.75Ω 푎푛푑 푍12 = 0.25 Ω (c) 푍11 = 3 Ω 푎푛푑 푍12 = 0.25Ω (b) 푍11 = 3 Ω 푎푛푑 푍12 = 0.5Ω (d) 푍11 = 2.25Ω 푎푛푑 푍12 = 0.5 Ω [GATE 2003: 2 Marks]

Soln. Using ∆ − 풀 conversion, the circuit reduces to

2 0.5 0.5 2

Z1 Z2

Z3 0.25

풁ퟏퟏ = 풁ퟏ + 풁ퟑ = ퟐ. ퟓ + ퟎ. ퟐퟓ = ퟐ. ퟕퟓΩ

풁ퟏퟐ = 풁ퟑ = ퟎ. ퟐퟓΩ

Option (a)

10. The h parameter of the circuit shown in the figure are

I1 I 10 2 + +

V1 20 V2

- -

0.1 0.1 30 20 (a) [ ] (c) [ ] −0.1 0.3 20 30 10 −1 10 1 (b) [ ] (d) [ ] 1 0.05 −1 0.05 [GATE 2005: 2 Marks]

Soln. 푽ퟏ = 풉ퟏퟏ푰ퟏ + 풉ퟏퟐ푽ퟐ

푰ퟐ = 풉ퟐퟏ푰ퟏ + 풉ퟐퟐ푽ퟐ Writing KVL in LHS and RHS Loop

푽ퟏ = ퟏퟎ푰ퟏ + ퟐퟎ(푰ퟏ + 푰ퟐ) − − − − − −(풊)

푽ퟐ = ퟐퟎ(푰ퟐ + 푰ퟏ) − − − − − −(풊풊)

Or 푽ퟏ = ퟏퟎ푰ퟏ + 푽ퟐ

푽ퟏ 풉ퟏퟏ = |푽ퟐ=ퟎ = ퟏퟎ 푰ퟏ

푰ퟐ 풉ퟐퟏ = |푽ퟐ=ퟎ = −ퟏ 푰ퟏ

푽ퟏ 풉ퟏퟐ = |푰ퟏ=ퟎ = ퟏ 푽ퟐ

푰ퟐ ퟏ

풉ퟐퟐ = |푰ퟏ=ퟎ = = ퟎ. ퟎퟓΩ 푽ퟐ ퟐퟎ

Option (d)

11. In the two network shown in the figure below Z12 and Z21 are, respectively

I1 I2

re βI1 r0

(a) 푟푒 푎푛푑 훽푟0 (c) 0 푎푛푑 훽푟0 (b) 0 푎푛푑 − 훽푟0 (d) 푟푒 푎푛푑 −훽푟0 [GATE 2006: 1 Mark]

Soln.

I1 I2

V1 re βI1 r0 V2

푽ퟏ = 풁ퟏퟏ푰ퟏ + 풁ퟏퟐ푰ퟐ

푽ퟐ = 풁ퟐퟏ푰ퟏ + 풁ퟐퟐ푰ퟐ

푽ퟏ 풁ퟏퟐ = |푰ퟏ=ퟎ 푰ퟐ

When 푰ퟏ = ퟎ, 푽ퟏ = ퟎ, 풔풐 풁ퟏퟐ = ퟎ

푽ퟐ 풁ퟐퟏ = |푽ퟐ=ퟎ 푰ퟏ

When 푰ퟐ = ퟎ, 푽ퟐ = −휷푰ퟏ풓ퟎ

풁ퟐퟏ = −휷풓ퟎ

Option (b)

12. For the two-port network shown, the short-circuit admittance parameter matrix is

0.5 1 2 yb

0.5 ya yc 0.5

1

4 −2 1 −0.5 (a) [ ] 푆 (c) [ ] 푆 −2 4 0.5 1 1 −0.5 4 2 (b) [ ] 푆 (d) [ ] 푆 −0.5 4 2 4 [GATE 2010: 1 Mark]

Soln. The short circuit admittance parameters of a two port π network:

ퟏ ퟏ 풚 = 풚 + 풚 = + = ퟒΩ ퟏퟏ 풂 풃 ퟎ.ퟓ ퟎ.ퟓ ퟏ 풚 = 풚 = −풚 = − = −ퟐΩ ퟏퟐ ퟐퟏ 풃 ퟎ.ퟓ ퟏ ퟏ 풚 = 풚 + 풚 = + = ퟒΩ ퟐퟐ 풃 풄 ퟎ.ퟓ ퟎ.ퟓ

Option (a)

13. If the scattering matrix [S] of a two port network is 0 0 [푆] = [0.2∠0 0.9∠0 ] 0.9∠00 0.1∠00 Then the network is (a) Lossless and reciprocal (c) Not lossless but reciprocal (b) Lossless but not reciprocal (d) Neither lossless not reciprocal [GATE 2010: 1 Mark]

Soln. For the reciprocal network |푺ퟏퟐ| = |푺ퟐퟏ|

ퟐ ퟐ For a loss less network |푺ퟏퟏ| + |푺ퟏퟐ| = ퟏ

(ퟎ. ퟐ)ퟐ + (ퟎ. ퟗ)ퟐ ≠ ퟏ

The network is lossy and reciprocal.

Option (c)

14. In the circuit shown below, the network N is described by the following Y matrix:

0.1푆 −0.01푆 푉2 푌 = [ ] The voltage gain is 0.01푆 0.1푆 푉1

(a) 1/90 (c) -1/99 (b) -1/90 (d) -1/11 [GATE 2011: 2 Marks]

Soln.

25 I1 I2 + + N 100 100V V -1 V-2

푰ퟐ = 풚ퟐퟏ푽ퟏ + 풚ퟐퟐ푽ퟐ

= ퟎ. ퟎퟏ 푽ퟏ + ퟎ. ퟏ 푽ퟐ − − − − − − − (풊)

푽ퟐ = −푰ퟐ푹푳 = −ퟏퟎퟎ푰ퟐ −푽 푰 = ퟐ ퟐ ퟏퟎퟎ

Substituting the value of I2 in equation (i)

−푽ퟐ = ퟎ. ퟎퟏ푽 + ퟎ. ퟏ푽 ퟏퟎퟎ ퟏ ퟐ

푽ퟐ −ퟏ Or = 푽ퟏ ퟏퟏ

Option (d)