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Home , Characteristic impedance, Standing wave ratio

Transmission Lines 1. A load impedance, (200 + j0) Ω is to be matched to a 50 Ω lossless by using a quarter line transformer (QWT). The of the QWT required is ______[GATE 1994: 1 Mark] Soln. For Quarter wave line transformer

ퟐ 풁ퟎ = 풁풊풏. 풁푳 ퟐ 풁ퟎ = ퟓퟎ × ퟐퟎퟎ

풁ퟎ = ퟏퟎퟎ 훀

2. A lossless transmission line having 50 Ω characteristic impedance and length 휆⁄4 is short circuited at one end and connected to an ideal source of 1V at the other end. The current drawn from the voltage sources is (a) 0 (c) ∞ (b) 0.02 A (d) None of the these [GATE 1996: 1 Mark] Soln. For quarter wave transformer (휆⁄4)

ퟐ 풁ퟎ 풁풊풏 = 풁푳

풁푳 = ퟎ (short circuit)

풁ퟐ 풁 = ퟎ = ∞ (open circuit) 풊풏 ퟎ

푽푺 푽푺 The current drawn from the voltage source 푰푺 = = = ퟎ 풁풊풏 ∞ Option (a)

3. The per unit length and the characteristic impedance of a lossless transmission line are C and Z0 respectively. The velocity of a travelling wave on the transmission line is

(a) Z0C (c) Z0/C (b) 1/(Z0C) (d) C/Z0 [GATE 1996: 1 Mark]

푳 푳 Soln. 풁 = √ , 풁ퟐ = ퟎ 푪 ퟎ 푪

ퟏ ퟏ ퟏ 풗풆풍풐풄풊풕풚 ( 푽) = = = √푳푪 ퟐ 풁ퟎ푪 √(풁ퟎ푪)(푪)

Option (b)

4. A transmission line of 50 Ω characteristic impedance is terminated with a 100 Ω resistance. The minimum impedance measured on the line is equal to (a) 0 Ω (c) 50 Ω (b) 25 Ω (d) 100 Ω [GATE 1997: 1 Mark]

Soln. 풁ퟎ = ퟓퟎ훀

풁푳 = ퟏퟎퟎ훀

풁푳 > 풁ퟎ

ퟐ 풁ퟎ 풁풊풏(풎풊풏) = 풁푳 ퟓퟎ×ퟓퟎ = = ퟐퟓ훀 ퟏퟎퟎ Option (b)

5. All transmission line section in Figure, have a characteristic impedance R0 + j0. The Zin equals

흀⁄ퟖ 흀⁄ퟐ

풁풊풏 ퟐ 푹ퟎ

푹ퟎ/ퟐ

흀⁄ퟒ

2 3 (a) 푅 (c) 푅 3 0 2 0 (b) 푅0 (d) 2 푅0 [GATE 1998: 1 Mark]

풁ퟐ 푹ퟐ ⁄ ퟎ ퟎ Soln. For 흀 ퟒ line, 풁풊풏ퟏ = = = ퟐ푹ퟎ 풁푳 푹ퟎ⁄ퟐ ⁄ For 흀 ퟐ line, 풁풊풏ퟐ = 풁푳ퟐ = ퟐ푹ퟎ

For 흀⁄ퟖ line, 풁푳 = (ퟐ푹ퟎ) ∥ ퟐ푹ퟎ

= 푹ퟎ

풁푳+풋풁ퟎ 풕풂풏 휷 풍 For transmission line of length 풍 , 풁풊풏 = 풁ퟎ [ ] 풁ퟎ+풋풁푳 풕풂풏 휷 풍

풁푳 + 풋풁ퟎ 풍 = 흀⁄ퟖ , 풁풊풏 = 풁ퟎ [ ] 풁ퟎ + 풋풁푳

푹ퟎ+풋푹ퟎ 풁풊풏 = 푹ퟎ [ ] = 푹ퟎ 푹ퟎ+풋푹ퟎ Option (b)

6. The magnitudes of the open – circuit and short – circuit input impedances of a transmission line are 100 Ω and 25 Ω respectively. The characteristic impedance of the line is.

(a) 25 Ω (c) 75 Ω (b) 50 Ω (d) 100 Ω [GATE 2000: 1 Mark]

Soln. 풁ퟎ = √풁ퟎ푪 . 풁푺푪 = √ퟏퟎퟎ × ퟐퟓ

풁ퟎ = ퟏퟎ × ퟓ = ퟓퟎ훀 Option (b)

7. A transmission line is distortion less if 1 (a) 푅퐿 = (c) 푅퐿 = 푅퐶 푅퐶 (d) 푅퐿 = 퐿퐶 (b) 푅퐿 = 퐺퐶 [GATE 2001: 1 Mark] 흎 Soln. For a distortion less line, velocity of propagation 풗 = must be 휷 independent of frequency. To achieve this 푳푮 = 푪푹 푳 푹 풐풓 = 푪 푮 Option (c)

8. The VSWR can have any value between (a) 0 and 1 (c) 0 and ∞ (b) – 1 and + 1 (d) 1 and ∞ [GATE 2002: 1 Mark]

ퟏ+|흆| Soln. 푽푺푾푹 = ퟏ−|흆| Where 흆 is 흆 can take values between 0 and 1 when 흆 = ퟎ, 푽푺푾푹 = ퟏ 흆 = ퟏ, 푽푺푾푹 = ∞ Option (d)

9. A transmission line has a characteristic impedance of 50Ω and a resistance of 0.1 Ω/m. If the line is distortion less, the attenuation constant (in Np/m) is (a) 500 (c) 0.014 (b) 5 (d) 0.002 [GATE 2010: 1 Mark] Soln. Attenuation constant α to be independent of frequency for distortion less transmission 휶 = √푹푮 For distortion less transmission: 푳 푹 = 푪 푮

푳 푹 풁 = √ = √ ퟎ 푪 푮

√푹 휶 = √푹푮 = √푹 풁ퟎ 푹 = 풁ퟎ ퟎ. ퟏ = ퟓퟎ = ퟎ. ퟎퟎퟐ 푵풑⁄풎

Option (d)

10. A transmission line of characteristic impedance 50Ω is terminated by a 50Ω load. When excited by a sinusoidal voltage source at 10 GHz, the phase difference between two points spaced 2 mm apart on the line is found to be π/4 radians. The phase velocity of the wave along the line is (a) 0.8 × 108 푚/푠 (c) 1.6 × 108 푚/푠 (b) 1.2 × 108 푚/푠 (d) 3 × 108 푚/푠 [GATE 2011: 1 Mark]

ퟐ흅 Soln. Phase difference 휷풍 = 퐩퐚퐭퐡 퐝퐢퐟퐟퐞퐫퐞퐧퐜퐞 흀 흅 ퟐ흅 = (ퟐ × ퟏퟎ−ퟑ) ퟒ 흀

흀 = ퟖ × ퟐ × ퟏퟎ−ퟑ = ퟏퟔ × ퟏퟎ−ퟑ 풎 Given , 풇 = ퟏퟎ푮푯풛 The phase velocity of the wave:

푽풑 = 풇흀 = ퟏퟎ × ퟏퟎퟗ × ퟏퟔ × ퟏퟎ−ퟑ = ퟏퟔퟎ × ퟏퟎퟔ 풎/풔풆풄 = ퟏ. ퟔ × ퟏퟎퟖ 풎/풔풆풄 Option (c)

11. The of a device is found to be 20 dB. The voltage ratio (VSWR) and magnitude of reflection coefficient are respectively (a) 1.22 and 0.1 (c) – 1.22 and 0.1 (b) 0.81 and 0.1 (d) 2.44 and 0.2 [GATE 2013: 1 Mark]

Soln. Return loss (dB) = −ퟐퟎ풍풐품ퟏퟎ|흆| Where 흆 is the reflection coefficient . For |흆| = ퟏ full reflection Return Loss = 0 dB If |흆| = ퟎ. ퟏ

푹. 푳풐풔풔 (풅푩) = −ퟐퟎ 풍풐품ퟏퟎ(ퟎ. ퟏ) = −ퟐퟎ × (−ퟏ) = 20 dB ퟏ+|흆| 푽푺푾푹 = ퟏ−|흆| ퟏ+ퟎ.ퟏ ퟏ.ퟏ = = = 1.22 ퟏ−ퟎ.ퟏ ퟎ.ퟗ Option (a)

12. To maximize power transfer, a lossless transmission line is to be matched to a resistive load impedance via a λ/4 transformer as shown. The characteristic impedance (in Ω) of the λ/4 transformer is ______. Soln. Input impedance for quarter wave transfer

풁푳 = ퟏퟎퟎ훀 풁풊풏 = ퟓퟎ훀

흀⁄ퟒ

ퟐ 풁ퟎ 풁풊풏 = 풁푳 ퟐ 풁ퟎ = 풁풊풏 풁푳

풁ퟎ = √풁풊풏 풁푳 = √ퟓퟎ × ퟏퟎퟎ = ퟕퟎ. ퟕퟐ 훀 Two Marks Questions

1. A transmission line of pure resistive characteristic impedance is terminated with an unknown load. The measured value of VSWR on the line is equal to 2 and a voltage minimum point is found to be at the load. The load impedance is then (a) Complex (c) Purely resistive (b) Purely capacitive (d) Purely inductive [GATE 1987: 2 Marks]

Soln. If Vmin or Vmax Occurs at the load for a lossless transmission line then load impedance ZL is purely resistive Option (c)

2. A two – wire transmission line of characteristic impedance Z0 is connected to a load of impedance 푍퐿(푍퐿 ≠ 푍0). cannot be achieved with (a) A quarter – wavelength transformer (b) A half – wavelength transformer (c) An open – circuited parallel (d) A short – circuited parallel stub [GATE 1988: 2 Marks]

Soln. If 풁푳 ≠ 풁ퟎ Then, impedance matching can be achieved by (i) a quarter wavelength transformer (흀⁄ퟒ). (ii) an open – circuited parallel stub. (iii) a short – circuited parallel stub. Half wave length transformer (흀⁄ퟐ) cannot be used for impedance matching Option (b)

3. A 50 ohm lossless transmission line has a pure reactance of (j 100) ohms as its load. The VSWR in the line is (a) 1/2 (c) 4 (b) 2 (d) (infinity) [GATE 1989: 2 Marks] Soln. Reflection coefficient 풁 − 풁 풋 ퟏퟎퟎ − ퟓퟎ 횪 = 푳 ퟎ = 풁푳 + 풁ퟎ 풋 ퟏퟎퟎ + ퟓퟎ

√ퟏퟎퟎퟐ + ퟓퟎퟐ 횪 = = ퟏ √ퟏퟎퟎퟐ + ퟓퟎퟐ

ퟏ + |횪| ퟏ + ퟏ ퟐ 푽푺푾푹 = = = = ∞ ퟏ − |횪| ퟏ − ퟏ ퟎ Option (d)

4. The input impedance of a short circuited lossless transmission line quarter wave long is (a) Purely reactive (b) Purely resistive (c) Infinite (d) Dependent on the characteristic impedance of the line [GATE 1991: 2 Marks] Soln. For a quarter wave line

ퟐ 풁ퟎ 풁풊풏 = 풁푳

풁푳 = ퟎ

풁ퟐ 풁 = ퟎ = ∞ 풊풏 ퟎ Option (c)

5. A transmission line whose characteristic impedance is a pure resistance (a) Must be a lossless line (b) Must be a distortion less line (c) May not be a lossless line (d) May not be a distortion less line [GATE 1992: 2 Marks] Soln. If the transmission line is to have neither frequency nor delay distortion, then α (attenuation constant) and velocity of propagation cannot be functions of frequency.

흎 풗 = 휷 휷 must be a direct function of frequency to achieve this condition 푳푮 = 푪푹

푳 푹 = 푪 푮

푹 + 풋흎푳 풁 = √ ퟎ 푮 + 풋흎푪

푳 For a lossless line, 풁 = √ ퟎ 푪

휶 = √푹푮 = ퟎ 풇풐풓 푹 = ퟎ, 푮 = ퟎ

휷 = 흎√푳푪 A loss less line is always a distortion less line

6. Consider a transmission line of characteristic impedance 50 ohms. Let it be terminated at one end by (+ j50) ohm. The VSWR produced by it in the transmission line will be (a) + 1 (c) ∞ (b) 0 (d) + j [GATE 19993: 2 Marks]

풁 −풁 Soln. Reflection coefficient = 횪 = 푳 ퟎ 풁푳+풁ퟎ

풋ퟓퟎ − ퟓퟎ −ퟓퟎ + 풋ퟓퟎ 횪 = = 풋ퟓퟎ + ퟓퟎ ퟓퟎ + 풋ퟓퟎ

√ퟓퟎퟐ + ퟓퟎퟐ 횪 = = ퟏ √ퟓퟎퟐ + ퟓퟎퟐ

ퟏ + |횪| ퟏ + ퟏ ퟐ 푽푺푾푹 = = = = ∞ ퟏ − |횪| ퟏ − ퟏ ퟎ Option (c)

7. If a pure resistance load, when connected to a lossless 75 ohm line, produce a VSWR of 3 on the line, then the load impedance can only be 25 ohms. True/False [GATE 1994: 2 Marks]

Soln. On a lossless line of 푹ퟎ = ퟕퟓ훀 with resistance load RL VSWR = S = 3

푹푳 푺 = 풊풇 푹푳 > 푹ퟎ 푹ퟎ

푹ퟎ = 풊풇 푹푳 < 푹ퟎ 푹푳

푹푳 = 푺푹ퟎ = ퟑ × ퟕퟓ = ퟐퟐퟓ 훀

푹 푹 = ퟎ 풊풇 푹 < 푹 푳 푺 푳 ퟎ

ퟕퟓ = = ퟐퟓ훀 ퟑ

ퟏ + |횪| 푹 − 푹 푽푺푾푹 = , 횪 = 푳 ퟎ ퟏ − |횪| 푹푳 + 푹ퟎ

풇풐풓 푹ퟎ = ퟕퟓ훀 , 푹푳 = ퟐퟓ훀

ퟐퟓ − ퟕퟓ −ퟓퟎ ퟏ 횪 = = = − ퟐퟓ + ퟕퟓ ퟏퟎퟎ ퟐ

푭풐풓 푹ퟎ = ퟕퟓ훀, 푹푳 = ퟐퟐퟓ훀

푹 − 푹 ퟏퟓퟎ ퟏ 횪 = 푳 ퟎ = = 푹푳 + 푹ퟎ ퟑퟎퟎ ퟐ

ퟏ ퟏ ퟏ + |횪| = 퐢퐧 퐞퐢퐭퐡퐞퐫 퐜퐚퐬퐞 퐚퐧퐝 퐒 = ퟐ = ퟑ ퟏ ퟐ ퟏ − ퟐ The statement, the load impedance can only be 25훀 is FALSE

8. In a twin – wire transmission line in air, the adjacent voltage maximum are at 12.5cm and 27.5cm. The operating frequency is (a) 300 MHz (c) 2 GHz (b) 1 GHz (d) 6.28 GHz [GATE 1999: 2 Marks]

Soln. Distance between adjacent voltage maximum = 흀⁄ퟐ 흀⁄ퟐ = ퟐퟕ. ퟓ − ퟏퟐ. ퟓ = 15 cm 흀 = ퟑퟎ 풄풎

Velocity of propagation on twin – wire TL line 풗 = ퟑ × ퟏퟎퟖ풎/풔풆풄

풗 ퟑ × ퟏퟎퟖ 풇 = = 흀 ퟑퟎ × ퟏퟎ−ퟐ

ퟑ × ퟏퟎퟏퟎ ퟑퟎ × ퟏퟎퟗ = = 푯풛 ퟑퟎ ퟑퟎ = 1 GHz Option (b)

9. In air, a lossless transmission line of length 50 cm with 퐿 = 10휇퐻/푚, 퐶 = 40푃퐹/푚 is operated at 25 MHz. It’s electrical path length is 휋 (a) 0.5 meters (c) radians 2 (b) 휆 meters (d) 180 degrees [GATE 1999: 2 Marks] Soln. Electrical path length = 휷 풍 radians

ퟏ ퟏ 풗풆풍풐풄풊풕풚 풗 = = √푳푪 √ퟏퟎ × ퟏퟎ−ퟔ × ퟒퟎ × ퟏퟎ−ퟏퟐ

ퟏ = = ퟎ. ퟓ × ퟏퟎퟖ 풎/풔 ퟏퟎ−ퟗ × ퟐퟎ 풗 = ퟎ. ퟓ × ퟏퟎퟖ 풎/풔

풗 ퟎ. ퟓ × ퟏퟎퟖ ퟓퟎ 흀 = = = 풇 ퟐퟓ × ퟏퟎퟔ ퟐퟓ = 2 meters

ퟐ흅 휷 풍 = × 풍 흀

ퟐ흅 ퟓퟎ = × ퟐ ퟏퟎퟎ

흅 풓풂풅풊풂풏풔 ퟐ Option (c)

10. A uniform plane electromagnetic wave incident normally on a plane surface of a dielectric material is reflected with a VSWR of 3. What is the percentage of incident power that is reflected (a) 10% (c) 50% (b) 25% (d) 75% [GATE 2001: 2 Marks] Soln. ퟏ + |횪| 푽푺푾푹 = ퟏ − |횪|

ퟏ + |횪| ퟑ = ퟏ − |횪| 횪 = ퟎ. ퟓ

푷 풓 = 횪ퟐ = (ퟎ. ퟓ)ퟐ 푷풊 = 0.25 25% of incident power is reflected. Option (b)

11. A short circuited stub is shunt connected to a transmission line as shown in figure. If 푍0 = 50Ω, the admittance Y seen at the junction of the stub and transmission line is

훌/ퟖ

풁푳 = ퟏퟎퟎ 풐풉풎 풁ퟎ

훌/ퟐ

(a) (0.01 – j 0.02) mho (c) (0.04 + j 0.02) mho (b) (0.02 – j 0.01) mho (d) (0.02 + j 0) mho [GATE 2003: 2 Marks]

Soln. For both Transmission line and stub, 풁ퟎ = ퟓퟎ훀

For 흀⁄ퟐ line input impedance 풁풊풍 = 풁푳

풁풊풍 = 풁푳 = ퟏퟎퟎ훀

풀풊풍 = ퟎ. ퟎퟏ 풎풉풐 For short circuited stub input impedance

풁풊ퟐ = 풋풁ퟎ 풕풂풏(휷풍)

ퟐ흅 흀 = 풋 풁 풕풂풏 ( ) ퟎ 흀 ퟖ

흅 = 풋 풁 풕풂풏 ( ) ퟎ ퟒ

= 풋 풁ퟎ = 풋 ퟓퟎ

ퟏ 풀 = = −풋 ퟎ. ퟎퟐ 풊ퟐ 풋 ퟓퟎ

풀 = 풀풊풍 + 풀풊ퟐ = (0.01 – j 0.02) mho Option (a)

12. Consider an impedance 푍 = 푅 + 푗푋 marked with point P in an impedance as shown in figure. The movement from point P along a constant resistance circle in the clockwise direction by an angle 450 is equivalent

r = 0.5

X = 0

X = - 0.5 P X = - 1

(a) Adding an in series with Z (b) Adding a capacitance in series with Z (c) Adding an inductance in shunt across Z (d) Adding a capacitance in shunt across Z [GATE 2004: 2 Marks] Soln. Point P ( 풁 = 푹 + 풋푿) on the Smith chart as shown in figure is the intersection of constant resistance circle 풓 = ퟎ. ퟓ and constant reactance circle 푿 = −ퟏ, Normalized impedance 풁 = ퟎ. ퟓ − 풋ퟏ The movement from point P along constant resistance circle of 0.5 by 450 in clockwise direction, resistance 0.5 is not changed but positive reactance is added. This is equivalent to adding inductance in series with Z. Option (a)

13. Characteristic impedance of a transmission line is 50Ω. Input impedance of the open circuited line is 푍0퐶 = 100 + 푗 150Ω. When the transmission line is short circuited then the value of the input impedance will be (a) 50Ω (c) 7.69 + 푗 11.54Ω (b) 100 + 푗 50Ω (d) 7.69 − 푗 11.54Ω [GATE 2005: 2 Marks]

Soln. 풁ퟎ = √풁푶푪 풁푺푪 ퟐ 풁푶 = 풁푶푪 풁푺푪

ퟐ 풁푶 풁푺푪 = 풁푶푪

ퟓퟎ × ퟓퟎ ퟓퟎ = = ퟏퟎퟎ + 풋 ퟏퟓퟎ ퟐ + 풋ퟑ

ퟓퟎ(ퟐ − ퟑ풋) = ퟒ + ퟗ

ퟏퟎퟎ − ퟓퟎ풋 = ퟏퟑ = ퟕ. ퟔퟗ − 풋 ퟏퟏ. ퟓퟒ Option (d)

Common data for Question 14 and 15. Voltage standing wave pattern in a impedance 50Ω and a resistive load is shown in the figure.

|푽(풛)|

4

1 Z 흀 흀⁄ퟐ load

14. The value of the load resistance is (a) 50Ω (c) 12.5Ω (b) 200Ω (d) 0Ω [GATE 2005: 2 Marks] Soln. 푽 ퟒ 푽푺푾푹(푺) = 풎풂풙 = 푽풎풊풏 ퟏ S = 4

풁풎풂풙 = 풁푶푺

풁 풁 = 푶 풎풊풏 푺 As minima is at load

풁 풁 = 풁 = 푶 푳 풎풊풏 푺

ퟓퟎ 풁 = = ퟏퟐ. ퟓ훀 푳 ퟒ Option (c)

15. The reflection coefficient is given by (a) – 0.6 (c) 0.6 (b) – 1 (d) 0 [GATE 2005: 2 Marks] Soln. The reflection coefficient 풁 − 풁 횪 = 푳 ퟎ 풁푳 + 풁ퟎ

ퟏퟐ. ퟓ − ퟓퟎ 횪 = = −ퟎ. ퟔ ퟏퟐ. ퟓ + ퟓퟎ Option (a) 16. A load of 50Ω is connected in shunt in a 2 – wire transmission line of 푍0 = 50Ω as shown in the figure. The 2 – port scattering parameter (s – matrix) of the shunt element is

풁ퟎ = ퟓퟎ훀 ퟓퟎ훀 풁ퟎ = ퟓퟎ훀

ퟏ ퟏ ퟏ ퟐ − − ퟐ ퟐ ퟑ ퟑ (a) [ ퟏ ퟏ] (c) [ ퟐ ퟏ] − − ퟐ ퟐ ퟑ ퟑ 0 1 ퟏ ퟑ (b) [ ] − 1 0 ퟒ ퟒ (d) [ퟏ ퟏ ]

ퟐ ퟒ

[GATE 2007: 2 Marks]

Soln. The line is terminated with 50 ohms at the ends, so matched on both the sides thus 푺ퟏퟏ = ퟎ, 푺ퟐퟐ = ퟎ and 푺ퟏퟐ = 푺ퟐퟏ = ퟏ Option (b)

17. The parallel branches of a 2 – wire transmission line are terminated in 100 Ω and 200 Ω resistors as shown in the figure. The characteristic 휆 impedance of the line is 50 and each section has a length of . The 4 voltage reflection coefficient Γ at the input is

ퟕ ퟓ (a) −풋 (c) 풋 ퟓ ퟕ ퟓ ퟓ (b) − (d) ퟕ ퟕ

[GATE 2007: 2 Marks] Soln.

푹ퟏ 푹ퟐ 풁ퟎ = ퟓퟎ훀 풁풊풏

흀⁄ퟒ

ퟐ 풁ퟎ 풁풊풏 = 풇풐풓 흀⁄ퟒ 풍풊풏풆 풁푳

ퟓퟎퟐ 푹 풅풖풆 풕풐 ퟏퟎퟎ 훀 = ퟏ ퟏퟎퟎ = 25 Ω

ퟓퟎퟐ 푹 풅풖풆 풕풐 ퟐퟎퟎ 훀 = ퟐ ퟐퟎퟎ

ퟐퟓ = 훀 ퟐ

ퟐퟓ ퟐퟓ 푹 ‖ 푹 = ퟐퟓ ‖ = ퟏ ퟐ ퟐ ퟑ

ퟐ 풁ퟎ 풁풊풏 = 풁푳

ퟓퟎ × ퟓퟎ = = ퟑퟎퟎ훀 ퟐퟓ/ퟑ Reflection coefficient 풁 − 풁 횪 = 풊풏 ퟎ 풁풊풏 + 풁ퟎ

ퟑퟎퟎ − ퟓퟎ 횪 = ퟑퟎퟎ + ퟓퟎ

ퟓ = ퟕ Option (d) 18. One end of a lossless transmission line having the characteristic impedance of 75 and length of 1 cm is short circuited. At 3 GHz, the input impedance at the other end of the transmission line is (a) 0 (c) Capacitive (b) Resistive (d) Inductive [GATE 2008: 2 Marks] Soln. 풇 = ퟑ푮푯풛

ퟐ흅 휷 = 흀

풄 ퟑ × ퟏퟎퟖ ퟏ 흀 = = = 풎 풇 ퟑ × ퟏퟎퟗ ퟏퟎ

ퟏ 휷풍 = ퟐ흅 × ퟏퟎ × ퟏퟎퟎ

흅 = ퟓ = ퟑퟔퟎ Input impedance of short circuited line

풁풊풏 = 풋 풁ퟎ 풕풂풏 휷풍 ퟎ = 풋 풁ퟎ 풕풂풏 ퟑퟔ = 풋 ퟕퟓ 풕풂풏 ퟑퟔퟎ = 풋 ퟓퟒ. ퟒퟗ 훀 Input impedance is inductive Option (d)

휆 19. A transmission line terminates in two branches each of length as 4 shown. The branches are terminated by 50 Ω loads. The lines are lossless and have the characteristic impedances shown. Determine the impedance 푍푖 as seen by the source

(a) 200 Ω (c) 50 Ω (b) 100 Ω (d) 25 Ω [GATE 2009: 2 Marks]

흀 Soln. For a line of characteristic impedance Z0 and terminated by ZL , ퟒ input impedance

ퟐ 풁ퟎ 풁ퟏ = 풁푳

ퟐ ퟐ 풁ퟎ ퟏퟎퟎ 풁ퟏ = = = ퟐퟎퟎ훀 풁푳ퟏ ퟓퟎ

ퟐ ퟐ 풁ퟎ ퟏퟎퟎ 풁ퟐ = = = ퟐퟎퟎ훀 풁푳ퟐ ퟓퟎ

풁푳 = 풁ퟏ‖ 풁ퟐ = ퟐퟎퟎ‖ ퟐퟎퟎ = ퟏퟎퟎ훀

ퟐ ퟐ 풁ퟎ ퟓퟎ 풁풊 = = = ퟐퟓ훀 풁푳 ퟏퟎퟎ Option (d)

20. In the circuit shown, all the transmission line sections are lossless. The voltage (VSWR) on the line Shot

풁ퟎ = ퟑퟎ훀 흀⁄ퟖ

풁ퟎ = ퟔퟎ 훀 풁ퟎ = ퟑퟎ √ퟐ 훀 풁푳 = ퟑퟎ 훀

흀⁄ퟒ

(a) 1.00 (c) 2.50 (b) 1.64 (d) 3.00 [GATE 0000: 2 Marks] Soln. The input impedance of a transmission line of length l of characteristic impedance Z0 and terminated by load ZL

풁푳 + 풋 풁ퟎ 풕풂풏 휷풍 풁풊풏 = 풁ퟎ ( ) 풁ퟎ + 풋 풁푳 풕풂풏 휷풍 흀 Input impedance of shorted line of 풁 = ퟑퟎ 훀 ퟖ ퟎ

ퟐ흅 흀 ퟎ + 풋 ퟑퟎ 풕풂풏 ( ) 풁 = ퟑퟎ ( 흀 ퟖ ) 풊 ퟑퟎ + ퟎ

흅 풁 = 풋 ퟑퟎ 풕풂풏 = 풋 ퟑퟎ 풊 ퟒ 흀 Input impedance of line of 풁 = ퟑퟎ√ퟐ 훀 and 풁 = ퟑퟎ훀 ퟒ ퟎ 푳

ퟐ 풁ퟎ 풁ퟐ = 풁푳

ퟐ (ퟑퟎ√ퟐ) ퟑퟎ√ퟐ × ퟑퟎ√ퟐ = = ퟑퟎ ퟑퟎ = ퟔퟎ 훀

Load impedance 풁푳 = 풁ퟏ + 풁ퟐ = 풋 ퟑퟎ + ퟔퟎ Reflection coefficient 풁 − 풁 흆 = 푳 ퟎ 풁푳 + 풁ퟎ

ퟔퟎ + 풋 ퟑퟎ − ퟔퟎ 흆 = ퟔퟎ + 풋 ퟑퟎ + ퟔퟎ

풋 ퟑퟎ 풋 ퟏ = = ퟏퟐퟎ + 풋 ퟑퟎ ퟒ + 풋 ퟏ

ퟏ ퟏ |흆| = = √ퟏퟔ + ퟏ √ퟏퟕ

ퟏ ퟏ + ퟏ + |흆| ퟏퟕ 푽푺푾푹 = = √ | | ퟏ ퟏ − 흆 ퟏ − √ퟏퟕ = 1.64 Option (b)

21. A transmission line of characteristic impedance 50 Ω is terminated in a load impedance ZL. The VSWR of the line is 5 and the first of the voltage 휆 maximum in the line is observed at a distance of from the load. The 4 value of ZL is (a) 10 Ω (c) 250 Ω (b) (19.23 + 푗46.15)Ω (d) (19.23 − 푗46.15)Ω [GATE 2011: 2 Marks]

Soln. For a transmission line, 풁ퟎ = ퟓퟎ 훀 , 푽푺푾푹 = ퟓ 흀 Distance of the first voltage maximum from the load = . The ퟒ 흀 distance between adjacent maxima and minima should be in a ퟒ standing wave pattern, Vmin should occur at the load.

Vmin occurs for a resistive termination. Vmin occurs at load if 풁 풁 ퟓퟎ 풁 = 풁 = ퟎ 풁 = ퟎ = = ퟏퟎ훀 푳 풎풊풏 푺 푳 푺 ퟓ Option (a)

22. A transmission line with a characteristic impedance of 100 Ω is used to match a 50 Ω section to a 200 Ω section. If the matching is to be done both at 429 MHz and 1 GHz. The length of the transmission line can be approximately. (a) 82.5 cm (c) 1.58 m (b) 1.05 m (d) 1.75 m [GATE 2012: 2 Marks]

Soln. 풁ퟎ = √풁ퟏ풁ퟐ ퟏퟎퟎ = √ퟓퟎ × ퟐퟎퟎ

ퟓퟎ 훀 ퟏퟎퟎ 훀 ퟐퟎퟎ 훀

풁ퟎ 풁ퟏ 풁ퟐ

This is quarter wave matching. The length would be odd multiples of 흀/ퟒ . 풍 = (ퟐ풎 + ퟏ)흀/ퟒ

풇ퟏ = ퟒퟐퟗ 푴푯풛

흀 ퟑ × ퟏퟎퟖ ퟏ = ퟒ ퟒퟐퟗ × ퟏퟎퟔ × ퟒ

퓵ퟏ = ퟎ. ퟏퟕퟒ 풎

풇ퟐ = ퟏ 푮푯풛

흀 ퟑ × ퟏퟎퟖ ퟐ = = ퟎ. ퟎퟕퟓ 풎 ퟒ ퟏퟎퟗ × ퟒ

퓵ퟐ = ퟎ. ퟎퟕퟓ 풎

ퟏ. ퟓퟖ ퟏ. ퟓퟖ (ퟐ풎 + ퟏ) = = = ퟗ 퓵ퟏ ퟎ. ퟏퟕퟒ

ퟏ. ퟓퟖ ퟏ. ퟓퟖ (ퟐ풎 + ퟏ) = = = ퟐퟏ 퓵ퟐ ퟎ, ퟎퟕퟓ

Only option (c) is odd multiples of both 퓵ퟏ 풂풏풅 퓵ퟐ .

23. The input impedance of a 휆/8 section of a lossless transmission line of characteristic impedance 50 Ω is found to be real when the other end is terminated by a load 푍퐿 = 푅 + 푗푋. If X is 30 Ω, the value of R is _____. [GATE 2014: 2 Marks] Soln. 풁푳 + 풋 풁ퟎ 풕풂풏 휷풍 풁풊풏 = 풁ퟎ ( ) 풁푳 + 풋 풁푳 풕풂풏 휷풍 풍 = 흀/ퟖ

ퟐ흅 흀 휷풍 = × 흀 ퟖ

흅 = ퟒ

흅 풕풂풏 (휷풍) = 풕풂풏 ( ) = ퟏ ퟒ

풁푳 + 풋 풁ퟎ 풁풊풏 = 풁ퟎ [ ] 풁ퟎ + 풋 풁푳

풁푳 = 푹 + 풋푿, 풁ퟎ = ퟓퟎ훀 = 푹 + 풋 ퟑퟎ

푹 + 풋ퟑퟎ + 풋ퟓퟎ 풁 = ퟓퟎ [ ] 풊풏 ퟓퟎ + 풋 (푹 + 풋 ퟑퟎ)

푹 + 풋ퟖퟎ = ퟓퟎ [ ] (ퟓퟎ − ퟑퟎ) + 풋푹

푹 + 풋ퟖퟎ 풁 = ퟓퟎ [ ] 풊풏 ퟐퟎ + 풋푹

(푹 + 풋ퟖퟎ)(ퟐퟎ − 풋푹) = ퟓퟎ [ ] ퟐퟎퟐ + 푹ퟐ

Since only real part of Zin exists so imaginary part of 풁풊풏 = ퟎ

ퟐퟎ푹 + ퟏퟔퟎퟎ풋 − 풋 푹ퟐ + ퟖퟎ 푹 풁 = ퟓퟎ [ ] 풊풏 ퟐퟎퟐ + 푹ퟐ

ퟏퟔퟎퟎ − 푹ퟐ 풁 = 풊풎풂품풊풏풂풓풚 ퟐퟎퟐ + 푹ퟐ ퟏퟔퟎퟎ − 푹ퟐ = ퟎ 푹ퟐ = ퟏퟔퟎퟎ 푹 = ퟒퟎ 훀

24. In the transmission line shown the impedance Zin between A and the ground is A

풁ퟎ = ퟓퟎ 훀, 퐋 = ퟎ. ퟓ훌 풁풊풏ퟏ ퟓퟎ 훀 ퟏퟎퟎ 훀

풁풊풏 = ?

Soln. Since line is of length 0.5 흀

풁푳 + 풋 풁ퟎ 풕풂풏 휷풍 풁풊풏ퟏ = 풁ퟎ = [ ] 풁ퟎ + 풋 풁푳 풕풂풏 휷풍

ퟐ흅 흀 풇풐풓 흀⁄ퟐ 풍풊풏풆 휷풍 = × 흀 ퟐ = 흅

풁푳 + 풋ퟎ 풁풊풏ퟏ = 풁ퟎ [ ] 풁ퟎ + 풋ퟎ

풁풊풏ퟏ = 풁푳 = ퟓퟎ훀

풁풊풏 = ퟏퟎퟎ‖ퟓퟎ

ퟏퟎퟎ × ퟓퟎ = ퟏퟓퟎ = ퟑퟑ. ퟑ훀