A 50 \(\Omega\) lossless transmission line is terminated with a load \( Z_L = (50 - j75) \, \Omega.\) { If the average incident power on the line is 10 mW, then the average power delivered to the load
(in mW, rounded off to one decimal place) is} _________.
A 50 \(\Omega\) lossless transmission line is terminated with a load \( Z_L = (50 - j75) \, \Omega.\) { If the average incident power on the line is 10 mW, then the average power delivered to the load
(in mW, rounded off to one decimal place) is} _________.
Show Hint
Solution and Explanation
\( P_{{incident}} = 10 \, {mW} \) is the incident power,
\( \Gamma \) is the reflection coefficient, which is given by: \[ \Gamma = \frac{Z_L - Z_0}{Z_L + Z_0}. \] Here, \( Z_0 = 50 \, \Omega \) is the characteristic impedance of the transmission line. Substituting \( Z_L = (50 - j75) \, \Omega \): \[ \Gamma = \frac{(50 - j75) - 50}{(50 - j75) + 50} = \frac{-j75}{100 - j75}. \] To simplify, multiply both the numerator and denominator by the complex conjugate of the denominator: \[ \Gamma = \frac{-j75(100 + j75)}{(100 - j75)(100 + j75)} = \frac{-j75(100 + j75)}{100^2 + 75^2} = \frac{-j7500 - 5625}{15625}. \] Now, calculate \( \left| \Gamma \right|^2 \): \[ \left| \Gamma \right|^2 = \frac{(-5625)^2 + (-7500)^2}{15625^2} = \frac{31890625 + 56250000}{244140625} = \frac{88140625}{244140625} = 0.36. \] Thus, the average power delivered to the load is: \[ P_{{load}} = \frac{10}{1 + 0.36} = \frac{10}{1.36} \approx 6.5 \, {mW}. \] Thus, the average power delivered to the load is 6.5 mW.
Top Questions on Flip-Flop
In the circuit shown below, the AND gate has a propagation delay of 1 ns. The edge-triggered flip-flops have a set-up time of 2 ns, a hold-time of 0 ns, and a clock-to-Q delay of 2 ns. The maximum clock frequency (in MHz, rounded off to the nearest integer) such that there are no setup violations is (answer in MHz).
- An ideal p-n junction germanium diode has a reverse saturation current of 10 \(\mu A\) at 300 K. The voltage (in Volts, rounded off to two decimal places) to be applied across the junction to get a forward bias current of 100 mA at 300 K is _________. (Consider the Boltzmann constant \( k_B = 1.38 \times 10^{-23} { J/K} \) and the charge of an electron \( e = 1.6 \times 10^{-19} { C} \).
In a 4-bit ripple counter, if the period of the waveform at the last flip-flop is 64 microseconds, then the frequency of the ripple counter in kHz is ___________. (Answer in integer)
- One example of the use of an S-R flip-flop is
- The characteristic equation of JK flip-flop is ______
Questions Asked in GATE EC exam
- Consider an additive white Gaussian noise (AWGN) channel with bandwidth \( W \) and noise power spectral density \( \frac{N_0}{2} \). Let \( P_{{av}} \) denote the average transmit power constraint. Which one of the following plots illustrates the dependence of the channel capacity \( C \) on the bandwidth \( W \) (keeping \( P_{{av}} \) and \( N_0 \) fixed)?
- GATE EC - 2025
- Channel Capacity
Consider a part of an electrical network as shown below. Some node voltages, and the current flowing through the \( 3\,\Omega \) resistor are as indicated.
The voltage (in Volts) at node \( X \) is _________.
- GATE EC - 2025
- Laplace, Fourier and z-transforms
The 12 musical notes are given as \( C, C^\#, D, D^\#, E, F, F^\#, G, G^\#, A, A^\#, B \). Frequency of each note is \( \sqrt[12]{2} \) times the frequency of the previous note. If the frequency of the note C is 130.8 Hz, then the ratio of frequencies of notes F# and C is:
A 4-bit weighted-resistor DAC with inputs \( b_3, b_2, b_1, \) and \( b_0 \) (MSB to LSB) is designed using an ideal opamp, as shown below. The switches are closed when the corresponding input bits are logic ‘1’ and open otherwise. When the input \( b_3b_2b_1b_0 \) changes from 1110 to 1101, the magnitude of the change in the output voltage \( V_o \) (in mV, rounded off to the nearest integer) is _________.
- GATE EC - 2025
- ADCs and DACs: Principles
- A square metal sheet of 4 m \( \times \) 4 m is placed on the x-y plane as shown in the figure below. If the surface charge density (in \( \mu \)C/m\(^2\)) on the sheet is \( \rho_s(x, y) = 4|y| \), then the total charge (in \( \mu \)C, rounded off to the nearest integer) on the sheet is _________.
- GATE EC - 2025
- Transistors