Test: ESE Electrical - 5 - Electrical Engineering (EE) MCQ

After killing all source equivalent resistance is R Open circuit voltage = v₁

After killing the source, R_TH = 6 Ω

Trying i_L (t)= At+ B,



A = 1, B = 1

α = ω_o critically damped
v(t) = 12 + (A + Bt)e^-5t
0 = 12 + A, 150 = -5A + B A = -12, B = 90
v(t) =12 + (90t -12)e^-5t 
i_L(t) = 0.02(-5) e^-5t(90t -12) +0.02(90)e^-5t = (3 -9t)e^-5t

  V₁ = 600I₁ + 100I₂ , V₂ = 100I₁ + 200I₂
V_s = 60I₁ + V₁ = 660I₁ + 100I₂ , V₂ = V_o = -300I₂

Concept
The kVA rating for a delta-connected load is given by:
S = 3V_PI_P
where, V_P = Phase Voltage
I_P = Phase Current
The phase current is given by: IP = V_P/Z_P
Calculation: 
Given, V_P = 415 V
Z_P = (86∠54A6°)Ω  
I_P = 415 / (86∠54.46)
I_P = 4.82 ∠-54.46
S = 3 x 415 x 4.82
S = 6.002 kVA 

The term power is defined as the product of square of current and the impedance. So the power in the Y phase = 10² x 20 = 2000W.

F = mg = 10 X 10-³ X 9.81 = 9.81 X 10-² N.
On calculating r by substituting charges, we get r = 0.3m.

F = q1q2/(4∏εor²) , substituting q1, q2 and F, r2 = q1q2/(4∏εoF) =
We get r = 0.09m.

Convolution in the time domain implies multiplication in S(or frequency) domain

The Laplace transform of any signal h(t) is given by

So we observe that the H(s) = 1/s³, corresponds to a unit step signal convoluted with itself thrice.

Therefore the correct answer is option 1

The function has poles at z = 1,3/4. Thus final value theorem applies.

Since the ROC is the exterior circle, we expect x(n) to be a causal signal. Thus we seek a power series expansion in negative powers of ‘z’. By dividing the numerator of X(z) by its denominator, we obtain the power series

So, we obtain x(n)= {1,3/2,7/4,15/8,31/16,….}.

The average value

Calculation:
f_s = Sampling Frequency = 56 MHz
The frequencies present at the input are:
f_m1 = 10 MHz, f_m2 = 50 MHz and, f_m3 = 70 MHz
Once sampled, the frequency content at the output will be: f_m ± f_s
For, f_m1 = 10 MHz, the output will contain frequencies of,
= f_m1, f_m1 ±, f_m1 ± 2f_s ….
= 10 MHz, 66 MHz, 122 MHz
For, f_m2 = 50 MHz, the output will contain frequencies of
= f_m2, f_m2 ± f_s, f_m2 ± 2f_s ….
= 50 MHz, 6 MHz, 106 MHz, 162 MHz
For, f_m3 = 70 MHz, the output will contain frequencies of
= f_m3, f_m3 ± f_s, f_m3 ± 2f_s ….
= 70 MHz, 14 MHz, 182 MHz ….
When these frequencies are passed through a low pass filter with a cutoff frequency of 15 MHz, the only remaining frequencies at its output will be 6MHz, 10 MHz and 14 MHz.

The number of additions to be performed in FFT are Nlog₂N. But in linear filtering of a sequence, we calculate DFT which requires Nlog₂N complex additions and IDFT requires Nlog₂N complex additions. So, the total number of complex additions to be performed in linear filtering of a sequence using FFT algorithm is 2Nlog₂N.

The transition width of the main lobe in the case of Hamming window is equal to 8π/M where M is the length of the window.

We know that the Fourier transform of a function w(n) is defined as

For a rectangular window, w(n)=1 for n=0,1,2….M-1
Thus we get

The closed loop transfer function is given by

∴ 
Now,   

∴ 

The given block diagram can be reduced as shown below.

Since ξ > 1, therefore the system is overdamped. Hence, there will be no peak overshoot in the output response of the system.

To determine which of the given properties is not true for a Liapunov function, let's analyze each option:

A: It is a unique function for a given system. This statement is true. A Liapunov function is unique for a given system. If there were multiple Liapunov functions for the same system, it would imply different stability properties, which would contradict the definition of a Liapunov function.

B: It is positive definite, at least in the neighborhood of the origin. This statement is generally true. A Liapunov function must be positive definite in a neighborhood of the origin, meaning it is greater than zero for all points other than the origin within a certain distance. This condition ensures that the function is decreasing towards the equilibrium point.

C: It is a scalar function. This statement is true. A Liapunov function is a scalar function, meaning it takes a single value for each point in the system's state space.

D: Its time derivative is non-positive. This statement is also true. The time derivative of a Liapunov function must be non-positive for all points within the system's state space, except at the origin. This condition guarantees that the function is decreasing along the system's trajectories towards the equilibrium point.

Therefore, the property that is not true for a Liapunov function is option A: It is not a unique function for a given system.

Given, s(s - 1) + K(s + 1) = 0 

The correct answer is D: relatively stable.

In control systems, the phase margin and gain margin are measures of the system's stability and robustness. The phase margin represents the amount of phase lag the system can tolerate before it becomes unstable, while the gain margin indicates how much additional gain the system can handle before instability occurs.

When the phase margin is close to zero or the gain margin is close to unity, it implies that the system is operating near the stability limits. However, it does not necessarily mean that the system is completely unstable or highly unstable.

Option A: unstable is incorrect because a system with a phase margin close to zero or gain margin close to unity is not necessarily unstable. It indicates that the system is close to the stability boundary but can still exhibit stability if the margins are slightly positive or slightly below unity.

Option B: highly stable is incorrect because operating near the stability limits does not necessarily imply higher stability. It suggests that the system is more susceptible to instability if there are any disturbances or changes in the system parameters.

Option C: oscillatory is incorrect because the phase margin and gain margin do not directly indicate oscillatory behavior. Oscillations depend on other factors like the system's poles and zeros.

Option D: relatively stable is the most appropriate answer. When the phase margin is close to zero or the gain margin is close to unity, it indicates that the system is relatively stable but operating near its stability limits. It suggests that the system is more sensitive to disturbances and parameter variations, making it less robust compared to systems with larger margins.