# Resonance and Locus Diagrams

 Question 1
The voltage v(t) across the terminals a and b as shown in the figure, is a sinusoidal voltage having a frequency $\omega$=100 radian/s. When the inductor current i(t) is in phase with the voltage v(t), the magnitude of the impedance Z (in $\Omega$) seen between the terminals a and b is ________ (up to 2 decimal places). A 25 B 50 C 100 D 150
GATE EE 2018   Electric Circuits
Question 1 Explanation:
At resonance imaginary part of $Z_{eq}=0$
Real of $Z_{eq}=\frac{R_1 X_c^2}{R_1^2+X-c^2}$
$\;\;=\frac{100 \times 100 \times 100}{100^2+100^2}=50\Omega$
 Question 2
A DC voltage source is connected to a series L-C circuit by turning on the switch S at time t=0 as shown in the figure. Assume i(0)=0, v(0)=0. Which one of the following circular loci represents the plot of i(t) versus v(t) ?  A A B B C C D D
GATE EE 2018   Electric Circuits
Question 2 Explanation: $I(s)=\frac{\frac{5}{s}}{s+\frac{1}{s}}$
$\;\;=\frac{5}{s^2+1}$
$i(t)=5 \sin t$
$v(t)=\frac{1}{C}\int_{0}^{t}i \; dt$
$v(t)=\int_{0}^{t}5 \sin t \; dt$
$v(t)=5[-\cos t]_0^t$
$v(t)=5[-\cos t+1]$
$v(t)=5-5 \cos t$ $\begin{matrix} t & i(t) & v(t)\\ 0 & 0 & 0\\ \frac{T}{4} & 5 & 5\\ \frac{T}{2}& 0 & 10\\ \frac{3T}{4} &-5 & 5\\ T& 0 & 0 \end{matrix}$

 Question 3
In the balanced 3-phase, 50 Hz, circuit shown below, the value of inductance (L) is 10 mH. The value of the capacitance (C) for which all the line currents are zero, in millifarads, is ___________. A 1.32 B 2.12 C 3.03 D 4.08
GATE EE 2016-SET-2   Electric Circuits
Question 3 Explanation:
Usingstar to delta conversion, Line current will be zero when the parallel pair of induction capacitor is resonant at f=50Hz
So, $50 \times 2\pi=\frac{1}{\sqrt{LC/3}}$
$100 \pi=\frac{1}{\sqrt{LC/3}}$
Since, L=10mH
C will be 3.03 mF.
 Question 4
The circuit below is excited by a sinusoidal source. The value of R, in $\Omega$, for which the admittance of the circuit becomes a pure conductance at all frequencies is _____________. A 14.14 B 8.62 C 22.46 D 12.18
GATE EE 2016-SET-1   Electric Circuits
Question 4 Explanation:
The resonant frequency for the circuit is
$\omega _0=\frac{1}{\sqrt{LC}}\sqrt{\frac{R_L^2-L/C}{R_C^2-L/C}}$
Since, $(R_L=R_C=R)$
So the circuit will have zero real part of admittance
when, $R=\sqrt{\frac{L}{C}}=\sqrt{\frac{0.02}{100\mu F}}=14.14\Omega$
 Question 5
An inductor is connected in parallel with a capacitor as shown in the figure. As the frequency of current i is increased, the impedance (Z) of the network varies as A A B B C C D D
GATE EE 2015-SET-1   Electric Circuits
Question 5 Explanation:
$Z=j\omega L||\frac{1}{j\omega C}$
$\;\;=\frac{L/C}{j\omega L+\frac{1}{j\omega C}}$
$\;\;=\frac{(L/C)j\omega C}{-\omega ^2 LC+1}$
$Z=\frac{j\omega L}{1-\omega^2 LC}$
$|Z|=\frac{\omega L}{1-\omega^2 LC}$
For, $1\gt \omega^2 LC;$
$Z=+ve$ For, $1 \lt \omega^2 LC;$
$Z=-ve$ There are 5 questions to complete.