Question

A series L-C-R circuit is made by taking R = 100 Ω, L=π2​H, C = π100​μF. This series combination is connected across an AC source of 220 V, 50 Hz. Calculate
(i) the impedance of the circuit and
(ii) the peak value of the current flowing in the circuit
(iii) the power factor of this circuit and compare this value with the one at its resonant frequency.

Answer 1

Hint: The L-C-R circuit is also said to be the resonant circuit, tuned circuit, or an RLC circuit. This L-C-R is an electrical circuit consisting of an inductor (L), capacitor (C), resonant (R) which are connected in series or parallel whereas the resistances, current and voltage phasors are always in phase.

Complete step-by-step answer:
\[
  given:R = 100\Omega \\
  L = \dfrac{2}{\pi }H \\
  C = \dfrac{{100}}{\pi }\mu F \\
  {V_{rms}} = 220V \\
  f = 50Hz \\
  (i):{\text{ }}{{\text{X}}_L} = 2\pi fL = 2\pi \times 50 \times \frac{2}{\pi } = 200\Omega \\
   \Rightarrow {X_C} = \dfrac{1}{{2\pi fC}} = \frac{1}{{2\pi \times 50 \times (100/\pi ) \times {{10}^{ - 6}}}} = 100\Omega \\
  impedence{\text{ of circuit }} \\
  {\text{Z = }}\sqrt {{R^2} + {{({{\text{X}}_L} - {X_C})}^2}} = \sqrt {{{100}^2} + {{(200 - 100)}^2}} = 100\sqrt 2 \Omega \\
\]
(ii) Peak value of voltage
     $
  {V_ \circ } = \sqrt 2 {V_{rms}} = 220\sqrt 2 V \\
  peak{\text{ }}value{\text{ of current }}{{\text{I}}_ \circ } = \dfrac{{V \circ }}{Z} = \dfrac{{220\sqrt 2 }}{{100\sqrt 2 }} = 2.2A \\
$
(iii) Power factor
$
  \cos \phi = \dfrac{R}{Z} = \dfrac{{100}}{{100\sqrt 2 }} = \dfrac{1}{{\sqrt 2 }} \\
  resonant{\text{ frequency }}{{\text{f}}_r} = \dfrac{1}{{2\pi \sqrt {LC} }} = \dfrac{1}{{2\pi \sqrt {\dfrac{2}{\pi } \cdot \dfrac{{100}}{\pi }} }} = \dfrac{1}{{20\sqrt 2 }} \\
  So, \\
  \dfrac{{\cos \phi }}{{{f_r}}} = \dfrac{{1/\sqrt 2 }}{{1/20\sqrt 2 }} = 20 \\
$
Power factor is the ratio of working power, measured in kilowatts to apparent power measured in kilovolt amperes. The lower power factor is usually is caused by inductive loads, such as: electric motors, transformers, arc welders, modelling equipment’s etc...In short power factor is a measure of how effectively incoming power is used in your electrical system and is define as the ratio of real (working) power to apparent (total) power.

PEAK VALUE OF VOLTAGE:
Peak value means the peak-to-peak value is the maximum voltage change occurring during one cycle of alternating the voltage or current. Therefore the peak value of AC voltage is defined as the difference between its positive peak and its negative peak.

Note: The use of LCR circuits are in radio and communication engineering. They can be used to select a narrow range of the frequency from the total spectrum of ambient radio waves such as: the AM/FM radio with analog tuners use a RLC circuit to tune a radio frequency.