Operation of A.C. Circuits
In an a.c. circuit, the value of the magnitude and the direction of the current and voltage changes at specific intervals of time. The current and voltage travel as sinusoidal waves whose complete cycle is made up of one positive half-cycle and one negative half-cycle.
These sinusoidal waves of current and voltage can either be in phase or out of phase. This variation is caused by a combination of the different components of the a.c. circuit such as the resistors, the capacitors, and the inductors.
In a.c. circuits, the opposition to the flow of current is as a result of:
- Resistance, $$R = \dfrac{V}{I}$$
- Inductive reactance, $$X_L = 2πfL$$ and
- Capacitive reactance, $$X_C = \dfrac{1}{2πfC}$$
The Waveform
Plotting the instantaneous values of the alternating quantities on the y-axis
and time or the angle on the x-axis
yields a waveform.
Purely resistive circuit
Important equation:
$$ I_{rms} = \dfrac{V_{rms}}{R} $$
Purely inductive circuit
Important equations:
$$ I_{L} = \dfrac{V_{L}}{X_L} $$
$$X_L = 2πfL$$
Purely capacitive circuit
Important equations:
$$ I_{rms} = \dfrac{V_{rms}}{X_C} $$
$$X_C = \dfrac{1}{2πfC}$$
Power factor
In a.c circuits, there are three important types of power:
- real power: this is the
true power
used to do work. $$= I^2R$$ - reactive power: this is the power due to reactive components in the circuit. $$= I^2X$$
- apparent power: this is the combination of
reactive power
andreal power
. $$= I^2Z$$
The ratio of real power to apparent power is the power factor whose value lies between 0 and 1.
Power factor (P.f) = $$ \dfrac{\textrm{real power (P) in Watts, W}}{\textrm{apparent power (S) in Volt-Amps, VA}} = cos ø $$
Parallel A.C. Circuits
R-L series a.c. circuits
$$V = \sqrt{V_{R}^{2}+V_{L}^{2}}$$
$$tanø = \dfrac{V_L}{V_R}$$
$$Z = \dfrac{V}{I} = \sqrt{R^2 + X_{L}^{2}}$$
$$cosø = \dfrac{R}{Z}$$
R-C series a.c. circuits
$$V = \sqrt{V_{R}^{2}+V_{C}^{2}}$$
$$tanø = \dfrac{V_C}{V_R} = \dfrac{X_C}{R}$$
$$Z = \dfrac{V}{I} = \sqrt{R^2 + X_{C}^{2}}$$
$$cosø = \dfrac{R}{Z}$$
R-L-C series a.c. circuits
When $$X_L > X_C$$:
$$ Z = \sqrt{R^2 + (X_L - X_C)^2} $$
$$ tanø = \dfrac{X_L - X_C}{R} $$
When $$X_C > X_L$$:
$$ Z = \sqrt{R^2 + (X_C - X_L)^2} $$
$$ tanø = \dfrac{X_C - X_L}{R} $$
When $$X_C = X_L$$, series resonance occur and $$V & I$$ are in phase.
Resonance frequency $$f_r = \dfrac{1}{2π\sqrt{LC}}$$ Hz
$$Q$$ factor $$ = \dfrac{1}{R}\sqrt{\dfrac{L}{C}}$$