## 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`

and`real 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 \( = \dfrac{1}{R}\sqrt{\dfrac{L}{C}}\)

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