AC Circuit

An AC (Alternating Current) circuit is an electrical circuit powered by an alternating current source, where the flow of electric charge periodically reverses direction. Unlike direct current (DC), which flows steadily in one direction, AC voltage and current alternate in a sinusoidal (or other wave-like) pattern over time. This type of circuit is the foundation of most modern power systems, as AC is more efficient for transmitting electricity over long distances. AC circuits are commonly found in household appliances, industrial equipment, and power distribution networks. Understanding how AC circuits work is key to mastering electrical systems and electronics.

1.0Definition of AC Circuits

To study the behavior of AC circuits, they are generally classified into two main categories:

(a) Simple Circuits – These contain only one basic component: either a resistor (R), an inductor (L), or a capacitor (C).

(b) Complex Circuits – These involve a combination of any two or all three components: resistor (R), inductor (L), and capacitor (C).

2.0AC Circuit Containing Pure Resistor

Alternating current developed in a pure resistance is also of the sinusoidal nature. In a.c. circuits containing pure resistance, the voltage and current are in the same phase. The vector or phasor diagram which represents the phase relationship between alternating current and alternating e.m.f.

In the a.c. circuit having R only, as current and voltage are in the same phase, hence in fig. both phasors $E_{0} an d I_{0}$ are in the same direction, making an angle t with OX. Their projections on the Y-axis represent the instantaneous values of alternating current and voltage.

$I = I_{0} sin (ω t) and E = E_{0} sin (ω t)$

$I_{0} = \frac{E _{0}}{R} hence I_{rms} = \frac{E _{rms}}{R}$

3.0AC Circuit Containing Pure Inductor

$E = E_{0} sin (ω t) and I = I_{0} sin (ω t - \frac{π}{2})$

In a pure inductive circuit current always lags behind the emf by $\frac{π}{2}$ or alternating emf leads the a.c by a phase angle of $\frac{π}{2}$ .

$I_{0} = \frac{E _{0}}{X _{L}} resembles the expression \frac{E}{I} = R$

This non-resistive opposition to the flow of A.C. circuit is called the inductive reactance ( $X_{L}$ ) of the circuit.

$X_{L} = ω L = 2 π f L$

$f$ is the frequency of A.C

Unit of $X_{L}$ is Ohm

Inductive Reactance $X_{L} \propto f$ .

Higher the frequency of A.C. higher is the inductive reactance offered by an inductor in an A.C. Circuit.

For d.c. circuit, $f = 0 ∴ X_{L} = ω L = 2 π f L = 0$ , Hence, the inductor offers no opposition to the flow of d.c. whereas a resistive path to a.c.

4.0AC Circuit Containing Pure Capacitor

$E = E_{0} sin (ω t) and I = I_{0} sin (ω t + \frac{π}{2})$

In a pure capacitive circuit the current always leads the emf by a phase angle of $\frac{π}{2}$ .The alternating emf lags behind the alternating current by a phase angle of $\frac{π}{2}$ .

$I = I_{0} sin (ω t + \frac{π}{2})$

$I_{0} = ω C E_{0}$

This non-resistive opposition to the flow of A.C. in a pure capacitive circuit is known as capacitive reactance $X_{C}, X_{C} = \frac{1}{ω C} = \frac{1}{2 π f C}$

Unit of $X_{C}$ : ohm

Capacitive Reactance $X_{C}$ is inversely proportional to frequency of A.C $X_{C}$ decreases as the frequency increases.

For d.c Circuit $f = 0 ∴ X_{C} = \frac{1}{ω C} = \frac{1}{2 π f C} = \infty$ but a very small value for a.c

This shows that capacitor blocks the flow of d.c but provides an easy path for a.c

5.0Series R-L AC Circuit

$E = E_{0} sin (ω t)$

Potential Differences across L and R are $V_{L} = I X_{L}$ and $V_{R} = I R$

$V_{R}$ is in phase with the current while $V_{L}$ Leads the current by $\frac{π}{2}$

$E = V_{R}^{2} + V_{L}^{2}$

$E^{2} = V_{R}^{2} + V_{L}^{2} = I^{2} R^{2} + I^{2} X_{L}^{2} I = \frac{E}{R ^{2} + X _{L}^{2}}$

$tan ϕ = \frac{V _{L}}{V _{R}} = \frac{X _{L}}{R} = \frac{ω L}{R} ϕ = tan^{- 1} (\frac{ω L}{R})$

Inductive Impedance $Z_{L}$ : In the L-R circuit the maximum value of current

$I_{0} = \frac{E _{0}}{R ^{2} + ω ^{2} L ^{2}}$

$R^{2} + ω^{2} L^{2}$ represents the effective opposition offered by L-R Circuit to the flow of a.c through it.It is known as impedance of L-R Circuit and is represented as

$Z_{L} = R^{2} + ω^{2} L^{2} = R^{2} + (2 π f L)^{2}$

Admittance-The reciprocal of impedance is called Admittance.

$Y_{L} = \frac{1}{Z _{L}} = \frac{1}{R ^{2} + ω ^{2} L ^{2}}$

6.0Series R-C AC Circuit

$E = E_{0} sin (ω t)$

Potential Differences across L and R are $V_{C} = I X_{C}$ and $V_{R} = I R$

$V_{R}$ is in phase with I , while $V_{C}$ lags behind I by $ \frac{π}{2}$ .

$V_{R}^{2} + V_{C}^{2} = E^{2}$

$E = V_{R}^{2} + V_{C}^{2} E^{2} = I^{2} R^{2} + I^{2} X_{C}^{2}$

$I = \frac{E}{R ^{2} + X _{C}^{2}}$

The terms $R^{2} + X_{C}^{2}$ represents the effective resistance of the R-C Circuit and called the capacitive impedance $Z_{C}$ of the circuit.

In C-R Circuit $Z_{C} = R^{2} + X_{C}^{2} = R^{2} + \frac{1}{ω ^{2} C ^{2}}$

Capacitive Impedance $Z_{C}$ : In R-C circuit $R^{2} + X_{C}^{2}$ effective opposition offered by the R-C circuit to the flow of a.c through it.It is known as impedance of R-C circuit and is represented by $Z_{C}$ .

$tan ϕ = \frac{V _{C}}{V _{R}} = \frac{X _{C}}{R} = \frac{1}{ω CR}$

$ϕ = tan^{- 1} (\frac{1}{ω CR})$

7.0Series LCR Circuit

A circuit containing a series combination of an resistance R, a coil of inductance L and a capacitor of capacitance C, connected with a source of alternating e.m.f. of peak value $E_{0}$ , as shown in figure.