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AC Circuits with Resistors, Inductors and Capacitors

Mar 27, 2021

—

John Mulindi

in Electrical Circuits & Networks

Contents

1 Purely Resistive Circuits
2 Purely Capacitive Circuits
3 Purely Inductive Circuits
4 R, L, and C in Series
5 R, L and C in parallel

Purely Resistive Circuits

In resistive elements the current and voltage are in phase.

Figure 1(a) purely resistive circuit

Figure 1(b) waveform of a purely resistive circuit (current in phase with voltage)

Figure 1(c) phasor diagram resistive circuit

Purely Capacitive Circuits

In capacitive circuits, the current leads the voltage by 90°.

Figure 2(a) purely capacitive circuit

Figure 2(b) waveform of a purely capacitive circuit

Figure 2(c) phasor diagram of a purely capacitive circuit

Purely Inductive Circuits

In inductive circuits the current lags the voltage by 90°.

Figure 3(a) purely inductive circuit

Figure 3(b) waveform of a purely inductive circuit

Figure 3(c) phasor diagram of a purely inductive circuit

Related: Inductance

R, L, and C in Series

Because the voltages and the currents are not in phase in capacitive and inductive AC circuits, these devices have impedance (Z) and not resistance (R) and therefore impedance and resistance cannot be directly added. Consider the following circuit where a resistor, capacitor and inductor are connected in in series:

Figure 4(a) AC circuit with R, C and L in series

The same current will flow through all the three devices, but the voltage in the capacitor and inductor will be 180° out of phase, and 90° out of phase with the voltage in the resistor.

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Figure 4(b) waveforms and phase relations in a R, C and L series circuit

These voltages can be combined using vectors as illustrated in figure 4(c) and 4(d) to give:

Where,

E = supply

V_R =voltage across the resistor

V_L= voltage across the inductor

V_C = voltage across the capacitor

Figure 4(c) voltage vectors of R, L & C series circuit

Figure 4(d) the resulting voltage E vector

The vector addition of the voltages is shown in figure 4(c) where the relationship between V_R, V_Land V_Care shown. In figure 4(d) the V_C – V_L vector and V_Rare shown vectors are shown with the resulting E vector which from the trigonometry function gives equation (a) above.

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The impedance triangle is illustrated below:

Figure 4(e) impedance triangle

Or if V_L is greater the V_C

The current flowing through the circuit can be obtained from ohm’s law as follows:

X_L and X_C are frequency dependent and as the frequency increases X_L increases and X_Cdecreases. A frequency is reached where X_L and X_C are equal and the voltages across each component are equal and opposite and cancel each other. At this frequency Z = R, E =IR and the current is maximum. This frequency is called the resonant frequency of the circuit.

At resonance

When the input frequency is below the resonant frequency, X_C is larger than X_L and the circuit is capacitive, and above the resonant frequency, X_Lis greater than X_C and the circuit is inductive. Plotting the input current against the frequency shows a peak in the input current at the resonant frequency as illustrated below:

Figure 4(f) current versus frequency in a series circuit

Figure 4(g) current versus frequency in a parallel circuit

R, L and C in parallel

Consider the circuit below:

Figure 4(h) parallel R, L and C circuit

In a parallel circuit shown in figure 4(h), each component receives the same voltage but not the same current as illustrated by the waveforms below:

Figure 4(I) waveforms and phase relations in a parallel R, L and C circuit

The source current is the sum of the currents in each component and is given by

The impedance of the circuit Z as seen by the input is given by:

At resonant frequency, IL and IC become equal and cancel each other so that E = IR. This can be noted in equation (b).

Below the resonant frequency, the circuit is inductive, and above the resonant frequency the circuit is capacitive. Plotting the current against frequency shows that the current is at minimum at the resonant frequency as shown in figure 4(g) above.

The frequency at resonance is given by

Current through the circuit is given by

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