Alternating Current in Capacitors: Phase Difference, Capacitive Reactance, and Power Explained

23.6 Alternating Currents Through Capacitors

23.6.1 Phase Difference between I and V for Pure Capacitor

Figure 23.25 The figure shows a capacitor connected to a sinusoidal voltage source, where the applied voltage produces an alternating current in the circuit.

A pure capacitor with capacitance C is connected to a sinusoidal voltage supply, as shown in Figure 23.25. The voltage V is given by

$V = V_C = V_0 \ sin \ ωt$

At a particular instant, the charge Q stored by the capacitor is

$Q = CV_C$

$= CV_0 \ sin \ ωt$

The instantaneous current I flowing in the circuit is given by

$I = \frac{dQ}{dt}$

$= \frac{d}{dt} (CV_0 \ sin \ ωt)$

$= (CV_0ω) \ cos \ ωt$

$= (CV_0ω) \ sin \ (ωt + \frac{1}{2}π)$

Let us make a comparison:

$I = (CV_0ω) \ sin \ (ωt + \frac{1}{2}π)$

$V_C = V_0 \ sin \ ωt$

Notice that I has a ‘$sin \ (ωt + \frac{1}{2}π)$’ term whereas $V_C$ has a ‘sin ωt’ term. This means that $V_C$ is changing not in step with I. In fact, $V_C$ is out of step with I by $\frac{1}{2}π$ radian. We express this fact by saying that there is a phase difference of ½π radian between I and $V_C$, with I leading $V_C$.

23.6.2 I–t graph and V_C–t graph for Pure Capacitor

Figure 23.26 The figure shows the I–t and V₍C₎–t graphs along with a phasor diagram, illustrating the phase relationship between current and capacitor voltage.

We can draw one I–t graph and one V_C–t graph using the I and $V_C$ functions given above. Figure 23.26 shows these two graphs. The phasor diagram is incorporated in the graphs.

We notice that:

(a) there are two arrows, one representing $V_C$ and the other representing I.

(b) the two arrows are $90^0$ apart, with I rotating ahead of $V_C$. This indicates that I leads $V_C$ by $\frac{1}{2}π$ radian.

(c) $I = 0$ when $V_C$ is maximum, and I is maximum when $V_C = 0$.

(d) It is possible to get values where ($+V_C$, $+I$), ($+V_C$, $−I$), ($−V_C$, $+I$), ($−V_C$, $−I$). This means that the product $VI$ for a pure capacitor could have a positive or a negative value.

23.6.3 Capacitive Reactance

1. Meaning

When alternating current flows in a circuit with a pure capacitor, the current experiences some opposition although there is no resistance in the circuit. This opposition, which arises due to the capacitance possessed by the capacitor, is the capacitive reactance.

2. Magnitude

We have

$I = (CV_0ω) \ cos \ ωt$

When the current becomes maximum, we have

$cos \ ωt = ±1$

and so,

$I_0 = CV_0ω$

or,

$\frac{V_0}{I_0} = \frac{1}{ωC}$

The ratio $\frac{V_0}{I_0}$ is the capacitive reactance, X_C. Hence, we have

$X_C = \frac{V_0}{I_0}$ (capacitive reactance)

$V_0 = I_0X_C$

$X_C = \frac{1}{ωC}$

where,

$ω = 2πf$

and f is the frequency of the alternating current or voltage.

We also have

$X_C = \frac{V_{rms}}{I_{rms}}$

3. Unit

The unit of capacitive reactance is the ohm.

4. Graph of X_C – f

$X_C = \frac{1}{ωC} = \frac{1}{(2πfC)}$

Hence,

$X_C ∝ \frac{1}{f}$

The graph in Figure 23.27 shows how $X_C$ varies inversely with f.

Figure 23.27 The figure shows that capacitive reactance ($X_C$) decreases inversely as the frequency increases.

EXAMPLE 23.20

A sinusoidal voltage of 240 V rms with frequency 50 Hz is connected to a capacitor of capacitance 10 μF. Determine:
(a) the capacitive reactance of the capacitor
(b) the rms current flowing in the circuit.

Answer

(a)
$X_C = \frac{1}{ωC}$

$= \frac{1}{(2πfC)}$

$= \frac{1}{[2π(50)(10 \times 10^{-6})]} = 318 \ Ω$

(b)
$I_{rms} = \frac{V_{rms}}{X_C}$

$= \frac{240}{318} = 0.75 \ A$

23.6.4 Power in Capacitive Circuit

1. Instantaneous power

Figure 23.28 The figure shows how instantaneous power varies with time in a capacitor, alternating between positive and negative values similar to an inductor.

Suppose that a current I given by

$I = I_0 \ cos \ (ωt)$

passes through a pure capacitor. Then the voltage V_C which is applied across the capacitor to produce this current is given by

$V_C = V_0 \ sin \ (ωt)$

since V_C lags I by ½π radian. The instantaneous power P delivered to the capacitor is given by

$P = IV_C$

$= (I_0 \ cos \ ωt)(V_0 \ sin \ ωt)$

$= I_0V_0 (cos \ ωt)(sin \ ωt)$

$= \frac{1}{2}I_0V_0 \ sin (2ωt)$

The maximum power available at any instant is $\frac{1}{2} I_0V_0$.

Figure 23.28 shows a graph of P against time t. This power graph for capacitor is similar to that for the inductor.

2. Average power

The average value of sin ($2ωt$) is zero. This means that the average power P_av is

$P_{av}= 0$

(a) During one half cycle of positive power, the capacitor absorbs electrical energy. This energy is stored in the electric field existing in the space between the plates of the capacitor.

(b) The positive power is followed by one half cycle of negative power. Now the electric field is diminishing in strength. Hence, the electrical energy stored is also diminishing with time. During this half cycle, the electrical energy is returned back to the circuit.

(c) Since the ideal capacitor has no resistance, no energy is dissipated in the form of heat.

Applications of Capacitors in AC Circuits

🔋 Power Supply Filtering

Capacitors are used in power supplies to smooth pulsating DC after rectification. They reduce ripple and provide a stable output voltage.

Smooth DC output → stable electronics ⚡

📡 Signal Coupling & Decoupling

Capacitors allow AC signals to pass while blocking DC, making them useful in amplifiers and communication circuits.

Pass AC, block DC ✔️

🔊 Audio Systems

In audio circuits, capacitors are used to filter noise and separate frequency components for better sound quality.

Improves sound clarity 🎵

⚡ Power Factor Correction

Capacitors are used to improve power factor in AC systems by reducing phase difference between voltage and current.

Reduces energy loss & improves efficiency

⏱ Timing Circuits

Capacitors charge and discharge over time, making them useful in timing and oscillator circuits.

Used in timers & oscillators ⏳

📺 Electronic Devices

Capacitors are found in almost all electronic devices such as TVs, radios, and computers to regulate voltage and store energy.

Essential component in electronics 💻

Conclusion

Capacitors in alternating current (AC) circuits exhibit unique behavior where the current leads the voltage by 90°. This phase difference results in a continuous exchange of energy between the electric field and the circuit without any net energy loss in an ideal capacitor.

The concept of capacitive reactance (X_C) explains how capacitors oppose alternating current, with its value depending on both frequency and capacitance. As frequency increases, the capacitive reactance decreases, allowing more current to flow.

Although instantaneous power varies with time, the average power over a complete cycle is zero. This indicates that energy is not dissipated but temporarily stored in the electric field and then returned to the circuit.

Key Insight: Capacitors do not consume energy — they store and release electrical energy, making them essential in filtering, signal processing, and AC circuit control.

Frequently Asked Questions (FAQ)

What is capacitive reactance (X_C)?

Capacitive reactance is the opposition offered by a capacitor to alternating current, given by X_C = 1 / (2πfC).

Opposition decreases as frequency increases ⚡

Why does current lead voltage in a capacitor?

In a capacitor, current leads voltage by 90° because the current depends on how fast the voltage changes over time.

Current leads voltage by 90° 🔁

Why is the average power in a capacitor zero?

The capacitor stores energy in its electric field during one half cycle and returns it during the next, resulting in zero net power consumption.

Energy stored ⇌ returned 🔋

What affects capacitive reactance?

Capacitive reactance depends on frequency (f) and capacitance (C).

X_C ∝ 1/f and 1/C

Where are capacitors used in real life?

Capacitors are used in power supplies, filters, audio systems, communication devices, and electronic circuits.

Used in almost all electronics 📱

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