Alternating Current in Inductors: Phase Difference, Inductive Reactance, and Power Explained

23.5 ALTERNATING CURRENTS THROUGH INDUCTORS

23.5.1 Phase Difference between I and V for Pure Inductors

Figure 23.21 The figure shows a pure inductor connected to a sinusoidal voltage source, producing a sinusoidal current in the circuit.

A pure inductor with inductance $L$ is connected to a sinusoidal voltage supply of voltage $V$, as shown in Figure 23.21. The voltage produces a sinusoidal current $I$ which is given by

$I = I_0 \ sin \ ωt$

where I₀ is the maximum current. Since the current is changing with time, self induction occurs in the inductor, producing a back emf $E_b$. The back emf is given by

$E_b = −L \frac{dI}{dt}$

$= −L \frac{d}{dt} (I_0 \ sin \ ωt)$

$= −ωLI_0 \ cos \ ωt$

Since the inductor is assumed to be ideal, we have

applied voltage = – back emf

$V_L = –(–ωLI_0 \ cos \ ωt)$

$= ωLI_0 \ cos \ ωt$

$= (ωLI_0) \ sin (ωt + \frac{1}{2}π)$

Let us make a comparison:

$I = I_0 \ sin \ ωt$

$V_L = (ωLI_0) \ sin (ωt + \frac{1}{2}π)$

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

23.5.2 I–t graph and V_L–t graph for Pure Inductor

Figure 23.22 The figure shows the I–t and Vₗ–t graphs along with a phasor diagram, illustrating the phase relationship between current and voltage in an inductor.

We can draw one I–t graph and one V_L–t graph using the I and V_L functions given above. Figure 23.22 shows these two graphs. The phasor diagram is incorporated in the graphs.

We notice that:

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

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

(c) $I = 0$ when $V_L$ is maximum, and I is maximum when $V_L = 0$. In other words, I varies out of step with V_L. This is the main characteristic of two quantities varying out of phase by 90°.

(d) It is possible to get values where ($+V_L$, $+I$), ($+V_L$, $–I$), ($–V_L$, $+I$), ($–V_L$, $–I$). This means that the product $V_LI$ for a pure inductor could have a positive or a negative value.

23.5.3 Inductive Reactance

1. Meaning

When alternating current flows through a pure inductor, the current experiences some opposition although there is no resistance in the circuit. This opposition, which arises due to the inductance possessed by the inductor, is the inductive reactance.

2. Magnitude

We have

$V_L = (ωLI_0) \ cos \ ωt$

When the voltage becomes maximum, we have

$cos \ ωt = ±1$

and so,

$V_0 = ωLI_0$

or,

$\frac{V_0}{I_0} = ωL$

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

$X_L = \frac{V_0}{I_0}$ (inductive reactance)

$V_0 = I_0X_L$

$X_L = ωL$

where

$ω = 2πf$

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

We have

$V_0 = \sqrt{2} V_{rms}$

$I_0 = \sqrt{2} I_{rms}$

Hence,

$X_L = \frac{V_{rms}}{I_{rms}}$

3. Unit

The unit of inductive reactance is the ohm.

4. Graph of X_L – f

$X_L = 2πfL$

$X_L ∝ f$

$X_L ∝ L$

The graph in Figure 23.23 shows how $X_L$ varies linearly with f.

This graph shows that the inductive reactance, $X_L$, increases linearly with frequency, f.
Since $X_L= 2\pi fL$, the relationship is directly proportional, meaning that as the frequency increases, the opposition offered by the inductor to the alternating current also increases.

EXAMPLE 23.17

What must be done in order to double the inductive reactance of a pure inductor?

Answer

$X_L ∝ f$

$X_L ∝ L$

Hence, we can:
(a) double the frequency of the current flowing through the inductor
(b) double the inductance of the inductor

EXAMPLE 23.18

A current I which flows through an ideal inductor with inductance of 200 mH is given by

$I = (2.0) \ sin \ (100πt)$

(a) Determine
(i) the inductive reactance of the inductor
(ii) the maximum voltage that can appear across the inductor

(b) Write down an expression for the voltage V which is applied across the inductor:

Answer

(a) (i) Given

$ω = 100π$

$X_L = ωL$

$= (100π)(0.200) = 62.8 \ Ω$

(ii)
$V_0 = I_0X_L$

$= (2.0)(62.8) = 126 \ V$

(b)
$V_L = V_0 \ sin (ωt + \frac{1}{2}π)$

$= (126 \ V) \ sin (100πt + \frac{1}{2}π)$

EXAMPLE 23.19

A coil with inductance 100 mH and negligible resistance is connected to a sinusoidal voltage of frequency 50 Hz. If the current flowing through the coil is 8.0 A (rms), determine:

(a) the inductive reactance

(b) the rms voltage applied across the coil

Answer

(a)
$X_L = 2πfL$

$= 2π(50)(0.100) = 31.4 \ Ω$

(b)
$V_{rms} = I_{rms} X_L$

$= (8.0)(31.4) = 251 \ V$

23.5.4 Power in Inductive Circuit

1. Instantaneous power

Suppose that a current I given by

$I = I_0 \ sin \ ωt$

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

$V_L = V_0 \ cos \ ωt$

since V leads I by $\frac{1}{2}π$ radian. The instantaneous power P delivered to the inductor is given by

$P = IV_L$

$= (I_0 \ sin \ ωt)(V_0 \ cos \ ωt)$

$= I_0V_0 \ (sin \ ωt)(cos \ ωt)$

$= \frac{1}{2} I_0V_0 \ sin \ (2ωt)$

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

Figure 23.24 The figure shows how instantaneous power varies with time, alternating between positive and negative values over one cycle.

Figure 23.24 shows a graph of P against time t. Notice that when the current has gone through one cycle,

(a) the power will have gone through two cycles

(b)
(i) the poweg has two regions of positive values and two regions of negative values

(ii) area of positive region = area of negative region

(iii) total area for one current cycle = 2 positive areas + 2 negative areas = 0. This implies that the average power is zero.

2. Average power

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

$P_{av} = 0$

Refer to the power curve shown in Figure 23.24 above. We notice that the positive power is cancelled by the negative power, producing a net power of zero over every one cycle of the power curve.

Reasons:

• (a) During one half cycle of positive power, the inductor absorbs electrical energy. This energy is converted to magnetic energy stored in the magnetic field produced by the current flowing through the inductor.
• (b) The positive power is followed by one half cycle of negative power. Now the magnetic field is shrinking and collapsing. The magnetic energy is diminishing with time. During this half cycle, the magnetic energy is converted into electrical energy and returned back to the circuit.
• (c) Over one power cycle, the energy of the system undergoes transformation from one form of energy into another, without any loss to the surroundings:
  electrical energy ⇌ magnetic energy
• (d) Since the ideal inductor has no resistance, no energy is dissipated in the form of heat.

(Note: We are assuming that there are no electromagnetic waves generated as the electrons oscillate in the circuit.)

Applications of Inductors in AC Circuits

⚡ Power Transmission Systems

Inductors are widely used in power transmission lines to control current and reduce sudden changes. They help stabilize the flow of alternating current and protect equipment from voltage spikes.

Inductive reactance (X_L) increases with frequency → helps control high-frequency disturbances.

🔊 Audio Equipment (Speakers & Filters)

Inductors are used in crossover circuits in speakers to filter frequencies. They allow low-frequency signals to pass while blocking high-frequency signals.

X_L ∝ f → higher frequency = more opposition → used for filtering sound.

📡 Radio and Communication Circuits

Inductors are essential in tuning circuits (LC circuits) used in radios, TVs, and wireless communication. They help select specific frequencies from signals.

Works with capacitors to create resonance at desired frequency.

🔌 Transformers

Transformers rely on inductance to transfer energy between coils. They are used to step up or step down voltage in electrical systems.

Based on electromagnetic induction → no direct electrical connection.

⚙️ Energy Storage (Magnetic Field)

Inductors store energy temporarily in their magnetic field when current flows through them. This energy is released back into the circuit when current decreases.

Energy alternates between electrical ↔ magnetic (no energy loss in ideal case).

💡 Power Factor Correction

Inductors affect the phase difference between voltage and current. They are used in power systems to manage power factor and improve efficiency.

Voltage leads current by 90° in pure inductors.

Conclusion

Alternating current flowing through a pure inductor exhibits unique characteristics where the voltage leads the current by 90°. This phase difference results in a continuous exchange of energy between the electrical and magnetic fields without any net energy loss in an ideal system.

The concept of inductive reactance (X_L) explains how inductors oppose alternating current, with its value depending on both frequency and inductance. As frequency increases, the opposition to current also increases.

Additionally, 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 and returned to the circuit. Key Insight: Inductors do not consume energy — they store and release it, making them essential components in AC circuits, power systems, and electronic devices.

Frequently Asked Questions (FAQ)

What is inductive reactance (X_L)?

Inductive reactance is the opposition offered by an inductor to alternating current. It is given by X_L = 2πfL.

Why does voltage lead current in an inductor?

Voltage leads current by 90° because the inductor produces a back emf that opposes the change in current, creating a phase difference.

Why is the average power in a pure inductor zero?

Energy is stored in the magnetic field during one half cycle and returned during the next, resulting in zero net energy consumption.

What factors affect inductive reactance?

It depends on frequency (f) and inductance (L).

Where are inductors used in real life?

Inductors are used in transformers, filters, power supplies, radios, and audio systems.

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