Phase Relation in a Pure Inductor

The voltage-current relation in the case of an inductor is given by:

$$v(t) = L\frac{di(t)}{dt}$$

where:

• $L$ = inductance
• $i(t)$ = current
• $v(t)$ = voltage

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Sinusoidal Current Through an Inductor

Consider the current function:

$$i(t) = I_m \sin \omega t$$

Differentiating with respect to time,

$$\frac{di(t)}{dt} = I_m \omega \cos \omega t$$

Substituting into the voltage equation,

$$v(t) = L I_m \omega \cos \omega t$$

Using the trigonometric identity,

$$\cos \omega t = \sin\left(\omega t + 90^\circ\right)$$

Therefore,

$$v(t) = \omega L I_m \sin\left(\omega t + 90^\circ\right)$$

Let,

$$V_m = \omega L I_m$$

Hence,

$$v(t) = V_m \sin\left(\omega t + 90^\circ\right)$$

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Phase Relationship

If we draw the waveforms for both voltage and current as shown in Fig. 2.11, we can observe the phase difference between these two waveforms.

In a pure inductor, voltage and current are out of phase.

The voltage leads the current by:

$$90^\circ$$

or equivalently,

The current lags behind the voltage by $90^\circ$.

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Impedance of a Pure Inductor

The impedance is the ratio of voltage to current.

$$Z = \frac{V}{I}$$

For a pure inductor,

$$Z = \frac{V_m}{I_m} = \omega L$$

The inductive reactance is represented by:

$$X_L = \omega L$$

where:

• $\omega = 2\pi f$
• $L$ = inductance

Thus,

$$X_L = 2\pi fL$$

The impedance of a pure inductor is represented in complex form as:

$$Z = jX_L$$

or

$$Z = j\omega L$$

Hence, a pure inductor has an impedance whose value is:

$$j\omega L$$

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Characteristics of Pure Inductive Circuit

• Current lags voltage by $90^\circ$.
• Impedance is purely imaginary.
• No real power is consumed.
• Energy is temporarily stored in the magnetic field.
• The average power over one complete cycle is zero.

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Power in a Pure Inductor

Instantaneous power is:

$$p(t) = v(t)i(t)$$

Average power is:

$$P = VI\cos\phi$$

For a pure inductor,

$$\phi = 90^\circ$$

Since,

$$\cos 90^\circ = 0$$

therefore,

$$P = 0$$

Thus, a pure inductor consumes no average power.

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Summary

• In a pure inductor, current lags voltage by $90^\circ$.
• Voltage leads current by $90^\circ$.
• The inductive reactance is:

$$X_L = \omega L$$

• The impedance of an inductor is:

$$Z = j\omega L$$

• A pure inductor stores energy in its magnetic field.
• The average power consumed by a pure inductor is zero.