Phase Relation in Pure Resistor

When a sinusoidal voltage of certain magnitude is applied to a resistor, a certain amount of sine wave current passes through it.

We know the relation between $v(t)$ and $i(t)$ in the case of a resistor.

The voltage-current relation in case of a resistor is linear, i.e.

$$v(t) = Ri(t)$$

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Sinusoidal Voltage Applied to a Resistor

Consider the sinusoidal voltage function:

$$v(t) = V_m \sin \omega t$$

Substituting this in the voltage-current relation equation,

$$V_m \sin \omega t = Ri(t)$$

Therefore,

$$i(t) = \frac{V_m}{R} \sin \omega t$$

Let,

$$I_m = \frac{V_m}{R}$$

Hence,

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

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Phase Relationship

If we draw the waveform for both voltage and current as shown in Fig. 2.10, there is no phase difference between these two waveforms.

The amplitudes of the waveforms may differ according to the value of resistance.

However, both waveforms reach:

• Maximum value at the same instant
• Minimum value at the same instant
• Zero value at the same instant

Therefore, in a pure resistive circuit:

Voltage and current are said to be in phase.

Thus, the phase angle between voltage and current is:

$$\phi = 0^\circ$$

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Impedance of a Pure Resistor

The impedance is defined as the ratio of voltage to current.

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

With AC voltage applied to elements, the ratio of voltage to the corresponding current consists of:

• Magnitude
• Phase angle

Since the phase difference is zero in case of a resistor, the phase angle is zero.

Therefore, the impedance of a pure resistor consists only of magnitude.

$$Z = R$$

or in polar form,

$$Z = R \angle 0^\circ$$

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Characteristics of Pure Resistive Circuit

• Voltage and current are in phase.
• Phase angle is zero.
• Impedance is purely real.
• No energy is stored in the resistor.
• Electrical energy is converted into heat.

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Power in Pure Resistive Circuit

Instantaneous power is given by:

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

Average power consumed by a resistor is:

$$P = VI \cos \phi$$

Since,

$$\phi = 0^\circ$$

and

$$\cos 0^\circ = 1$$

therefore,

$$P = VI$$

Thus, all the supplied power is consumed by the resistor.

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Summary

• In a pure resistor, voltage and current are in phase.
• The phase difference between voltage and current is zero.
• The impedance of a resistor is purely real.
• A resistor does not store electrical energy.
• Electrical energy supplied to the resistor is dissipated as heat.