Series RL Circuit

Consider the RL series circuit shown in Fig. 2.13.

If we apply the real function

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

to the circuit, the response may be

$$i(t) = I_m \cos(\omega t - \phi)$$

Similarly, if we apply the imaginary function

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

to the same circuit, the response is

$$i(t) = jI_m \sin(\omega t - \phi)$$

If we apply a complex function, which is a combination of real and imaginary functions, we will get a complex response.

This complex function is:

$$v(t) = V_m e^{j\omega t}$$

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Applying Kirchhoff’s Voltage Law

Applying Kirchhoff’s law to the circuit shown in Fig. 2.13, we get:

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

The solution of this differential equation is:

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

Substituting $i(t)$ in the above equation, we get:

$$v(t) = I_m R \sin(\omega t - \phi) + \omega L I_m \cos(\omega t - \phi)$$

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Impedance of Series RL Circuit

Impedance is defined as the ratio of the voltage to current function.

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

Complex impedance is the total opposition offered by the circuit elements to AC current and can be displayed on the complex plane.

The impedance is denoted by $Z$.

Here:

• Resistance $R$ is the real part of impedance.
• Reactance $X_L$ is the imaginary part of impedance.

The resistance $R$ is located on the real axis.

The inductive reactance $X_L$ is located on the positive $j$ axis.

The impedance of a series RL circuit is:

$$Z = R + jX_L$$

where:

$$X_L = \omega L$$

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Magnitude of Impedance

The magnitude of impedance is:

$$|Z| = \sqrt{R^2 + X_L^2}$$

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Phase Angle

The phase angle is:

$$\phi = \tan^{-1}\left(\frac{X_L}{R}\right)$$

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Current in Series RL Circuit

The circuit current is given by:

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

or

$$I = \frac{V}{\sqrt{R^2 + X_L^2}}$$

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Phase Relation in Series RL Circuit

In Fig. 2.13:

• The resistor voltage $V_R$ and current $I$ are in phase with each other.
• The inductor voltage $V_L$ leads the current by $90^\circ$.
• The current lags behind the source voltage $V_S$.

The phase angle between current and voltage in a pure inductor is always:

$$90^\circ$$

The amplitudes of voltages and currents in the circuit are completely dependent on the values of:

• Resistance $R$
• Inductive reactance $X_L$

In a series RL circuit, the phase angle is somewhere between:

$$0^\circ \text{ and } 90^\circ$$

depending on the relative values of $R$ and $X_L$.

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Voltage Relations

The voltage across the resistor is:

$$V_R = IR$$

The voltage across the inductor is:

$$V_L = IX_L$$

From Kirchhoff’s Voltage Law, the source voltage is the phasor sum of $V_R$ and $V_L$.

Thus,

$$V_S = \sqrt{V_R^2 + V_L^2}$$

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Power Factor of RL Circuit

The power factor is:

$$\cos\phi = \frac{R}{Z}$$

Since current lags voltage, the RL circuit has a lagging power factor.

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Characteristics of Series RL Circuit

• Current lags voltage.
• Impedance has both real and imaginary parts.
• The resistor consumes real power.
• The inductor stores energy in magnetic field.
• The phase angle depends on $R$ and $X_L$.

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Summary

• A series RL circuit contains resistance and inductance connected in series.
• The impedance of the circuit is:

$$Z = R + jX_L$$

• The inductive reactance is:

$$X_L = \omega L$$

• Current lags voltage in a series RL circuit.
• The phase angle is:

$$\phi = \tan^{-1}\left(\frac{X_L}{R}\right)$$

• Source voltage is the phasor sum of resistor and inductor voltages.