Series RLC Circuit

A series RLC circuit is the series combination of:

• Resistance
• Inductance
• Capacitance

If we observe the impedance diagrams of series RL and series RC circuits as shown in Fig. 2.15(a) and (b):

• The inductive reactance $X_L$ is displayed on the positive $j$ axis.
• The capacitive reactance $X_C$ is displayed on the negative $j$ axis.

These reactances are $180^\circ$ apart and tend to cancel each other.

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Inductive and Capacitive Reactance

The inductive reactance is:

$$X_L = \omega L$$

The capacitive reactance is:

$$X_C = \frac{1}{\omega C}$$

where:

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

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Net Reactance in Series RLC Circuit

The magnitude and type of reactance in a series RLC circuit is the difference between the two reactances.

Thus,

$$X = X_L - X_C$$

• If $X_L > X_C$, the circuit behaves inductively.
• If $X_C > X_L$, the circuit behaves capacitively.
• If $X_L = X_C$, the circuit behaves purely resistively.

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Impedance of Series RLC Circuit

The impedance of a series RLC circuit is given by:

$$Z = R + j(X_L - X_C)$$

where:

• $R$ = resistance
• $X_L$ = inductive reactance
• $X_C$ = capacitive reactance

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Magnitude of Impedance

The magnitude of impedance is:

$$|Z| = \sqrt{ R^2 + (X_L - X_C)^2 }$$

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Phase Angle

The phase angle for a series RLC circuit is:

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

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Current in Series RLC Circuit

The current flowing through the circuit is:

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

or

$$I = \frac{V}{ \sqrt{ R^2 + (X_L - X_C)^2 } }$$

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Nature of Series RLC Circuit

Inductive Circuit

If

$$X_L > X_C$$

then,

• The circuit behaves inductively.
• Current lags voltage.

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Capacitive Circuit

If

$$X_C > X_L$$

then,

• The circuit behaves capacitively.
• Current leads voltage.

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Resonance Condition

If

$$X_L = X_C$$

then,

$$X_L - X_C = 0$$

Therefore,

$$Z = R$$

At this condition:

• The circuit behaves like a pure resistor.
• Voltage and current are in phase.
• The impedance becomes minimum.
• The current becomes maximum.

This condition is called resonance.

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Voltage Relations in Series RLC Circuit

The resistor voltage is:

$$V_R = IR$$

The inductor voltage is:

$$V_L = IX_L$$

The capacitor voltage is:

$$V_C = IX_C$$

The source voltage is the phasor sum of these voltages.

Thus,

$$V = \sqrt{ V_R^2 + (V_L - V_C)^2 }$$

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Power Factor

The power factor of a series RLC circuit is:

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

• Lagging for inductive circuits
• Leading for capacitive circuits
• Unity at resonance

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Characteristics of Series RLC Circuit

• Contains resistance, inductance, and capacitance in series.
• Reactances oppose each other.
• Circuit may behave inductively or capacitively.
• Impedance depends on net reactance.
• Resonance occurs when $X_L = X_C$.

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Summary

• A series RLC circuit contains resistance, inductance, and capacitance connected in series.
• The impedance is:

$$Z = R + j(X_L - X_C)$$

• The phase angle is:

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

• Inductive and capacitive reactances oppose each other.
• Resonance occurs when:

$$X_L = X_C$$

• At resonance, impedance is minimum and current is maximum.