Series RC Circuit

Consider the RC series circuit shown in Fig. 2.14.

If we apply the complex function

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

to the circuit, we get a complex response.

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Applying Kirchhoff’s Voltage Law

Applying Kirchhoff’s law to the circuit, we get:

$$v(t) = Ri(t) + \frac{1}{C} \int i(t)\,dt$$

Solving this equation, we get:

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

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Impedance of Series RC Circuit

Impedance is defined as the ratio of voltage to current.

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

For a series RC circuit,

$$Z = R - jX_C$$

where:

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

Here, impedance $Z$ consists of:

• Resistance $R$, which is the real part
• Capacitive reactance $X_C$, which is the imaginary part

The resistance $R$ is located on the real axis, and the capacitive reactance $X_C$ is located on the negative $j$ axis.

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Magnitude of Impedance

The magnitude of impedance is:

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

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Phase Angle

The phase angle is:

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

Since the circuit is capacitive, current leads voltage.

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Phase Relation in Series RC Circuit

When a sinusoidal voltage is applied to an RC series circuit, the current in the circuit and voltages across each of the elements are sinusoidal.

Here:

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

The phase angle between the current and capacitor voltage is always:

$$90^\circ$$

The amplitudes and phase relations between voltages and current depend on the values of:

• Resistance $R$
• Capacitive reactance $X_C$

The circuit is a series combination of both resistance and capacitance; therefore, the phase angle between the applied voltage and total current is somewhere between:

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

depending on the relative values of resistance and reactance.

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Voltage Relations

In a series RC circuit, the current is the same through the resistor and capacitor.

The resistor voltage is:

$$V_R = IR$$

The capacitor voltage is:

$$V_C = IX_C$$

Here:

• $V_R$ and $I$ are in phase.
• Current $I$ leads $V_C$ by $90^\circ$.

From Kirchhoff’s Voltage Law, the sum of voltage drops must be equal to the applied voltage.

Therefore, the source voltage is:

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

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Current in Series RC Circuit

The circuit current is:

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

or

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

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Power Factor of RC Circuit

The power factor is:

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

Since current leads voltage, the RC circuit has a leading power factor.

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Characteristics of Series RC Circuit

• Current leads voltage.
• Impedance has both real and imaginary parts.
• The resistor consumes real power.
• The capacitor stores energy in electric field.
• The phase angle depends on $R$ and $X_C$.

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Summary

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

$$Z = R - jX_C$$

• Capacitive reactance is:

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

• Current leads voltage in a series RC circuit.
• The phase angle is:

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

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