RL Series AC Circuit

This section covers the behavior of alternating current (AC) in circuits containing capacitors and resistors, specifically focusing on RL and RC series configurations. It introduces the concepts of capacitive reactance and impedance.

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Key Concepts

A.C Through a Capacitor

When an alternating voltage $V = V_{0} \sin \omega t$ is applied across a capacitor, the charge on the capacitor at any instant is $q = CV = CV_{0} \sin \omega t$. The resulting current is the rate of change of charge:
$$I = \frac{\Delta q}{\Delta t} = CV_{0} \omega \cos(\omega t) = I_{0} \sin\left(\omega t + \frac{\pi}{2}\right)$$
where the peak current is $I_{0} = CV_{0} \omega$.

• Phase Relationship: In a purely capacitive AC circuit, the voltage lags the current by $\frac{\pi}{2}$ radians or $90^{\circ}$.

Figure 20.12 (b): Voltage lags the current by $90^{\circ}$.

• Phasor Diagram: The phasor diagram shows the voltage phasor $90^{\circ}$ behind the current phasor.

• Capacitive Reactance ($X_C$): This is the opposition offered by a capacitor to the flow of AC. It is analogous to resistance.
  $$X_{C} = \frac{V_0}{I_0} = \frac{1}{C\omega} = \frac{1}{2\pi f C}$$ Capacitive reactance is measured in Ohms () and is inversely proportional to the frequency () and capacitance (). For DC (), is infinite, meaning a capacitor blocks DC current.

Impedance (Z)

Impedance is the total opposition to the flow of alternating current in a circuit, combining the effects of resistance (R) and reactance (X). It is measured in Ohms ($\Omega$).

RL Series AC Circuit

A circuit with a resistor (R) and an inductor (L) connected in series.

• The voltage drop across the resistor, $V_R = IR$, is in phase with the current.
• The voltage drop across the inductor, $V_L = IX_L$, leads the current by .

Figure 20.13 (b): Phasor diagram of RL series circuit.

The total voltage $V$ is the phasor sum of $V_R$ and $V_L$. The impedance $Z$ is calculated using the Pythagorean theorem:
$$V$$ where is the inductive reactance.

• Impedance Triangle: A right-angled triangle with sides R, $X_L$, and Z.

Figure 20.13 (c): RL Series Impedance triangle.

RC Series AC Circuit

A circuit with a resistor (R) and a capacitor (C) connected in series.

• The voltage drop across the resistor, $V_R = IR$, is in phase with the current.
• The voltage drop across the capacitor, $V_C = IX_C$, lags the current by .

Figure 20.14 (b): Phasor diagram of RC series circuit.

The total voltage $V$ is the phasor sum of $V_R$ and $V_C$. The impedance $Z$ is:
$$V=$$ where is the capacitive reactance.

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Possible Questions/Answer

• Q: An AC voltage across a $0.5 \mu F$ capacitor is $V = 16 \sin(2 \times 10^3 t)$ V. Find the capacitive reactance and the peak current.
  A: From the voltage equation, $V_0 = 16$ V and rad/s.

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Summary

• In a pure capacitive AC circuit, the voltage lags the current by $90^{\circ}$ ($\pi/2$).
• Capacitive reactance ($X_C$) is the opposition to AC by a capacitor, inversely proportional to frequency and capacitance.
• In an RL series circuit, the voltage leads the current by an angle between $0^{\circ}$ and .

The analysis of impedance and phase relationships in RL and RC circuits is fundamental to designing and understanding filters, power supplies, and many other electronic systems.

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References

(Derived from FBISE textbook)

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