Unit – 9 Electrical circuits and Electronics | AC circuits with R, L and C –complex impedance and phase diagram

AC Circuits with R, L and C – Complex Impedance and Phase Diagram

Alternating Current (AC) circuits are electrical circuits in which the current and voltage vary sinusoidally with time. Practical AC circuits consist of resistance (R), inductance (L) and capacitance (C). In AC circuits, current and voltage are generally not in the same phase. To analyze such circuits, the concept of complex impedance is used.

Sinusoidal AC Voltage and Current

A sinusoidal alternating voltage is given by

$$ v(t) = V_0 \sin(\omega t) $$

where $V_0$ is the peak voltage and $\omega = 2\pi f$ is the angular frequency. The corresponding current is

$$ i(t) = I_0 \sin(\omega t + \phi) $$

where $(\phi)$ is the phase difference between voltage and current.

AC Circuit with Resistance (R)

For a pure resistor, the voltage across it is proportional to the current:

$$ v(t) = i(t) R $$

If the current is

$$ i(t) = I_0 \sin(\omega t) $$

then the voltage becomes

$$ v(t) = I_0 R \sin(\omega t) $$

Voltage and current are in the same phase. The impedance of a resistor is

$$ Z_R = R $$

AC Circuit with Inductance (L)

The voltage across an inductor is

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

For a current

$$ i(t) = I_0 \sin(\omega t) $$

the voltage is

$$ v(t) = \omega L I_0 \cos(\omega t) $$

or

$$ v(t) = V_0 \sin\left(\omega t + \frac{\pi}{2}\right) $$

Thus, voltage leads current by 90°. The inductive reactance is

$$ X_L = \omega L $$

The impedance of an inductor is

$$ Z_L = j \omega L $$

AC Circuit with Capacitance (C)

The current through a capacitor is

$$ i(t) = C \frac{dv}{dt} $$

For a voltage

$$ v(t) = V_0 \sin(\omega t) $$

the current is

$$ i(t) = \omega C V_0 \cos(\omega t) $$

or

$$ i(t) = I_0 \sin\left(\omega t + \frac{\pi}{2}\right) $$

Hence, current leads voltage by 90°. The capacitive reactance is

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

The impedance of a capacitor is

$$ Z_C = -\frac{j}{\omega C} $$

Complex Impedance

In AC circuits, resistance and reactance are combined using complex numbers. The general expression for impedance is

$$ Z = R + jX $$

where R is resistance and X is reactance.

Series R–L Circuit

The impedance of a series R–L circuit is

$$ Z = R + jX_L $$

The magnitude of impedance is

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

The phase angle is

$$ \tan \phi = \frac{X_L}{R} $$

Current lags voltage by angle $\phi$.

Series R–C Circuit

The impedance of a series R–C circuit is

$$ Z = R - jX_C $$

The magnitude of impedance is

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

The phase angle is

$$ \tan \phi = \frac{X_C}{R} $$

Current leads voltage by angle $\phi$.

Series R–L–C Circuit

The net reactance of a series R–L–C circuit is

$$ X = X_L - X_C $$

The impedance is

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

The magnitude of impedance is

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

The phase angle is

$$ \tan \phi = \frac{X_L - X_C}{R} $$

Phase Diagram

A phase (phasor) diagram represents the phase relationship between voltage and current. In an R–L–C circuit:

• Voltage across resistor is in phase with current
• Voltage across inductor leads current by 90°
• Voltage across capacitor lags current by 90°

Conclusion

Complex impedance and phasor diagrams provide a powerful method to analyze AC circuits containing resistance, inductance and capacitance. This method simplifies calculations and helps in understanding phase relationships in AC circuits.

Comparison of AC Circuits with Resistance, Inductance and Capacitance

Property	Resistive Circuit (R)	Inductive Circuit (L)	Capacitive Circuit (C)
Basic Element	Resistor	Inductor	Capacitor
Opposition to AC	Resistance (R)	Inductive Reactance (XL)	Capacitive Reactance (XC)
Impedance (Z)	Z = R	Z = jωL	Z = −j / (ωC)
Phase Relation	Voltage and current in phase	Current lags voltage by 90°	Current leads voltage by 90°
Phase Angle (φ)	0°	+90°	−90°
Frequency Dependence	Independent of frequency	Directly proportional to frequency	Inversely proportional to frequency
Power Consumption	Consumes real power	No real power consumed	No real power consumed
Energy Storage	No energy storage	Energy stored in magnetic field	Energy stored in electric field
Instantaneous Power	Always positive	Positive and negative	Positive and negative
Average Power	Maximum	Zero	Zero
Applications	Heaters, bulbs, resistors	Chokes, transformers	Filters, tuning circuits

Multiple Choice Questions – AC Circuits with R, L and C

1. In a purely resistive AC circuit, the phase difference between voltage and current is
   (A) 90°
   (B) 45°
   (C) 0°
   (D) 180°


2. The impedance of a pure inductor in an AC circuit is given by
   (A) Z = R
   (B) Z = ωL
   (C) Z = jωL
   (D) Z = 1 / ωC


3. In an AC circuit containing only a capacitor, the current
   (A) lags the voltage by 90°
   (B) leads the voltage by 90°
   (C) is in phase with voltage
   (D) has no phase relation with voltage


4. The inductive reactance of an inductor is
   (A) inversely proportional to frequency
   (B) independent of frequency
   (C) directly proportional to frequency
   (D) proportional to square of frequency


5. The capacitive reactance of a capacitor decreases when
   (A) capacitance decreases
   (B) frequency decreases
   (C) frequency increases
   (D) voltage increases


6. The impedance of a series R–L circuit is
   (A) Z = R − jX_L
   (B) Z = R + jX_L
   (C) Z = R + jX_C
   (D) Z = R


7. In a series R–C circuit, the current
   (A) lags the applied voltage
   (B) leads the applied voltage
   (C) is in phase with voltage
   (D) becomes zero


8. The magnitude of impedance of a series R–L–C circuit is
   (A) R + X_L + X_C
   (B) √(R² + X²)
   (C) √(R² + (X_L − X_C)²)
   (D) R² + X²


9. At resonance in a series R–L–C circuit, the impedance is
   (A) maximum
   (B) infinite
   (C) minimum and equal to R
   (D) zero


10. Which one of the following does NOT consume average power in an AC circuit?
    (A) Resistor
    (B) Inductor
    (C) Capacitor
    (D) Both inductor and capacitor

Answer Key

1. C
2. C
3. B
4. C
5. C
6. B
7. B
8. C
9. C
10. D