AC Voltage Applied to an Inductor: Definition, Derivation, Benefits & Class 12 Notes

When AC voltage is applied to an ideal inductor, it generates a self-induced EMF (electromotive force), due to Lenz's Law. This led to the current lagging the voltage by 90 degrees (π/2 radians). 

Mathematical Derivation

Here we will derive the current response when an AC voltage is applied to an inductor.

• AC voltage: $v\left(t\right)=V_m\sin \left(\omega t\right)$
• $V_m$ peak voltage
• $\omega$ is angular frequency
• $L\; is$ Inductance of the coil

Using the inductor voltage-current relationship:

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

Voltage expression:

$V_m\sin \left(\omega t\right)=L\frac{di\left(t\right)}{dt}$

$\frac{di\left(t\right)}{dt}=\frac{V_m}{L}\sin \left(\omega t\right)$

Integrating both sides:

$i\left(t\right)=\int \frac{V_m}{L}\sin \left(\omega t\right)dt$

$i\left(t\right)=-\frac{V_m}{\omega L}\cos \left(\omega t\right)+C$

Let $C = 0$ for simplicity:

$i\left(t\right)=\frac{V_m}{\omega L}\sin (\omega t-\frac{\pi }{2})$

Conclusion: 

Current lag the voltage by 90°, and the amplitude is: $I_m=\frac{V_m}{\omega L}$

Impedance of an Inductor

In the frequency domain, the inductor’s impedance is: $Z_L=j\omega L$

• $j$: Imaginary unit (phase shift)
• $\omega L$: Inductive reactance, measured in ohms (Ω)

Benefits of applying AC Voltage to an Inductor

• An inductor stores the energy in a magnetic field. These energy are used for power supply and smoothing circuits.
• Inductor block high frequency and allow the low-frequency signals to pass. Due to this behavior of inductor it is used in audio and radio circuit to filter signals.
• Inductors cause the current to lag the voltage, which helps to create a phase shift in AC circuits for motors and oscillators.
• The key components in a transformer are the inductor. It is essential to change the voltage level in AC transmission.

Also Read:

In a pure inductor:

1. Average power (P) = 0.
2. Because the power alternates between positive and negative due to a phase difference (90°).
3. There is no loss of energy by its return to the source in each cycle.

This led to the concept of reactive power (Q), measured in volt-ampere reactive. 

Important Link: NCERT Solutions

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