Alternating Current Circuits

Construction and Working of an AC Generator

Construction of an AC Generator
An AC generator converts mechanical energy into electrical energy based on electromagnetic induction.

It consists of:
1. Armature Coil ( ): A rectangular coil rotates inside a magnetic field.
2. Magnetic Field: Created using permanent magnets or electromagnets.
3. Slip Rings ( ): Connected to the coil, providing continuous contact.
4. Brushes: Connect the slip rings to the external circuit.

Working of an AC Generator
- The coil rotates in the magnetic field due to external mechanical energy.
- According to Faraday’s Law of Electromagnetic Induction, an EMF is induced: - The induced current alternates because the coil changes orientation every half cycle.
- The output AC voltage follows:

Factors Affecting Resonant Frequency
- The resonant frequency of a series LCR circuit is given by: It depends on:
1. Inductance (L): Increasing decreases the resonant frequency.
2. Capacitance (C): Increasing decreases the resonant frequency. Graph: Variation of Impedance with Frequency - The impedance of a series LCR circuit is: where:
- (Inductive reactance),
- (Capacitive reactance).

At resonance ( ):
- → Minimum impedance (Only remains).
- High current flows through the circuit.

Understanding Rectifier Frequency Response
- The given AC voltage is: - The general form of an AC signal is: Comparing, we get:

Half-Wave Rectifier Output Frequency:
- A half-wave rectifier allows only one half-cycle of the input AC voltage.
- The output voltage still follows the same fundamental frequency as the input, i.e., 50 Hz. Full
-Wave Rectifier Output Frequency:
- A full-wave rectifier inverts the negative half-cycles, making the output frequency twice the input frequency.
- Thus, the frequency of the output becomes: Thus, the correct answer is 50 Hz for the half-wave rectifier and 100 Hz for the full-wave rectifier.

Difference between Peak Value and RMS Value:
The peak value of an alternating current (AC) is the maximum value of the current during a cycle.
The root mean square (RMS) value of an AC is the square root of the average of the squares of the instantaneous values of current over a cycle.

Expression for RMS Value: For a sinusoidal current , where is the peak value of the current, the RMS value is given by: Substituting : The average value of over a full cycle is . Therefore: Thus, the RMS value of the current is , where is the peak value.

Step-by-Step Explanation: Current and Voltage in an Inductor

Step 1: Relationship Between Current and Voltage

In an inductor, the voltage across the inductor is related to the rate of change of current through the following equation: where:
is the voltage across the inductor,
is the inductance,
is the derivative of the current with respect to time.

When the current varies with time in a triangular waveform, the voltage waveform will be proportional to the rate of change of the current. This means that the voltage will depend on how quickly the current changes over time.

Step 2: Analyzing the Waveform

If the current waveform is triangular, its rate of change is constant during the rise and fall of the triangle, which results in a **constant voltage** during these periods. The voltage will follow the shape of the derivative of the triangular waveform.

The current waveform can be represented by a triangular waveform, and the voltage waveform will correspond to the derivative of the current waveform. If the current increases linearly over time, the voltage will be proportional to the rate of change, which corresponds to the graph of , where is the period of the triangular waveform.

Conclusion

The correct representation of the voltage waveform for a triangular current waveform is derived from the fact that voltage is proportional to the derivative of current, and this gives a voltage waveform that follows the graph of .

The given question involves calculating the average power consumed in an AC circuit over one complete cycle. In an AC circuit, the voltage and current are represented as:

Voltage:
Current:

To find the average power consumed, the expression is derived from:

Where is the period of the waveform. Substituting the voltage and current expressions:

Using the trigonometric identity for product of sine functions:

We have:

Substitute back into the power expression:

Since is constant with respect to and the integral of a cosine function over a complete period is zero:

So:


Therefore, the average power consumed in the circuit over a cycle is:

The impedance of the LCR circuit is given by: where is the resistance, is the inductance, and is the capacitance. The phase difference between the applied voltage and the current is given by: The current lags the voltage by the phase angle , which can be visualized in the phasor diagram.