Question

Is the superposition theorem applicable to AC circuits?

Answer 1

Yes, the superposition theorem is applicable to AC circuits, just as it is to DC circuits. The superposition theorem states that in a linear circuit with multiple independent sources (voltage or current), the response (voltage or current) in any element of the circuit is the algebraic sum of the responses caused by each independent source acting alone, while all other independent sources are replaced by their internal impedances.

### Steps to Apply Superposition Theorem to AC Circuits:
1. **Turn off all sources except one**:
   - For voltage sources, replace them with a short circuit (0V).
   - For current sources, replace them with an open circuit (0A).
2. **Solve the circuit** using the remaining source and determine the voltage or current of interest.
3. **Repeat the process** for each independent source in the circuit.
4. **Sum the effects**: After analyzing the circuit for each source, sum up the individual voltages or currents (considering both magnitude and phase in AC circuits, as they are typically represented as phasors).

### Considerations for AC Circuits:
- **Phasor Representation**: AC voltages and currents are represented as phasors (complex numbers), meaning you have to account for both magnitude and phase shifts when applying the superposition theorem.
- **Impedance**: In AC circuits, resistances are generalized to impedances, which take into account resistors, capacitors, and inductors. The impedance affects how the circuit responds to different frequencies.

### Example:
If you have an AC circuit with two voltage sources \( V_1 \) and \( V_2 \), to find the total current through a resistor:
1. Set \( V_2 \) to 0 (short circuit), solve the circuit for the current due to \( V_1 \).
2. Set \( V_1 \) to 0, solve for the current due to \( V_2 \).
3. Sum the individual current contributions, considering both magnitude and phase angle.

In summary, the superposition theorem is fully applicable to AC circuits, but requires careful handling of complex quantities like phasors and impedance.

Answer 2

Yes, the **superposition theorem** is applicable to **AC (Alternating Current) circuits** as well as DC (Direct Current) circuits. However, there are some important aspects to keep in mind when applying it to AC circuits.

### What is Superposition Theorem?
The **superposition theorem** states that in a linear circuit with multiple independent sources (like voltage or current sources), the total current or voltage at any point in the circuit is the algebraic sum of the currents or voltages caused by each independent source acting alone, with all other independent sources turned off (deactivated).

To deactivate:
- **Voltage sources**: Replace them with short circuits (i.e., set the voltage to 0).
- **Current sources**: Replace them with open circuits (i.e., set the current to 0).

### Applicability to AC Circuits
The superposition theorem works for AC circuits because these circuits are typically **linear**—the relationship between current and voltage follows Ohm's law, and the elements (resistors, capacitors, inductors) are linear over their operating ranges. In AC circuits, the sources and signals are usually time-varying and sinusoidal.

When applying the superposition theorem to an AC circuit, here are the key points:

1. **Phasor Representation**: In AC circuits, voltages and currents are typically sinusoidal and represented by phasors (complex numbers that represent magnitude and phase). To apply the superposition theorem, it's easier to work in the **frequency domain** using phasors instead of time-domain functions.
   - AC signals like \( V(t) = V_m \sin(\omega t + \phi) \) are converted to phasors like \( \tilde{V} = V_m \angle \phi \).
  
2. **Reactance and Impedance**: In AC circuits, inductors and capacitors introduce frequency-dependent reactance. This means when dealing with different sources, the impedance of inductors and capacitors needs to be taken into account.
   - For an inductor, \( Z_L = j\omega L \)
   - For a capacitor, \( Z_C = \frac{1}{j\omega C} \)
   
   Here, \( Z \) represents impedance, \( \omega \) is the angular frequency, \( L \) is inductance, and \( C \) is capacitance. These complex quantities must be included in the circuit analysis.

3. **Independent Sources**: In an AC circuit, each independent source can have its own frequency. When applying the superposition theorem, only sources with the **same frequency** can be added together. Sources of different frequencies are treated separately, and the final response is the combination of the effects of each source after applying superposition.
   - For example, if one source has a frequency of 60 Hz and another source has 100 Hz, solve the circuit for each frequency independently and then combine the results after converting back to the time domain.

### Steps to Apply Superposition Theorem in AC Circuits

1. **Convert to Phasors**: Convert all sinusoidal voltages and currents to their phasor equivalents.
   
2. **Deactivate Sources**:
   - For each source, deactivate all other independent sources (set AC voltage sources to 0, which means replacing them with short circuits, and set AC current sources to 0, replacing them with open circuits).
   
3. **Solve for Each Source**: Solve the circuit for the contribution of each independent source individually, taking into account the impedance of all components (resistors, inductors, capacitors).
   
4. **Add the Phasor Results**: After finding the individual responses (phasors) for each source, sum them up algebraically (vector addition) to get the total voltage or current phasor in the circuit.
   
5. **Convert Back to Time Domain**: If necessary, convert the final phasor result back into the time domain to get the actual sinusoidal waveform.

### Example of Superposition in AC Circuit
Imagine a simple AC circuit with two sources, \( V_1(t) \) at 50 Hz and \( V_2(t) \) at 100 Hz, connected to a resistor \( R \) and an inductor \( L \).

- First, deactivate \( V_2(t) \) (replace it with a short circuit) and calculate the current and voltage in the circuit using only \( V_1(t) \).
- Then, deactivate \( V_1(t) \) and calculate the current and voltage due to \( V_2(t) \).
- Finally, combine the two solutions to get the overall current and voltage in the circuit. However, because the frequencies are different, you'll need to combine the results separately and convert them back to the time domain for a complete picture.

### Summary
Yes, the **superposition theorem** can be applied to AC circuits, but you need to:
- Work with **phasors** in the frequency domain.
- Handle **impedances** (including inductors and capacitors).
- Treat sources with **different frequencies** separately.

This approach simplifies the analysis of AC circuits with multiple independent sources and helps in understanding the contribution of each source individually.