1.2 Network Theorems

1. Superposition Theorem

The Superposition Theorem states that in a linear network with multiple independent sources, the response (voltage or current) at any point in the circuit is the sum of the responses caused by each source individually, with all other sources replaced by their internal resistances. Steps to apply the theorem:

• Consider each independent source one at a time while deactivating the other sources (replace voltage sources with short circuits and current sources with open circuits).

• Find the contribution to the response from each source.

• Sum all the contributions to get the total response.

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2. Thevenin’s Theorem

Thevenin’s Theorem simplifies a complex linear circuit with multiple voltage sources, current sources, and resistors into a simple equivalent circuit consisting of a single voltage source (Thevenin voltage, $V_{th}$) in series with a resistance (Thevenin resistance, $R_{th}$). Steps to apply Thevenin’s Theorem:

• Remove the load resistance and calculate the open-circuit voltage, $V_{oc}$, to find $V_{th}$.

• Find $R_{th}$ by calculating the resistance seen from the load terminals with all independent sources turned off (voltage sources shorted, current sources opened).

• Reconnect the load resistance to the Thevenin equivalent circuit.

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3. Norton’s Theorem

Norton’s Theorem is similar to Thevenin's theorem but replaces the voltage source with a current source. A complex linear circuit is simplified into an equivalent circuit with a single current source (Norton current, $I_{N}$) in parallel with a resistance (Norton resistance, $R_{N}$). Steps to apply Norton’s Theorem:

• Find the short-circuit current, $I_{sc}$, to determine $I_{N}$.

• Find $R_{N}$ by calculating the resistance seen from the load terminals with all independent sources turned off.

• Reconnect the load resistance to the Norton equivalent circuit.

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4. Maximum Power Transfer Theorem

The Maximum Power Transfer Theorem states that maximum power is delivered to the load when the load resistance ($R_L$) is equal to the Thevenin resistance ($R_{th}$) of the source circuit. Mathematically: $R_L = R_{th}$

For maximum power, the load should be matched with the internal resistance of the source.

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5. R-L, R-C, and R-L-C Circuits

• R-L Circuit: A circuit consisting of a resistor (R) and an inductor (L) in series or parallel. The inductor introduces inductive reactance that opposes changes in current.

• R-C Circuit: A circuit consisting of a resistor (R) and a capacitor (C) in series or parallel. The capacitor introduces capacitive reactance that opposes changes in voltage.

• R-L-C Circuit: A circuit with a resistor (R), inductor (L), and capacitor (C) connected in series or parallel. These circuits are used to filter signals or control the frequency response, with different behavior depending on the frequency of the applied signal.

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6. Resonance in AC Series and Parallel Circuits

• Series Resonance: Occurs when the inductive reactance and capacitive reactance are equal in magnitude but opposite in phase, resulting in the total impedance being at a minimum. At this point, the circuit resonates, and the current is at its maximum.

• Parallel Resonance: Occurs when the total impedance of the parallel LC circuit reaches its maximum, and the current through the circuit is at its minimum. Resonance occurs when the inductive reactance equals the capacitive reactance.

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7. Active and Reactive Power

• Active Power (Real Power, P): The real power consumed by the circuit, responsible for doing work. It is measured in watts (W). In AC circuits, it is given by: $P = V \times I \times \cos \theta$ Where $\theta$ is the phase angle between the voltage and current.
• Reactive Power (Q): The power that oscillates between the source and reactive components (inductors and capacitors) but does no real work. It is measured in volt-amperes reactive (VAR). $Q = V \times I \times \sin \theta$
• Apparent Power (S): The total power supplied by the source, combining both active and reactive power, measured in volt-amperes (VA). $S = V \times I$

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Conclusion

• Superposition, Thevenin's, and Norton's Theorems: These methods simplify complex circuits; Superposition adds individual responses from each source, Thevenin's reduces a circuit to a voltage source with resistance, and Norton's replaces it with a current source in parallel.

• Maximum Power Transfer Theorem & R-L, R-C, R-L-C Circuits: Maximum power is transferred when the load resistance equals the source's Thevenin resistance, and R-L, R-C, and R-L-C circuits control frequency response and filter signals.

• Resonance & Power Types: Series resonance occurs when inductive and capacitive reactance are equal, and active (real) power does work, while reactive (imaginary) power oscillates, with apparent power combining both.