Network Theorems in Electric Circuits

Network theorems are analytical tools used to simplify the analysis of electrical networks. These theorems help in reducing complex circuits into simpler equivalent forms, thereby making the calculation of current, voltage, and power easier.

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Superposition Theorem

The Superposition theorem states that:

In any linear bilateral network containing multiple independent sources, the current through or voltage across any element is equal to the algebraic sum of the currents or voltages produced by each independent source acting alone.

While considering one source at a time:

• Voltage sources are replaced by short circuits.
• Current sources are replaced by open circuits.

The theorem is applicable only to linear circuits.

Procedure

1. Consider one independent source at a time.
2. Replace all other independent voltage sources by short circuits.
3. Replace all other independent current sources by open circuits.
4. Find the required current or voltage.
5. Repeat the process for all sources.
6. Add all individual responses algebraically.

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Thevenin’s Theorem

Thevenin’s theorem states that:

Any linear two-terminal network can be replaced by an equivalent circuit consisting of a single voltage source in series with a resistance.

The equivalent circuit consists of:

• Thevenin voltage $V_{th}$
• Thevenin resistance $R_{th}$

Thevenin Voltage

The Thevenin voltage is the open-circuit voltage across the terminals.

$$V_{th} = V_{oc}$$

Thevenin Resistance

The Thevenin resistance is the equivalent resistance seen from the terminals after removing all independent sources.

$$R_{th} = \frac{V_{th}}{I_{sc}}$$

where:

• $I_{sc}$ = short-circuit current

Procedure

1. Remove the load resistance.
2. Find the open-circuit voltage across the terminals.
3. Remove all independent sources.
4. Calculate the equivalent resistance seen from the terminals.
5. Replace the original network with the Thevenin equivalent circuit.

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Norton’s Theorem

Norton’s theorem states that:

Any linear two-terminal network can be replaced by an equivalent current source in parallel with a resistance.

The equivalent circuit consists of:

• Norton current $I_N$
• Norton resistance $R_N$

Norton Current

The Norton current is the short-circuit current across the terminals.

$$I_N = I_{sc}$$

Norton Resistance

The Norton resistance is equal to Thevenin resistance.

$$R_N = R_{th}$$

Relation Between Thevenin and Norton Equivalents

$$V_{th} = I_N R_N$$

Procedure

1. Remove the load resistance.
2. Find the short-circuit current.
3. Determine the equivalent resistance.
4. Replace the network with the Norton equivalent circuit.

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Maximum Power Transfer Theorem

The Maximum Power Transfer theorem states that:

Maximum power is transferred from a source to a load when the load resistance is equal to the source resistance.

Thus,

$$R_L = R_{th}$$

where:

• $R_L$ = load resistance
• $R_{th}$ = Thevenin resistance

Maximum Power

$$P_{max} = \frac{V_{th}^2}{4R_{th}}$$

This theorem is widely used in communication systems and impedance matching applications.

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Summary

• Network theorems simplify complex circuit analysis.
• Superposition theorem analyzes one source at a time.
• Thevenin’s and Norton’s theorems reduce networks into equivalent forms.
• Maximum Power Transfer theorem determines optimum load conditions.
• These theorems are widely used in electrical and electronic circuit analysis.