Compensation Theorem: Working Principle, Examples, Advantages, and Applications

Introduction

In network analysis, studying the impact of a change in impedance in one branch of a circuit is very important. Any change in impedance directly affects the current and voltage distribution throughout the network. The compensation theorem is used to analyze these changes effectively. This theorem is based on Ohm’s law, which states that a voltage drop appears across a resistor when current flows through it. This voltage drop opposes the applied source voltage. To compensate for this effect, an additional voltage source of equal magnitude and opposite polarity is introduced.

What Is Compensation Theorem?

The compensation theorem states that any resistance in a network can be replaced by a voltage source having zero internal resistance. The value of this voltage source is equal to the voltage drop across the replaced resistance due to the current flowing through it, and its polarity is opposite to the original voltage.

Explanation of Compensation Theorem

Assume that a resistor R carries a current I, producing a voltage drop
V = I × R.
According to the compensation theorem, this resistor can be replaced by a voltage source that produces the same voltage but in the opposite direction to the original current flow. This replacement helps determine the change in current or voltage when the resistance value in the circuit is altered.

Compensation Theorem in AC Circuits

The compensation theorem can also be applied to AC circuits to find the change in current when a resistance value is modified. Consider an AC circuit where a 3 ohm resistor is replaced by a 7 ohm resistor. Initially, the current flowing through the 3 ohm branch is determined using the current divider rule.

Initial Current Calculation

Using the current divider rule:

I = (8 × 7) ÷ (7 + 3) = 56 ÷ 10 = 5.6 A

However, the actual current flowing through the 3 ohm resistor is 7 A. When the resistor is replaced with a 7 ohm resistor, the current distribution changes.

Compensation Voltage Calculation

To determine this change, a compensation network is formed by suppressing all independent sources. The compensation voltage is given by:

V_C = I × ΔZ

V_C = 7 × (7 − 3) = 28 V

Change in Current

The compensation circuit now contains a single loop with two 7 ohm resistors.
The change in current is:

ΔI = V_C ÷ (7 + 7) = 28 ÷ 14 = 2 A

Verification of Compensation Theorem

Now, calculate the new current using the current divider rule:

I″ = (8 × 7) ÷ (7 + 7) = 56 ÷ 14 = 4 A

The change in current is:

ΔI = I − I″ = 7 − 4 = 3 A

This verifies the compensation theorem.

Why Do We Need Compensation Theorem?

The compensation theorem is useful because it provides precise information about how a network responds to changes in one of its elements. It allows engineers to determine new current and voltage values without reanalyzing the entire circuit.

Advantages of Compensation Theorem

• Provides information about changes in network parameters
• Based on the fundamental principle of Ohm’s law
• Helps determine changes in current or voltage due to resistance variation
• Simplifies analysis when small changes occur in circuit elements

Applications of Compensation Theorem

• Determining the effect of small parameter variations in electrical networks
• Analyzing sensitivity of bridge networks
• Studying tolerance effects of circuit components
• Solving electrical networks and bridge circuits efficiently
• Evaluating current and voltage changes after circuit modification

Conclusion

The compensation theorem is a powerful tool in network analysis that helps engineers understand the effect of changes in circuit components. It simplifies complex recalculations and is especially useful in sensitivity analysis
and bridge network evaluation.