Coupled Circuits

Why it matters

Coupled circuits are fundamental in understanding how electrical circuits interact with each other through mutual inductance. This concept is crucial in the design and operation of transformers, which are essential components in power distribution and various electronic devices.

Key ideas

• Mutual Inductance: When two circuits are placed close to each other, the magnetic field of one can induce a voltage in the other. This is known as mutual inductance.
• Coefficient of Coupling (k): This is a measure of how effectively two inductors are magnetically coupled. It ranges from 0 (no coupling) to 1 (perfect coupling).
• Dot Convention: A method to indicate the relative polarity of the voltages in coupled inductors. Dots are placed on the terminals of inductors to show the direction of the induced voltage.
• Energy Transfer: Coupled circuits allow for energy transfer between circuits without direct electrical connection, which is essential in transformers.

Formulas

• V2 = M * (dI1/dt)
  • V2: Induced voltage in the second circuit (Volts)
  • M: Mutual inductance (Henry)
  • dI1/dt: Rate of change of current in the first circuit (Amperes per second)
• M = k * sqrt(L1 * L2)
  • M: Mutual inductance (Henry)
  • k: Coefficient of coupling (dimensionless)
  • L1, L2: Inductances of the two circuits (Henry)

For positive M, if i1 enters its dotted terminal, the mutual voltage at winding 2 is positive at its dot for increasing i1. The displayed V2 is the mutual contribution only; a varying i2 also creates L2 di2/dt. In the example winding 2 is open and V2 is measured dot-to-undotted terminal.

Worked example

Given:

• L1 = 4 H
• L2 = 9 H
• k = 0.8
• dI1/dt = 3 A/s

Find: Induced voltage V2 in the second circuit.

1. Calculate mutual inductance M:

   • Formula: M = k * sqrt(L1 * L2)
   • Calculation: M = 0.8 * sqrt(4 * 9) = 0.8 * 6 = 4.8 H

2. Calculate induced voltage V2:

   • Formula: V2 = M * (dI1/dt)
   • Calculation: V2 = 4.8 * 3 = 14.4 V

Final Answer: 14.4 V

Common mistakes

• Confusing the direction of induced voltage due to incorrect application of the dot convention.
• Miscalculating mutual inductance by not correctly applying the coefficient of coupling.
• Forgetting to consider the units, especially when dealing with rates of change.

For GATE EE

Questions on coupled circuits often involve calculating mutual inductance, induced voltages, and understanding the dot convention. Practice problems involving transformers and energy transfer between circuits.

Quick check

1. What is mutual inductance?
2. How does the coefficient of coupling affect mutual inductance?
3. What does the dot convention indicate?

Answers: 1. Induction of voltage in one circuit due to the magnetic field of another. 2. It determines the effectiveness of magnetic coupling. 3. The relative polarity of induced voltages in coupled inductors.