Equivalent Inductance of Inductors in Parallel

Introduction

• Derivation of equivalent inductance for two inductors connected in parallel
• Inductors connected between points A and B

Circuit Setup

• Voltage source: E_total
• Current entering point A: I
• Current splits at A: I1 (through first inductor) and I2 (through second inductor)
• Current recombines at B: I = I1 + I2

Definitions

• L1: Self inductance of first inductor
• L2: Self inductance of second inductor
• m: Mutual inductance between the two inductors

Key Concepts

• For parallel connections, EMF across each inductor is the same: E1 = E2 = E_total
• Total current at A: I = I1 + I2
• Differentiate with respect to time:
  • dI/dt = dI1/dt + dI2/dt

EMF Expressions

1. EMF across first inductor (E1):
   • E1 = L1 * (dI1/dt) + m * (dI2/dt)
2. EMF across second inductor (E2):
   • E2 = L2 * (dI2/dt) + m * (dI1/dt)

Equating EMFs

• Set E1 = E2:
  • L1 * (dI1/dt) + m * (dI2/dt) = L2 * (dI2/dt) + m * (dI1/dt)

Rearranging Terms

• Collect terms involving dI1/dt and dI2/dt:
  • (L1 - m)(dI1/dt) = (L2 - m)(dI2/dt)
• Rearrangement leads to:
  • dI1/dt = [(L2 - m) / (L1 - m)] * (dI2/dt)*

Total Current Expression

• Substitute back into dI/dt:
  • dI/dt = [(L2 - m) / (L1 - m) + 1] * (dI2/dt)*

Final Equivalent Inductance Expression

• Total EMF expression:
  • E_total = L_equiv * (dI/dt)
• Set equal to E2:
  • E_total = L2 * (dI2/dt) + m * (dI1/dt)*

Equating Inductance Values

• Cross multiply and simplify to find L_equiv:
  • L_equiv = (L1 * L2 - m^2) / (L1 + L2 - 2m)*

Conclusion

• The equivalent inductance of two inductors in parallel:
  • L_equiv = (L1 * L2 - m^2) / (L1 + L2 - 2m)*