Energy Stored in a Magnetic Field — Question 5

Question 5

Series and parallel inductor energies reverse their comparison under a voltage-pulse constraint

Two identical uncoupled ideal inductors each have inductance L. Compare series and parallel arrangements. All branch currents start at zero.

1. Find their energies when the same total terminal current I is established.

2. Instead apply the same signed voltage-time area S to each arrangement from rest. Compare final energies.

Question 5 - Solution

Series and parallel inductor energies reverse their comparison under a voltage-pulse constraint

(a) Method and calculation

$$L_s=2L,\quad L_p=L/2\Rightarrow U_s=LI^2,\quad U_p=LI^2/4.$$$$\boxed{U_s/U_p=4}\quad\text{at fixed terminal current}.$$

(b) Interpretation and further calculation

$$I=S/L_{\mathrm{eq}}\Rightarrow U=S^2/(2L_{\mathrm{eq}}).$$$$\boxed{U_p/U_s=4}\quad\text{at fixed voltage-time area}.$$

Physical check

The comparison changes because the same applied voltage-time area produces a larger current in the smaller equivalent inductance.