Inductors in Series

Series inductors carry the same current, and their inductances add. L1 is $400\,\mathrm{mH}$ and L2 is $600\,\mathrm{mH}$. Together they behave like one $1\,\mathrm{H}$ inductor when viewed from the pair's terminals.

$$\begin{aligned} L_{\mathrm{eq}} &= L_1+L_2 = 1\,\mathrm{H} \\ \tau &= \frac{L_{\mathrm{eq}}}{R_1} = \frac{1\,\mathrm{H}}{100\,\Omega} = 10\,\mathrm{ms} \end{aligned}$$

Read the circuit

V1 drives the two inductors through R1. Both inductors start with zero current. The pulse rises from 0 to 5 V at 5 ms and returns to zero at about 55 ms. R1 limits the eventual current to $5/100 = 50\,\mathrm{mA}$ and provides a decay path when the source returns to zero.

• VL — green: voltage across the complete inductor pair, measured from its top to ground.
• VL2 — red: voltage across L2, whose lower terminal is grounded.
• I — blue: the common current through R1, L1, and L2. The arrow defines positive current.

The voltage across L1 is the difference $V_{L1}=V_L-V_{L2}$. These are ground-referenced voltage probes; VL2 does not measure the upper inductor.

Why the voltages differ

The same changing current passes through both inductors. Each voltage is proportional to its inductance:

$$\begin{aligned} V_{L1} &= L_1\frac{dI}{dt}=0.4V_L \\ V_{L2} &= L_2\frac{dI}{dt}=0.6V_L \end{aligned}$$

Just after the rising edge, VL is nearly 5 V and VL2 nearly 3 V; L1 takes the remaining 2 V. As current builds, both voltages approach zero. Just after the falling edge they reverse polarity, while current remains continuous and flows in the same direction as it decays.

Observe in CircuitSim

Transient simulation runs automatically over 0–110 ms with a 50 µs maximum step. The chart separates voltage from current. Open Table to inspect Time, VL, VL2, and I in that order.

At about 15 ms, one time constant after the rising edge, I is approximately 31.6 mA. Just before 55 ms, it is approximately 49.7 mA. At about 65 ms, one time constant into decay, it is approximately 18.3 mA. The 10 µs source edges make the displayed numbers differ slightly from ideal instantaneous-step calculations.

Try two changes

1. Set L1 → Inductance to 600m (600 mH), then run. Equal inductors split the voltage equally; the total becomes 1.2 H and the time constant becomes 12 ms. The long-term current is still 50 mA.
2. Restore L1 to 400m, then set R1 → Resistance to 200 Ω. The time constant becomes 5 ms and the long-term current becomes 25 mA. Restore R1 to 100 Ω afterward.

Model assumptions

The inductors are ideal and not magnetically coupled. Real coils have winding resistance and parasitic capacitance. Nearby coupled coils require a mutual-inductance model; their series equivalent can depend on winding direction. The 50 mA limit here is set by R1, not by the inductors' current ratings.