Thevenin Equivalent Circuit of Inductor

Electrical Circuit Analysis > Inductors

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in Feb 2026

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Introduction

In circuit analysis, simplifying complex networks into simpler equivalent circuits makes solving for voltages, currents, and responses easier. One powerful method of simplification is Thevenin’s theorem, which states that any linear network with voltage sources and resistances can be replaced by an equivalent voltage source in series with an equivalent resistance. This theorem is especially useful when analyzing circuits containing inductors, as it allows us to focus on the part of the network seen by the inductor without dealing with the entire complexity of the original circuit.

In this article, we explore how to find the Thevenin equivalent circuit of an inductor in a given inductive network. Using this equivalent, we derive the transient expressions for inductor current and inductor voltage, and show how the transient response behaves over time.

Thevenin’s Theorem: A Quick Review

Thevenin’s theorem states that any two-terminal linear electrical network containing voltage sources and resistances can be replaced by a single voltage source $V_{Th}$ in series with a resistance $R_{Th}$, such that the behavior at the terminals is unchanged.

The steps to find the equivalent are:

Find the open-circuit voltage across the two terminals. This is the Thevenin voltage $V_{Th}$.
Turn off all independent sources (short voltage sources and open current sources) and find the equivalent resistance seen at those terminals. This is the Thevenin resistance $R_{Th}$.
Replace the original network with a voltage source $V_{Th}$ in series with the resistance $R_{Th}$.

Standard Inductive Circuit

In general, an inductive circuit consists of resistances and one or more inductors connected to a power source. To analyze the transient response of the inductor after a switching event, it is often helpful to reduce the rest of the network to its Thevenin equivalent as seen by the inductor.

Example 1: For the network of [Fig. 2]:
a. Find the mathematical expression for the transient behavior of the current iL and the voltage $v_L$ after the closing of the switch ($I_i = 0 mA$).
b. Draw the resultant waveform for each.

Fig. 2: For Example 1.

Solution:
a. Applying Thevenin's theorem to the $80mH$ inductor ([Fig. 3]) yields

$$ \bbox[10px,border:1px solid grey]{R_{TH} = {R \over N} = {20kΩ \over 2} = 10kΩ}$$

Fig. 3: Determining $R_{Th}$ for the network of Fig. 2

Applying the voltage divider rule ([Fig. 4]),

$$E_{TH} = {(R_2 + R_3) E \over R_1 + R_2 + R_3}$$ $$ = {(4kΩ + 16kΩ) 12 \over 20kΩ+4kΩ + 16kΩ}$$

$$ \bbox[10px,border:1px solid grey]{E_{TH}= 6V}$$

Fig. 4: Determining $E_{Th}$ for the network of Fig. 2.

The thevenin equivalent circuit is shown in [Fig. 5].

Fig. 5: The resulting thevenin equivalent circuit for the network of [Fig. 2].

$$\begin{split} i_L &= I_m(1-e^{-t/\tau}) = {E_{Th} \over R_{TH}}(1-e^{-t/\tau})\\ \tau &={ L \over R_{TH}} = {80mH \over 10kΩ} = 8\mu s\\ i_L &= {6 \over 10kΩ}(1-e^{-t/8\mu s})\\ \end{split} $$

$$\bbox[10px,border:1px solid grey]{i_L = 0.6 \times 10^{-3}(1-e^{-t/8\mu s})}$$

and

$$ \bbox[10px,border:1px solid grey]{v_L = E_{TH} e^{-t/\tau} = 6e^{-t/8\mu s}}$$

Fig. 6: The resulting waveforms for $i_L$ and $v_L$ for the network of [Fig. 2].

Waveform Behavior

The inductor current rises exponentially from zero to its steady-state value, while the inductor voltage decreases exponentially from its initial maximum to zero.

Conclusion

The Thevenin equivalent circuit of an inductor simplifies the analysis of transient behavior in inductive networks. By replacing complex resistive networks with a single voltage source and resistance, engineers can easily determine the inductor current, voltage, and time constant. This approach is essential in electrical circuit analysis, power systems, and electronic design.

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