Circuit laws and energy storage

Intent

• Apply Kirchhoff’s current and voltage laws to a switched circuit, one interval at a time.
• Compute how much an inductor’s current and a capacitor’s voltage change under a constant voltage or current.
• Explain why inductor current and capacitor voltage cannot change instantly, and what that means for a switch.

Theory

Kirchhoff’s laws

The currents entering a node sum to zero (Kirchhoff’s current law), and the voltages around a closed loop sum to zero (Kirchhoff’s voltage law) [Alexander & Sadiku 2017, Ch. 2]. A switching converter is a different linear circuit in each interval of the switching period. Within one interval both laws apply, and the element laws below join the intervals together [Erickson & Maksimović 2020, Ch. 2].

The inductor under a constant voltage

The voltage across an inductor is its inductance times the rate of change of its current [Alexander & Sadiku 2017, Ch. 6]. With a constant voltage across it, the current changes linearly in time:

$$\Delta i = \frac{V_L t}{L}$$

$\Delta i$

change of inductor current

$L$

inductance

$V_L$

inductor voltage

$t$

duration

Current change of an inductor under a constant voltage✓ verified by 7 tests

Assumptions, convention and source

Assumptions

• ideal (lossless) switches and passive components
• constant voltage (inductor) or current (capacitor) during the interval t

Notes

integrates the inductor law; the current changes linearly while the voltage is constant. This is the full change, not the half-ripple $\Delta i_L$ of the converter pages

Source

Alexander & Sadiku 2017, Ch. 6 (Capacitors and Inductors)

Derivation

step-by-step sympy derivation

Equation id

ind.di · equations.yaml entry

This is the whole change during the interval; the converter pages use the half-ripple $\Delta i_L$, half the peak-to-peak change (Erickson’s convention).

The capacitor under a constant current

The dual law: the current into a capacitor is its capacitance times the rate of change of its voltage, so a constant current changes the voltage linearly [Alexander & Sadiku 2017, Ch. 6]:

$$\Delta v_C = \frac{I_C t}{C}$$

$\Delta v_C$

change of capacitor voltage

$C$

capacitance (output capacitor)

$I_C$

capacitor current

$t$

duration

Voltage change of a capacitor under a constant current✓ verified by 7 tests

Assumptions, convention and source

Assumptions

• ideal (lossless) switches and passive components
• constant voltage (inductor) or current (capacitor) during the interval t

Notes

integrates the capacitor law, the dual of the inductor law

Source

Alexander & Sadiku 2017, Ch. 6 (Capacitors and Inductors)

Derivation

step-by-step sympy derivation

Equation id

cap.dv · equations.yaml entry

Stored energy

An inductor stores energy in its magnetic field, a capacitor in its electric field [Alexander & Sadiku 2017, Ch. 6]:

$$E_L = \frac{L I_L^{2}}{2}$$

$E_L$

energy in the inductor

$I_L$

dc inductor current

$L$

inductance

Energy stored in an inductor✓ verified by 7 tests

Assumptions, convention and source

Assumptions

• ideal (lossless) switches and passive components
• linear (unsaturated) magnetic material

Source

Alexander & Sadiku 2017, Ch. 6 (Capacitors and Inductors)

Derivation

step-by-step sympy derivation

Equation id

ind.E · equations.yaml entry

$$E_C = \frac{C V_C^{2}}{2}$$

$E_C$

energy in the capacitor

$C$

capacitance (output capacitor)

$V_C$

capacitor voltage

Energy stored in a capacitor✓ verified by 7 tests

Assumptions, convention and source

Assumptions

• ideal (lossless) switches and passive components

Source

Alexander & Sadiku 2017, Ch. 6 (Capacitors and Inductors)

Derivation

step-by-step sympy derivation

Equation id

cap.E · equations.yaml entry

Inductor current and capacitor voltage cannot change instantly: a jump would change the stored energy in zero time [Alexander & Sadiku 2017, Ch. 6]. When a transistor turns off, the inductor current therefore has to continue somewhere, and a converter gives it a path through a diode or a second switch [Erickson & Maksimović 2020, Ch. 4]. In steady state the current also ends every period where it started, which is volt-second balance.

Worked example

An example inductor and capacitor, each under a constant excitation:

Inductor and capacitor example

Symbol	Value	How it is obtained
$V_L$ inductor voltage	12 V	given
$t$ duration	5 µs	given
$L$ inductance	100 µH	given
$I_C$ capacitor current	1.2 A	given
$C$ capacitance (output capacitor)	10 µF	given
$I_L$ dc inductor current	2 A	given
$V_C$ capacitor voltage	12 V	given
$\Delta i$ change of inductor current	600 mA	Current change of an inductor under a constant voltage
$\Delta v_C$ change of capacitor voltage	600 mV	Voltage change of a capacitor under a constant current
$E_L$ energy in the inductor	200 µJ	Energy stored in an inductor
$E_C$ energy in the capacitor	720 µJ	Energy stored in a capacitor

During the interval the inductor current rises by 600 mA and the capacitor voltage by 600 mV. At the given current the inductor stores 200 µJ; at the given voltage the capacitor stores 720 µJ.

Try it

▶ Open in the equation explorer: Current change of an inductor under a constant voltage
Preset from the example lc-elements

Sweep $L$ over its logarithmic range: the current change falls in inverse proportion to the inductance. This is why a larger inductor gives a smaller current ripple at the same voltages and switching frequency.

Bench exercise

Drive a resistor in series with an inductor from a function generator with a square wave whose period is much shorter than the inductance divided by the resistance. The inductor current, seen with a current probe, is then close to a triangle: during each half-period the inductor sees an almost constant voltage. Measure the slope of each ramp and compare it with the inductor equation at the measured voltage across the inductor. Repeat with a capacitor in place of the inductor, a period much shorter than the resistance times the capacitance, and the voltage across the capacitor.

Gotchas

• The half-ripple is half. The change on this page is the whole change during the interval; the converter pages’ $\Delta i_L$ is half of the peak-to-peak ripple. Mixing the two conventions doubles or halves a result.
• Never leave an inductor’s current without a path. Its current cannot stop instantly [Alexander & Sadiku 2017, Ch. 6], so opening the only path forces a large voltage across the switch until something conducts.
• A discharged capacitor across a stiff source draws a surge. Its voltage cannot jump [Alexander & Sadiku 2017, Ch. 6], so connecting it directly to a voltage source draws a current limited only by the resistance in the loop.

Go deeper

• Course 6.002 Circuits and Electronics (Spring 2007) — A. Agarwal, MIT OpenCourseWare (course content by A. Agarwal and J. H. Lang) · retrieved 2026-09-23
  MIT's first circuits course, with video lectures, notes and problem sets.
• Lecture notes 6.002 Lecture Notes — A. Agarwal, MIT OpenCourseWare · retrieved 2026-09-23
  Lecture-by-lecture notes (from the Fall 2000 offering) on circuit laws, energy-storage elements and sinusoidal steady state.

Quiz

1. A $100\,\mu\mathrm{H}$ inductor has a constant $5\,\mathrm{V}$ across it for $4\,\mu\mathrm{s}$. By how much does its current change?

   1. $0.02\,\mathrm{A}$
   2. $0.2\,\mathrm{A}$
   3. $2\,\mathrm{A}$
   4. $20\,\mathrm{A}$
   Answer and explanation

   Correct answer: B The current changes at the rate of the voltage divided by the inductance, $5 \times 10^4\,\mathrm{A/s}$, for $4\,\mu\mathrm{s}$.

2. A $10\,\mu\mathrm{F}$ capacitor is charged by a constant $2\,\mathrm{mA}$ for $1\,\mathrm{ms}$. How much does its voltage rise?

   1. $0.02\,\mathrm{V}$
   2. $0.2\,\mathrm{V}$
   3. $2\,\mathrm{V}$
   4. $20\,\mathrm{V}$
   Answer and explanation

   Correct answer: B The charge delivered is the current times the time, $2\,\mu\mathrm{C}$; divided by the capacitance it gives the voltage change.

3. How much energy does a $1\,\mathrm{mH}$ inductor store at $3\,\mathrm{A}$?

   1. $1.5\,\mathrm{mJ}$
   2. $3\,\mathrm{mJ}$
   3. $4.5\,\mathrm{mJ}$
   4. $9\,\mathrm{mJ}$
   Answer and explanation

   Correct answer: C Half the inductance times the square of the current: $0.5 \times 10^{-3} \times 9$.

4. Which quantity of an ideal inductor cannot change instantly?

   1. Its voltage
   2. Its current
   3. Both its voltage and its current
   4. Neither
   Answer and explanation

   Correct answer: B The stored energy depends on the current, and a jump in current would need an infinite voltage. The voltage, by contrast, can jump, as it does at every switching edge.

5. The constant voltage across an inductor is doubled. What happens to the rate of change of its current?

   1. It halves
   2. It is unchanged
   3. It doubles
   4. It quadruples
   Answer and explanation

   Correct answer: C The rate of change of the current is the voltage divided by the inductance.

6. In a buck converter the inductor current rises by $0.4\,\mathrm{A}$ during the on-interval. What is the half-ripple $\Delta i_L$ used on the converter pages?

   1. $0.1\,\mathrm{A}$
   2. $0.2\,\mathrm{A}$
   3. $0.4\,\mathrm{A}$
   4. $0.8\,\mathrm{A}$
   Answer and explanation

   Correct answer: B The rise during the on-interval is the peak-to-peak ripple, and $\Delta i_L$ is half of it (Erickson's convention).