Capacitance and Inductance: How Circuits Behave Over Time
Resistance explains steady-state current and voltage, but it does not explain how a circuit gets from one state to another. Capacitors and inductors store energy, so they make circuit behavior depend on time as well as on the present input.
Learning objectives
After this lesson, you will be able to:
- define capacitance and inductance in physical and mathematical terms;
- predict how capacitors and inductors respond to DC steps and changing signals;
- calculate stored energy in electric and magnetic fields;
- estimate first-order timing using RC and RL time constants;
- avoid unsafe assumptions about charged capacitors and inductive kickback.
Why these components matter
In a purely resistive circuit, changing the source ideally changes current and voltage immediately. In real electronics, that is often not what we want.
- A power rail should not jump and ring violently.
- A sensor input should not carry switching noise directly into an ADC.
- A relay coil should not destroy a transistor when current is interrupted.
- A timing network should create a deliberate delay.
Capacitors and inductors are the components that let designers shape those behaviors.
Capacitors store energy in an electric field
A capacitor consists of two conductors separated by an insulator called a dielectric. Its defining equation is:
$$
q = Cv
$$
where q is charge, C is capacitance, and v is the voltage across the capacitor.
Current is the rate of change of charge, so for a capacitor:
$$
i_C = C\frac{dv_C}{dt}
$$
This equation is the key idea:
- if capacitor voltage is changing quickly, current can be large;
- if capacitor voltage is constant, ideal capacitor current is zero;
- capacitor voltage cannot change instantaneously unless infinite current is available.
Units and practical ranges
Capacitance is measured in farads (F). Practical electronics usually uses:
| Unit | Value |
|---|---|
1 mF |
10^{-3} F |
1 µF |
10^{-6} F |
1 nF |
10^{-9} F |
1 pF |
10^{-12} F |
Typical roles:
| Capacitor type | Typical range | Typical use |
|---|---|---|
| Ceramic | 1 pF to 1 µF |
decoupling, RF, timing |
| Film | 1 nF to tens of µF |
filters, audio, timing |
| Electrolytic | 1 µF to thousands of µF |
bulk energy storage, supply smoothing |
Inductors store energy in a magnetic field
An inductor is usually a coil of wire. When current flows, it creates a magnetic field. The inductor's defining equation is:
$$
v_L = L\frac{di_L}{dt}
$$
This is the complementary idea to the capacitor:
- if inductor current is changing quickly, a large voltage appears across it;
- if inductor current is constant, ideal inductor voltage is zero;
- inductor current cannot change instantaneously unless infinite voltage is applied.
Inductance is measured in henries (H), with practical values commonly expressed in mH and µH.
| Inductor type | Typical range | Typical use |
|---|---|---|
| RF inductors | 10 nH to 100 µH |
matching and tuned circuits |
| Power inductors | 1 µH to 10 mH |
buck/boost converters |
| Relay and motor windings | application dependent | magnetic actuation |
What each one resists
| Component | Cannot change instantly | Governing equation | Ideal DC behavior after settling |
|---|---|---|---|
| Resistor | neither voltage nor current by itself | v = iR |
constant v and i allowed |
| Capacitor | voltage | i = C dv/dt |
open circuit |
| Inductor | current | v = L di/dt |
short circuit |
That last column applies only after the transient has settled in an ideal DC circuit. Real components always include parasitic resistance, leakage, and breakdown limits.
Time-domain intuition: RC and RL step response
RC charging circuit
BT1: Device:Battery value="5 V step" rotate=0
SW1: Switch:SW_SPST value="Close at t = 0"
R1: Device:R value="10 kΩ" rotate=270
C1: Device:C value="100 µF" rotate=0
layout direction=LR gap=70
group RC label="First-order RC charging path" direction=LR gap=70 {
BT1 SW1 R1 C1
}
BT1.1 --> SW1.1 color=#b91c1c
SW1.2 --> R1.1 color=#b91c1c
R1.2 --> C1.1 color=#2563eb
C1.2 --> BT1.2 color=#334155
Read the loop clockwise: the source is on the left, the switch and resistor are in the upper charging path, the capacitor is the right-hand branch, and the lower wire returns to the negative terminal.
When the switch closes, the capacitor initially behaves almost like a short circuit because its voltage is still near 0 V. Current starts at:
$$
i(0^+) = \frac{V}{R}
$$
Then the capacitor voltage rises gradually toward the source voltage while the current decays toward zero.
The time constant is:
$$
\tau_{RC} = RC
$$
For R = 10 kΩ and C = 100 µF:
$$
\tau = 10{,}000 \times 100 \times 10^{-6} = 1\ \text{s}
$$
After about 1τ, the capacitor reaches about 63.2% of its final voltage. After about 5τ, it is effectively settled for most practical work.
RL current build-up
BT1: Device:Battery value="12 V step" rotate=0
SW1: Switch:SW_SPST value="Close at t = 0"
R1: Device:R value="6 Ω winding resistance" rotate=270
L1: Device:L value="30 mH" rotate=270
layout direction=LR gap=70
group RL label="First-order RL current rise" direction=LR gap=70 {
BT1 SW1 R1 L1
}
BT1.1 --> SW1.1 color=#b91c1c
SW1.2 --> R1.1 color=#b91c1c
R1.2 --> L1.1 color=#2563eb
L1.2 --> BT1.2 color=#334155
Here the source is again on the left. Current follows the upper path through the switch, winding resistance, and inductor before returning along the lower wire.
When the switch closes, the inductor initially resists the change in current, so current starts at 0 A and rises gradually toward the final value V/R.
The RL time constant is:
$$
\tau_{RL} = \frac{L}{R}
$$
For L = 30 mH and R = 6 Ω:
$$
\tau = \frac{0.03}{6} = 5\ \text{ms}
$$
Waveform view
These idealized waveforms show the core behavior after a step input. They are explanatory sketches, not oscilloscope captures.
title "RC and RL step-response intuition"
time start=0 end=5 unit=tau divisions=10
VIN: step low=0 high=1 at=0.2 label="input step" unit=norm color=#1d4ed8
VC: exponential from=0 to=1 tau=1 delay=0.2 label="capacitor voltage" unit=norm color=#16a34a
IC: exponential from=1 to=0 tau=1 delay=0.2 label="capacitor current" unit=norm color=#dc2626
IL: exponential from=0 to=1 tau=1 delay=0.2 label="inductor current" unit=norm color=#7c3aed
VL: exponential from=1 to=0 tau=1 delay=0.2 label="inductor voltage" unit=norm color=#ea580c
marker SW at=0.2 label="step applied"
- Capacitor voltage rises gradually.
- Capacitor current is largest at the switching instant and then decays.
- Inductor current rises gradually.
- Inductor voltage is largest when current is changing fastest and then decays.
Energy storage
These components store energy rather than merely dissipating it.
For a capacitor:
$$
E_C = \frac{1}{2}CV^2
$$
For an inductor:
$$
E_L = \frac{1}{2}LI^2
$$
Worked example 1: capacitor energy
Find the energy stored in a 470 µF capacitor charged to 12 V.
$$
E_C = \frac{1}{2} \times 470 \times 10^{-6} \times 12^2
$$
$$
E_C = 0.03384\ \text{J}
$$
That is about 33.8 mJ. The number seems small, but higher-voltage capacitors can store hazardous energy.
Worked example 2: inductor current slope
A 10 mH inductor has 5 V across it. How fast is current changing?
Using
$$
v = L\frac{di}{dt}
$$
$$
\frac{di}{dt} = \frac{v}{L} = \frac{5}{0.01} = 500\ \text{A/s}
$$
So the current increases at 500 A/s, or 0.5 A/ms, while that voltage is maintained.
Capacitors and inductors in AC
These components react to frequency as well as to amplitude.
- A capacitor passes higher-frequency content more easily than DC.
- An inductor opposes higher-frequency current changes more strongly than DC.
Their ideal reactances are:
$$
X_C = \frac{1}{2\pi f C}
$$
$$
X_L = 2\pi f L
$$
You do not need to master AC analysis yet, but these relationships explain why capacitors are used for bypassing and inductors are used for filtering and energy transfer.
Safety notes that matter in real hardware
A capacitor can remain charged after power is removed. Large electrolytics, photoflash capacitors, and mains-side capacitors can deliver dangerous shock or arc energy.
An inductor, relay coil, solenoid, or motor winding can generate a high reverse voltage when current is interrupted. That is why flyback diodes, snubbers, or clamp circuits are used.
Common applications
| Application | Main component | Why it works |
|---|---|---|
| Power-supply decoupling | capacitor | supplies short bursts of current locally |
| RC delay or debounce | resistor + capacitor | sets a predictable time constant |
| Buck/boost converter | inductor | stores and transfers energy each switching cycle |
| Relay driver protection | diode + inductor load | safely handles flyback energy |
| Resonant tuning | capacitor + inductor | exchanges energy between electric and magnetic fields |
Common mistakes
- Saying a capacitor “stores current.” It stores energy in an electric field and charge on its plates.
- Saying an inductor “blocks current.” It opposes changes in current; steady DC can flow through an ideal inductor.
- Assuming a capacitor is always open circuit or an inductor is always short circuit without checking whether the circuit is still in transient.
- Ignoring time constants and expecting a node to settle instantly.
- Forgetting discharge paths, leakage, ESR, winding resistance, and flyback protection in real circuits.
Summary
- Capacitors oppose sudden voltage change and follow
i = C dv/dt. - Inductors oppose sudden current change and follow
v = L di/dt. - Capacitors store energy in electric fields; inductors store energy in magnetic fields.
- RC and RL circuits introduce time constants that shape how circuits start, stop, and filter signals.
- Real hardware requires attention to stored energy, voltage rating, current rating, ESR, saturation, and switching transients.
Next: Voltage Divider and Current Divider.