Capacitors and Inductors
Energy storage components. Capacitors store energy in electric fields, inductors in magnetic fields. Unlike resistors, their behavior depends on frequency, making them essential for filtering, timing, and power conversion.
Why It Matters
Every IC needs decoupling capacitors or it glitches. Every switching regulator needs an inductor or it cannot convert voltage. Every analog signal chain needs filtering. Understanding reactive components is the bridge between DC circuits (Voltage Current Resistance) and real-world AC behavior.
How It Works
Capacitor Fundamentals
I = C x dV/dt (current flows only when voltage changes)
Energy stored: E = 1/2 x C x V²
A 100uF cap charged to 5V stores 0.5 x 100e-6 x 25 = 1.25 mJ.
| Type | Capacitance | ESR | Use |
|---|---|---|---|
| Ceramic (MLCC) | 1pF - 100uF | Very low | Decoupling, high frequency |
| Electrolytic | 1uF - 10,000uF | Higher | Bulk storage, power supply |
| Tantalum | 0.1uF - 1000uF | Low | Space-constrained, stable |
| Film | 100pF - 10uF | Very low | Audio, precision |
RC Time Constant
tau = R x C
After one tau, a charging capacitor reaches 63% of final voltage. After 5 tau, it is 99% charged.
Vcc
|
[R]
|
+──── Vcap (measured here)
|
[C]
|
GND
Charge: Vcap(t) = Vcc x (1 - e^(-t/tau))
Discharge: Vcap(t) = Vcc x e^(-t/tau)
| Time | % of Vcc (charging) |
|---|---|
| 1 tau | 63.2% |
| 2 tau | 86.5% |
| 3 tau | 95.0% |
| 5 tau | 99.3% |
Inductor Fundamentals
V = L x dI/dt (voltage appears only when current changes)
Energy stored: E = 1/2 x L x I²
An inductor opposes changes in current. This is why cutting current through an inductor creates a voltage spike — the energy has to go somewhere (see flyback diode in Transistors as Switches).
Inductor saturation: every inductor has a maximum current. Above it, the core saturates, inductance drops sharply, and current spikes. Always check the saturation current in the datasheet — it is not the same as the rated current.
RL Time Constant
tau = L / R
Same exponential behavior as RC, but for current instead of voltage.
Impedance (AC Behavior)
Reactance is frequency-dependent resistance:
Capacitive: Xc = 1 / (2 x pi x f x C) (decreases with frequency)
Inductive: XL = 2 x pi x f x L (increases with frequency)
This is why capacitors pass AC and block DC (Xc → infinity at f=0), while inductors do the opposite.
LC Resonance
When a capacitor and inductor are combined, they resonate at:
f_res = 1 / (2 x pi x sqrt(L x C))
At resonance, energy sloshes between the capacitor’s electric field and the inductor’s magnetic field. This is the basis of oscillators, tuned filters, and radio receivers. Maps to second-order systems in Transfer Functions.
Practical Decoupling Pattern
The standard decoupling strategy for ICs:
Vcc rail
|
[10uF] ← bulk cap, within 20mm of IC
|
[100nF] ← high-freq cap, within 3mm of IC power pin
|
IC Vcc pin ── IC ── IC GND pin
|
GND plane
The 100nF ceramic handles fast transients (MHz). The 10uF electrolytic handles slower current demands. Some designs add a 1nF for very high frequencies. See PCB Design Basics for placement rules.
Calculation Example
import math
# RC timing: how long to charge to 3.3V from 5V supply?
R, C = 10e3, 100e-6 # 10k, 100uF
tau = R * C # 1.0 second
# Vcap = Vcc * (1 - e^(-t/tau))
# Solve for t: t = -tau * ln(1 - Vcap/Vcc)
vcc, v_target = 5.0, 3.3
t = -tau * math.log(1 - v_target / vcc) # 1.08 seconds
# Energy stored in capacitor at 3.3V
E = 0.5 * C * v_target**2 # 0.5445 mJ
# Impedance at 1kHz
f = 1000
Xc = 1 / (2 * math.pi * f * C) # 1.59 ohm
# At 1MHz
Xc_1M = 1 / (2 * math.pi * 1e6 * C) # 0.00159 ohm (basically a short)
# LC resonance
L = 10e-6 # 10uH inductor
C2 = 100e-9 # 100nF capacitor
f_res = 1 / (2 * math.pi * math.sqrt(L * C2)) # 159 kHzRelated
- Voltage Current Resistance — DC fundamentals
- Power Supply Design — inductors in buck converters, capacitor selection
- Op Amps — active filters using RC networks
- Signal Integrity — decoupling for high-speed circuits