Capacitors and inductors: energy storage in circuits

Lesson 1 introduced capacitors and inductors through their defining equations and their DC behaviour as open and short circuits. This lesson develops them further, because almost every real circuit stores energy somewhere: in the filter capacitors of a power supply, the inductor of a buck converter, the hold-up capacitors that keep a server running through a brief power dip, or the parasitic capacitance of a long cable. We find voltages and currents from changing waveforms, combine capacitors and inductors in series and parallel, solve circuits in DC steady state, determine initial conditions at the moment a switch operates, and build op amp integrators. These are the foundations on which ELE-203 builds transient and AC analysis.

From current to voltage, and back

The capacitor equation \(i = C\,dv/dt\) can be integrated to give the voltage from the current:

The capacitor remembers its past: its present voltage depends on all the current that has ever flowed into it, summarized by the initial voltage \(v(t_{0})\). The dual statements hold for an inductor:

Graphically, the change in capacitor voltage over an interval is the area under the current curve divided by \(C\). For piecewise-constant currents, each segment gives a straight-line ramp in voltage.

import numpy as np

C = 100e-6                         # 100 uF
dt = 1e-5
t = np.arange(0, 0.006 + dt / 2, dt)
i = np.where(t < 0.002, 0.05, np.where(t < 0.004, -0.025, 0.0))   # 50 mA, then -25 mA, then 0

v = 1.0 + np.concatenate([[0.0], np.cumsum(i[:-1] * dt)]) / C   # v(0) = 1 V; v = v0 + (1/C) * integral of i
w = 0.5 * C * v**2
for tk in [0, 0.002, 0.004, 0.006]:
    k = int(round(tk / dt))
    print(f"t = {tk * 1e3:.0f} ms: v = {v[k]:.3f} V, stored energy = {w[k] * 1e3:.3f} mJ")

t = 0 ms: v = 1.000 V, stored energy = 0.050 mJ
t = 2 ms: v = 2.000 V, stored energy = 0.200 mJ
t = 4 ms: v = 1.500 V, stored energy = 0.112 mJ
t = 6 ms: v = 1.500 V, stored energy = 0.112 mJ

Numerical integration like this is how simulators handle capacitors and inductors in transient analysis: they replace the derivative with a small time step and solve a resistive circuit at each step (ELE-203).

Series and parallel combinations

Capacitors combine like conductances, and inductors like resistances:

Capacitors in series share the same charge, so the voltage divides in inverse proportion to capacitance: the smaller capacitor takes the larger voltage. This matters when capacitors are stacked to reach a higher voltage rating, as in supercapacitor modules and the DC bus of power converters: unequal capacitance or leakage would overstress one of them, so balancing resistors or active balancing circuits are added.

DC steady state

When a circuit containing capacitors and inductors has been connected to DC sources for a long time, all voltages and currents stop changing. Then every capacitor carries no current (an open circuit) and every inductor has no voltage (a short circuit). DC steady-state analysis replaces them accordingly and solves the remaining resistive circuit with the methods of Lessons 2 to 6. The capacitor voltages and inductor currents found this way give the stored energy.

Switching and initial conditions

Circuits change suddenly when switches open or close, breakers trip or transistors turn on. The key facts for analysing the instant of switching, at \(t = 0\), come from the defining equations:

• Capacitor voltage cannot change instantaneously: \(v_{C}(0^{+}) = v_{C}(0^{-})\).
• Inductor current cannot change instantaneously: \(i_{L}(0^{+}) = i_{L}(0^{-})\).
• Everything else, capacitor current, inductor voltage, resistor voltages and currents, can jump.

The procedure is: (1) find the DC steady state before switching, giving \(v_{C}(0^{-})\) and \(i_{L}(0^{-})\); (2) at \(t = 0^{+}\), replace each capacitor by a voltage source of value \(v_{C}(0^{-})\) and each inductor by a current source of value \(i_{L}(0^{-})\); (3) solve that resistive circuit for any quantity at \(0^{+}\); (4) find the new final steady state as \(t \to \infty\). ELE-203 then shows how the circuit moves from \(0^{+}\) to its final state, exponentially with time constant \(RC\) or \(L/R\).

Op amp integrators and differentiators

Placing a capacitor in the feedback path of an inverting op amp (Lesson 8) gives an integrator. The input current \(v_{i}/R\) flows into the capacitor, so

A constant input produces a linear ramp at the output, at a rate of \(-v_{i}/RC\) volts per second, until the op amp reaches its supply limit. Integrators form the core of analog control loops (the I in PID control, ELE-390), waveform generators and dual-slope ADCs. Swapping the resistor and capacitor gives a differentiator, \(v_{o} = -RC\,dv_{i}/dt\), which is rarely used as drawn because it amplifies high-frequency noise.

Real capacitors and inductors

Practical components depart from the ideal in ways that often decide a design:

• Capacitors have equivalent series resistance (ESR), which heats them under ripple current, and equivalent series inductance (ESL), which limits their usefulness at high frequency. Their capacitance varies with voltage (strongly for high-k ceramic dielectrics), temperature and age.
• Aluminium electrolytic capacitors offer large capacitance at low cost but have limited life: their electrolyte slowly dries out, and as a rule of thumb life doubles for every 10 °C reduction in operating temperature. They are a leading cause of failure in power supplies and fans.
• Supercapacitors (electric double-layer capacitors) store hundreds to thousands of farads at 2.7 V per cell, bridging the gap between capacitors and batteries.
• Inductors have winding resistance and core losses, and their inductance falls when the core saturates (ELEC-110, Lesson 9).