Circuit Theory 1 · Energy Storage Elements

#37 Energy Storage Elements #37 — Capacitor and inductor fundamentals

Compares the physical models, differential relations, stored energy, continuity rules, and DC behavior of capacitors and inductors.

Question

Fundamental duality table for capacitors and inductors. — Voltage-current, energy, continuity, and DC behavior side by side.

Derive the charge/flux, voltage-current, integral, and energy relations for capacitors and inductors; explain their DC steady-state equivalents and identify which state variable cannot change instantaneously.

Written solution and narration transcript(shows the full solution)

Below are all the lines written in the notebook together with the full narration transcript.

1. Capacitor structure
The fundamental charge-voltage relation q=Cv and the farad unit.
Capacitor: stores energy in an electric field
Two conducting plates + dielectric
q=Cv
$\displaystyle C=\frac{q}{v}$
Unit: farad (F)
Practical values: µF, nF, pF
C measures charge stored per volt
Narration transcript
Up until now, we've only worked with resistors and op-amps. But real circuits need elements that can store energy. The first of these is the capacitor. A capacitor is made of two conducting plates separated by an insulating material called a dielectric. The charge stored is proportional to voltage: q equals C times V, where C is the capacitance measured in Farads. Typical values range from picofarads to microfarads — a one-Farad capacitor is enormous.
2. Capacitor voltage-current relation
The relation q=Cv gives i=C·dv/dt and w=½Cv².
Differentiate q=Cv with respect to time
dq/dt=i
i=C·dv/dt
Constant voltage ⇒ dv/dt=0 ⇒ i=0
$\displaystyle v\left(t\right)=v\left(t_{0}\right)+\left(\frac{1}{C}\right)\int i\left(\tau \right)\mathrm{d}\tau$
Current accumulates capacitor voltage over time
The passive sign convention is assumed
Narration transcript
Let's derive the voltage-current relationship. Starting from q equals C V, we differentiate both sides with respect to time. Since d q d t is current, we get the fundamental equation: i equals C times d V d t. This tells us something very important — current flows through a capacitor only when voltage is changing. If the voltage is constant, the current is zero. The inverse relationship says: voltage equals the initial voltage plus one over C times the integral of current. This means voltage across a capacitor accumulates over time as current flows through it.
3. Energy in a capacitor
The relation q=Cv gives i=C·dv/dt and w=½Cv².
Capacitor power and energy
p=vi=vC·dv/dt
w=∫pdt=C∫v·dv
w_C=½Cv²
$\displaystyle C=10 µF, v=5 V$
$\displaystyle w=½\left(10\times 10^{-6}\right)\left(25\right)$
$\displaystyle w=125 µJ$
Narration transcript
Now let's find the energy stored. Power absorbed by the capacitor is P equals V times i, which is V times C times d V d t. Integrating power over time gives us energy: W equals one-half C V squared. Notice that energy depends only on the voltage, not on how we got there. Capacitors are lossless energy storage elements — they store energy in the electric field between the plates. As a quick example: a 10 microfarad capacitor charged to 5 volts stores one-half times 10 micro times 25, which equals 125 microjoules.
4. DC behavior and continuity
It is open at DC and preserves v_C(0−)=v_C(0+).
Capacitor at DC steady state
dv/dt=0 ⇒ i=0
An ideal capacitor is an open circuit
Voltage cannot jump instantaneously
$\displaystyle v_{C}\left(0^{-}\right)=v_{C}\left(0^{+}\right)$
A jump would require infinite current
This continuity rule anchors switching analysis
Narration transcript
Two critical rules for capacitors. First: in DC steady state, d V d t is zero, so current is zero. This means a capacitor acts as an open circuit at DC. Second, and most importantly: capacitor voltage cannot change instantaneously. V at time zero-minus equals V at time zero-plus. This is the continuity condition, and it will be essential when we analyze switching circuits. Physically, changing voltage instantly would require infinite current, which is impossible.
5. Inductor structure
Flux linkage φ=Li; an ideal inductor is a short at DC steady state.
Inductor: stores energy in a magnetic field
Basic structure: a conducting coil
Flux linkage φ=Li
L: inductance
Unit: henry (H)
Practical values: mH and µH
An ideal inductor is a short at DC steady state
Narration transcript
The second energy storage element is the inductor. An inductor is a coil of wire that stores energy in its magnetic field. The magnetic flux linkage is proportional to current: phi equals L times I, where L is the inductance measured in Henrys. Typical values range from microhenrys to millihenrys. Its impedance increases with frequency: X L equals 2 pi f L. At DC, the impedance is zero — meaning the inductor acts as a short circuit.
6. Inductor relations and energy
The relation φ=Li gives v=L·di/dt and w=½Li².
Faraday's law and φ=Li
v=dφ/dt=L·di/dt
Constant current ⇒ di/dt=0 ⇒ v=0
$\displaystyle i\left(t\right)=i\left(t_{0}\right)+\left(\frac{1}{L}\right)\int v\left(\tau \right)\mathrm{d}\tau$
w_L=½Li²
$\displaystyle L=0.1 H, i=2 A$
$\displaystyle w=½\left(0.1\right)\left(2²\right)=0.2 J$
Narration transcript
By Faraday's law of electromagnetic induction, the voltage across an inductor equals the rate of change of flux. Since phi equals L I, we get: V equals L times d I d t. This is the mirror image of the capacitor equation. Voltage appears across an inductor only when current is changing. If the current is constant, the voltage is zero. The inverse relationship is: current equals the initial current plus one over L times the integral of voltage. Current through an inductor accumulates as voltage is applied across it. For energy, power is P equals V times I, which gives L times I times d I d t. Integrating: W equals one-half L I squared. Energy depends only on the current, not on how we got there. Inductors store energy in the magnetic field around the coil. Example: an inductor of 0.1 Henry carrying 2 amps stores one-half times 0.1 times 4, which equals 0.2 joules, or 200 millijoules.
7. Capacitor-inductor duality
Voltage-current, energy, continuity, and DC behavior side by side.
Capacitors and inductors are duals
v ↔ i
C ↔ L
i=C·dv/dt ↔ v=L·di/dt
½Cv² ↔ ½Li²
v_C is continuous ↔ i_L is continuous
DC: capacitor open ↔ inductor short
Narration transcript
Capacitors and inductors are beautiful duals of each other. Every capacitor equation has a mirror inductor equation. Swap voltage and current, swap C and L, and every formula transforms perfectly. The capacitor stores energy in an electric field as one-half C V squared. The inductor stores energy in a magnetic field as one-half L I squared. Capacitor voltage is continuous — it cannot jump. Inductor current is continuous — it cannot jump. At DC, the capacitor becomes an open circuit while the inductor becomes a short circuit. These dual relationships will appear again and again throughout circuit analysis.
8. Summary and analysis rules
Voltage-current, energy, continuity, and DC behavior side by side.
Energy-storage-element summary
A capacitor produces current against voltage change
An inductor produces voltage against current change
w_C=½Cv²
w_L=½Li²
State variables: v_C and i_L
Continuity + DC equivalents start transient analysis
Narration transcript
Let's summarize. A capacitor obeys i equals C d V d t, storing energy as one-half C V squared. An inductor obeys V equals L d I d t, storing energy as one-half L I squared. Capacitor voltage and inductor current are always continuous — they cannot change instantaneously. At DC steady state, capacitors are open circuits and inductors are short circuits. In the next lecture, we'll explore typical waveforms like the unit step, ramp, and impulse functions that we'll use to analyze these circuits.