Notes › Fundamentals of Electric Circuits (Sadiku) Lecture 6

Capacitors and Inductors

calendar_today July 3, 2025 article 402 words access_time 3 min replay Modified July 4, 2025

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Capacitors

Passive element that stores energy in its E-field
Consists of two conducting plates separated by a dielectric
When a voltage source is connected to the capacitor, each plate stores a charge $q$

Definition 06.1 (Capacitance).

The ratio of a capacitor’s plate charge to the potential difference between the plates, measured in farads (F):

$$q = Cv.$$

In practice, $C$ is dependent on neither $q$ or $v$; it is dependent on capacitor’s physical dimensions (surface area, spacing, and permittivity). For a parallel plate capacitor:

$$ C = \frac{\epsilon A}{d}.$$

Two important properties:

Acts as an open circuit when the voltage is no longer changing with time; for ideal DC circuits, this occurs instantaneously.
The voltage across the capacitor is continuous.

Taking the derivative of both sides:

$$ i = C \frac{\mathrm{d}v}{\mathrm{d}t}$$

Taking the integral of that:

$$v = \frac{1}{C}\int_{t_{0}}^{t} i(\tau) \, \mathrm{d}\tau + v(t_{o})$$

Power delivered:
- If $p > 0$, the capacitor is charging (and vice-versa)

$$ p = vi = Cv \frac{\mathrm{d}v}{\mathrm{d}t}$$

Energy stored:

$$w = \frac{1}{2}Cv^2 = \frac{q^2}{2C}$$

Equivalent Capacitance

Series:
- Share $q$ and $i$

$$C_{\text{eq}} = \left(\sum_{n=1}^{N} \frac{1}{C_{n}}\right)^{-1}$$

Parallel:

$$C_{\text{eq}} = \sum_{n = 1}^{N} C_{n}$$

Inductors

Passive element that stores energy in its B-field
Consists of a coil of conducting wire
If current passes through the inductor, the potential difference is directly proportional to the change in current

Definition 06.2 (Inductance).

The property of an inductor to oppose the change of current through it, measured in henrys (H):

$$v = L \frac{\mathrm{d}i}{\mathrm{d}t}.$$

In practice, $L$ is dependent on neither $v$ nor $i$:

$$L = \frac{N^2 \mu A}{\ell},$$

where $N$ is the turns, $\ell$ is length, $A$ is cross-sectional area, and $\mu$ is permeability of the core.

Two important properties:

Acts as a closed circuit when the current is no longer changing with time; for ideal DC circuits, this occurs instantaneously.
The current through an inductor is continuous.

Integrating both sides:

$$ i = \frac{1}{L} \int_{t_{0}}^{t} v(\tau) \, \mathrm{d}\tau + i(t_{0})$$

Power delivered:

$$p = vi = \left( L \frac{\mathrm{d}i}{\mathrm{d}t} \right) i$$

Energy stored:

$$w = \frac{1}{2} Li^2$$

Equivalent Inductance

Series

$$L_{\text{eq}} = \sum_{n=1}^{N} L_{n}$$

Parallel

$$L_{\text{eq}} = \left(\sum_{n=1}^{N} \frac{1}{L_{n}}\right)^{-1}$$

Integrator and Differentiator

Integrator

Definition 06.3 (Integrator).

An op amp with $v_o \propto \int v_{i}$ by replacing $R_f$ with a capacitor in an inverting amplifier.

$$ v_{o} = -\frac{1}{RC} \int_{0}^{t} v_{i}(\tau) \, \mathrm{d}\tau$$

Differentiator

Definition 06.4 (Differentiator).

An op amp with $v_o \propto \frac{\mathrm{d}v_{i}}{\mathrm{d}t}$ by replacing $R_i$ with a capacitor in an inverting amplifier.

$$v_{o} = -RC \frac{\mathrm{d}v_{i}}{\mathrm{d}t}$$

Graph

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