Chapter 5 — Passive Components: Resistors, Capacitors, and Inductors
Chapter Overview (click to expand)
Beyond the resistor, two essential passive components — capacitors and inductors — store energy rather than dissipating it, giving circuits memory and frequency-dependent behavior. This chapter explains the physical construction, V-I relationships, and energy storage properties of all three passive component types, laying the groundwork for transient and AC analysis. **Key Takeaways** 1. A capacitor stores energy in an electric field and its current is proportional to the rate of change of voltage: i = C dv/dt. 2. An inductor stores energy in a magnetic field and its voltage is proportional to the rate of change of current: v = L di/dt. 3. Under DC steady-state conditions, capacitors behave as open circuits and inductors behave as short circuits, which is a key simplification for initial analysis.5.1 The Capacitor
A capacitor is a component that stores energy in an electric field. In its simplest form, it consists of two conductive plates separated by an insulating material called a dielectric. When a voltage is applied across the plates, charge accumulates — positive charge on one plate, negative on the other — and an electric field forms between them.
The relationship between charge and voltage defines capacitance \(C\):
where \(C\) is capacitance in farads (F), \(Q\) is charge in coulombs (C), and \(V\) is voltage in volts (V).
For a parallel plate capacitor, capacitance depends on physical dimensions:
where \(\varepsilon_0 = 8.854 \times 10^{-12}\) F/m is the permittivity of free space, \(\varepsilon_r\) is the relative permittivity of the dielectric, \(A\) is the plate area, and \(d\) is the plate separation.
The unit of capacitance is the farad (F), named after Michael Faraday. One farad is enormous in practice — most capacitors are measured in microfarads (μF, 10⁻⁶ F), nanofarads (nF, 10⁻⁹ F), or picofarads (pF, 10⁻¹² F).
The V-I relationship for a capacitor:
Since \(Q = CV\), differentiating both sides with respect to time:
This is the defining equation of a capacitor. Two immediate consequences:
- A capacitor passes no DC current. If \(dv/dt = 0\) (constant voltage), then \(i = 0\). Capacitors block DC.
- Voltage across a capacitor cannot change instantaneously. An instantaneous voltage change (\(dv/dt \to \infty\)) would require infinite current, which is physically impossible.
Diagram: Capacitor
Diagram: Parallel-Plate Capacitor
5.2 Energy Stored in a Capacitor
Unlike a resistor, which dissipates energy as heat, a capacitor stores energy and can return it to the circuit later. The energy stored in a charged capacitor is:
where \(E\) is in joules (J), \(C\) in farads (F), and \(V\) in volts (V).
This is analogous to the kinetic energy of a moving mass (\(\frac{1}{2}mv^2\)), with capacitance analogous to mass and voltage analogous to velocity. The capacitor "resists" changes in voltage just as a mass resists changes in velocity.
Example: A 100 μF capacitor charged to 12 V stores: \(E = \frac{1}{2} \times 100 \times 10^{-6} \times 144 = 7.2\) mJ
5.3 Capacitors in Series and Parallel
Capacitors combine differently from resistors — the formulas are swapped.
Capacitors in parallel — plates effectively add together:
Capacitors in series — same charge on each, voltages add:
For two capacitors: \(\displaystyle C_{eq} = \frac{C_1 C_2}{C_1 + C_2}\)
Memory Aid
Capacitors combine like resistors but with the series/parallel rules reversed: parallel capacitors add directly (like series resistors), and series capacitors use the reciprocal formula (like parallel resistors).
Series combination increases the effective plate separation (lower capacitance, higher voltage rating). Parallel combination increases effective plate area (higher capacitance).
Diagram: Capacitor Combinations
5.4 The Inductor
An inductor is a component that stores energy in a magnetic field. In its basic form, it's a coil of wire. When current flows through the coil, it creates a magnetic field. If the current changes, the changing magnetic flux induces a voltage that opposes the change — this is Lenz's Law in action.
The defining relationship for an inductor:
where \(L\) is inductance in henrys (H), \(v_L\) is the voltage across the inductor in volts, and \(i_L\) is the current through the inductor in amperes.
Two key consequences, perfectly dual to the capacitor:
- An inductor has no voltage across it in DC steady state. If \(di/dt = 0\) (constant current), then \(v = 0\). Inductors are short circuits to DC.
- Current through an inductor cannot change instantaneously. An instantaneous current change would require infinite voltage, which is physically impossible.
Inductance formula for a solenoid:
where \(N\) is the number of turns, \(A\) is the cross-sectional area, \(\ell\) is the coil length, and \(\mu_r\) is the relative permeability of the core.
Diagram: Inductor
5.5 Energy Stored in an Inductor
The energy stored in an inductor's magnetic field:
The duality between capacitors and inductors is complete:
| Property | Capacitor | Inductor |
|---|---|---|
| Stores energy in | Electric field | Magnetic field |
| Energy stored | \(\frac{1}{2}CV^2\) | \(\frac{1}{2}LI^2\) |
| V-I relation | \(i = C\,dv/dt\) | \(v = L\,di/dt\) |
| Cannot change instantly | Voltage | Current |
| Blocks | DC | AC |
| Passes | AC | DC |
5.6 Inductors in Series and Parallel
Inductors combine exactly like resistors (assuming no mutual coupling):
Inductors in series:
Inductors in parallel:
For two: \(\displaystyle L_{eq} = \frac{L_1 L_2}{L_1 + L_2}\)
5.7 Mutual Inductance
When two inductors are placed near each other, the magnetic field of one can influence the other. This is mutual inductance \(M\), and it is the operating principle of transformers.
If current \(i_1\) changes in inductor 1, it induces a voltage in inductor 2:
The coupling coefficient \(k\) describes how tightly the inductors are coupled:
- \(k = 0\): No coupling (inductors completely isolated)
- \(k = 1\): Perfect coupling (all flux of one links the other) — ideal transformer behavior
- Practical transformers achieve \(k\) between 0.95 and 0.999
Diagram: Mutual Inductance
For inductors in series with mutual inductance:
\(+2M\) when fields aid (series-aiding), \(-2M\) when fields oppose (series-opposing)
5.8 Real vs. Ideal Components
The components you've been analyzing are ideal: a capacitor is purely capacitive, an inductor is purely inductive. Real components are more complicated.
Real capacitors have: - Equivalent Series Resistance (ESR): A small resistance in series with the capacitor, representing losses in the plates and leads. Matters in power supply filtering and high-current applications. - Equivalent Series Inductance (ESL): At high frequencies, the leads act as inductors, limiting the capacitor's effectiveness. - Leakage resistance: A large resistance in parallel, causing slow self-discharge.
Real inductors have: - Winding resistance (DCR): The wire forming the coil has finite resistance. A real inductor looks like an ideal inductor in series with a small resistor. - Parasitic capacitance: At high frequencies, capacitance between adjacent windings dominates. Every inductor has a self-resonant frequency (SRF) above which it acts like a capacitor. - Core losses: If a ferromagnetic core is used, hysteresis and eddy current losses dissipate energy, modeled as a resistance in parallel with the inductor.
Component selection tips: - For high-frequency bypass (decoupling), choose capacitors with low ESL — chip capacitors over lead-type. - For RF inductors, use air-core to avoid core losses. - Derate voltage and current ratings in design (typically to 80% of maximum).
Diagram: Real Capacitor Model
5.9 Signal Fundamentals
Before diving into AC circuits, we need the vocabulary to describe signals that change over time.
A sinusoidal signal (the most important signal in electrical engineering) is described by:
where \(V_m\) is the peak (amplitude), \(\omega = 2\pi f\) is the angular frequency (rad/s), \(f\) is the frequency (Hz), and \(\phi\) is the phase angle (rad or degrees).
Key signal parameters:
| Parameter | Symbol | Definition | Unit |
|---|---|---|---|
| Period | \(T\) | Time for one complete cycle | seconds (s) |
| Frequency | \(f\) | Cycles per second, \(f = 1/T\) | hertz (Hz) |
| Angular frequency | \(\omega\) | \(\omega = 2\pi f\) | rad/s |
| Peak value | \(V_m\) | Maximum amplitude | V |
| Peak-to-peak | \(V_{pp}\) | \(V_{pp} = 2V_m\) | V |
| Phase angle | \(\phi\) | Time offset from reference | degrees or rad |
RMS value: For a sinusoid, the Root Mean Square (RMS) value is:
The RMS value is what meters measure (for sinusoids), and it's what you use to calculate power. US household voltage is 120 V RMS — the peak voltage is actually \(120\sqrt{2} \approx 170\) V.
Phase relationships:
When \(\phi > 0\), the signal leads the reference (arrives earlier in time).
When \(\phi < 0\), the signal lags the reference (arrives later in time).
Diagram: Signal Parameters
5.10 The Decibel Scale
The decibel (dB) is a logarithmic unit for expressing ratios of power, voltage, or current. It's used everywhere in audio and communications because human hearing is approximately logarithmic and because gains/losses cascade multiplicatively (which becomes addition in dB).
Power ratio:
Voltage ratio (assuming equal impedances):
Key dB values to memorize:
| dB | Power ratio | Voltage ratio |
|---|---|---|
| +3 dB | ×2 | ×1.41 |
| +6 dB | ×4 | ×2 |
| +10 dB | ×10 | ×3.16 |
| +20 dB | ×100 | ×10 |
| 0 dB | ×1 | ×1 |
| -3 dB | ÷2 | ÷1.41 |
| -20 dB | ÷100 | ÷10 |
The human hearing range spans approximately 20 Hz to 20,000 Hz (20 kHz). In amplitude, the ear can handle a power range of about 10¹² (120 dB) from threshold of hearing to threshold of pain.
Diagram: Decibel Scale
5.11 Chapter Summary
Capacitors: - Store energy in electric fields: \(E = \frac{1}{2}CV^2\) - V-I: \(i = C\,dv/dt\) — block DC, voltage cannot change instantaneously - Parallel: \(C_{eq} = C_1 + C_2\); Series: \(1/C_{eq} = 1/C_1 + 1/C_2\)
Inductors: - Store energy in magnetic fields: \(E = \frac{1}{2}LI^2\) - V-I: \(v = L\,di/dt\) — short to DC, current cannot change instantaneously - Series: \(L_{eq} = L_1 + L_2\); Parallel: \(1/L_{eq} = 1/L_1 + 1/L_2\) - With mutual inductance: \(L_{eq} = L_1 + L_2 \pm 2M\)
Real components: Have parasitic ESR, ESL (capacitors) and DCR, capacitance, core losses (inductors)
Signal fundamentals: - Sinusoid: \(v(t) = V_m \sin(\omega t + \phi)\) - RMS: \(V_{rms} = V_m/\sqrt{2}\) for sinusoids - Decibel: \(\text{dB} = 20\log_{10}(V_2/V_1)\) for voltage