6.3 Resistance, Capacitance & Inductance

6.3 Resistance, Capacitance & Inductance

Quick Answer: R in ohms, C in farads, L in henrys. $X_L = 2\pi f L$, $X_C = 1/(2\pi f C)$. Series L values add; parallel C values add. Inductors store magnetic energy; capacitors store electric energy. A bypass capacitor shunts AC to ground through a low impedance; fewer turns means less inductance.

If §6.1 named power and §6.2 named materials, this section quantifies the three passive properties you adjust every time you tune a tank, set a coupling network, or troubleshoot an RF deck.

Units and physical meaning

Practical values: RF caps are often pF–µF; inductors µH–mH; resistors from fractions of an ohm to megaohms. The farad and henry are large base units—prefixes dominate lab talk—but the exam still wants the base unit names.

Resistance opposes current and dissipates true power as heat: $P = I^2 R$. It does not return energy to the circuit each cycle the way pure L and C do.

Reactance: frequency-dependent opposition

Reactance is measured in ohms but is not resistance. It causes phase shift and contributes to impedance (Z = R + jX) without the continuous heat of a pure resistor (ideal case).

Inductive reactance

$$X_L = 2\pi f L$$

• (f) in hertz, (L) in henrys → $X_L$ in ohms.
• Double frequency or double L → double $X_L$.
• Coils look more open at higher RF frequencies (until self-resonance and parasitics intervene).

This is the exact formula tested in Topic 3-A: if frequency and coil inductance are known, $X_L = 2\pi f L$ (not $\pi f L$, not the capacitive formula).

Capacitive reactance

$$X_C = \frac{1}{2\pi f C}$$

• Higher frequency or larger C → smaller $X_C$.
• Capacitors look more like shorts to high-frequency AC and opens to DC—foundation of coupling and bypass practice.

Worked example — coil reactance

An antenna loading coil is 50 µH at 2.0 MHz (near the 160 m / marine MF neighborhood for scale).

$$X_L = 2\pi (2.0\times 10^6)(50\times 10^{-6}) \approx 628\,\Omega$$

At 4.0 MHz, the same coil presents about 1.26 kΩ of $X_L$. Frequency doubled → reactance doubled. That is why a coil that “looks modest” at low HF can dominate impedance higher up.

Worked example — capacitor reactance

A 0.01 µF bypass at 1 MHz:

$$X_C = \frac{1}{2\pi (1\times 10^6)(0.01\times 10^{-6})} \approx 16\,\Omega$$

At audio (say 1 kHz) the same capacitor is roughly 16 kΩ—poor as a bypass, fine as a coupling thought experiment. Frequency agility of $X_C$ is the whole point of choosing bypass values.

Series and parallel combination formulas

Resistance

• Series: $R_T = R_1 + R_2 + R_3 + \cdots$ (always larger than any single resistor).
• Parallel: $\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots$, or for two resistors $R_T = \frac{R_1 R_2}{R_1 + R_2}$ (always smaller than the smallest branch).

Inductance

• Series inductors (no mutual coupling): $L_T = L_1 + L_2 + \cdots$ — same pattern as series R. Element 3 explicitly asks for $L_T = L_1 + L_2$ as the series formula.
• Parallel inductors (ideal, no mutual coupling): reciprocal sum, analogous to parallel R.

Capacitance

• Parallel capacitors: $C_T = C_1 + C_2 + C_3$ — add, like series R. Element 3 tests $C_T = C_1 + C_2 + C_3$ for three capacitors in parallel.
• Series capacitors: reciprocal sum; total C is less than the smallest capacitor.

Memory hook: Series L like series R; parallel C like series R (they add).

Energy storage and time-domain behavior (overview)

Inductor: energy $E_L = \frac{1}{2} L I^2$ in the magnetic field. Current through an inductor cannot change instantaneously; the coil “resists” current change with induced voltage. At switch-on, an inductor initially behaves like an open; after many time constants in a resistive circuit, DC current settles as if the inductor were a short (wire resistance only).

Capacitor: energy $E_C = \frac{1}{2} C V^2$ in the electric field. Voltage across a capacitor cannot change instantaneously. At switch-on to a DC source through resistance, the capacitor initially looks like a short (charging surge), then approaches an open once charged.

Time-constant sketch (RC / RL): $\tau = RC$ or $\tau = L/R$. After about 5τ, transients are essentially settled for exam-level work. Detailed RC time-constant math returns in Topic 3-B; here you only need the qualitative “L magnetic / C electric” storage picture and that reactance—not resistance—sets sinusoidal AC phase shift.

Practical RF component behaviors from the pool

Bypass capacitor purpose: it removes alternating current by providing a low-impedance path to ground (for the AC component), keeping DC bias undisturbed. It does not primarily “remove DC by shunting DC to ground”—that would destroy bias supplies. Think: AC short, DC open.

Reducing the inductance of an antenna coil: reduce the number of turns. Adding turns, raising core permeability, or compressing turns generally increases inductance (or coupling), not decreases it.

Conductors again (RLC topic overlap): good conductors → many free electrons; best low-resistance metals → gold, silver, copper. Same material literacy as §6.2, now tied to minimizing R losses in coils and tank circuits.

Putting R, L, and C together on one mental schematic

Imagine a series R–L–C filter feeding an RF stage:

1. R sets damping and true-power loss.
2. L contributes $X_L = 2\pi f L$ and magnetic energy storage.
3. C contributes $X_C = 1/(2\pi f C)$ and electric energy storage.
4. Net reactance $X = X_L - X_C$ (series convention) goes to zero at resonance—preview of later RLC circuit topics—but even before resonance math, you must compute each reactance correctly.
5. Parallel bypass caps on supply pins use large enough C (low enough $X_C$ at the frequencies of interest) so AC noise sees ground.

When Element 3 shows a formula stem, match symbols carefully: $2\pi f L$ is inductive; $1/(2\pi f C)$ is capacitive; series L adds; parallel C adds. Those four facts clear most of key topic 004 and the reactance item from 001.