1.3 Inductance, Capacitance, Reactance, and Time Constants

1.3 Inductance, Capacitance, Reactance, and Time Constants

Inductors and capacitors represent the two fundamental reactive components in electronics. Unlike resistors, which convert electrical energy permanently into heat, inductors and capacitors store energy temporarily in magnetic and electrostatic fields, releasing it back to the circuit during alternate portions of the AC cycle. Their frequency-dependent opposition to alternating current—known as reactance—underpins resonant circuits, RF impedance matching networks, filters, oscillators, and transmission line mechanics.

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1. Inductance and Magnetic Energy Storage

The Henry and Back-EMF

Inductance ($L$) is the property of an electrical conductor or coil that opposes any change in the electric current flowing through it. When current traverses a conductor, it establishes a surrounding magnetic field proportional to the current magnitude. If that current changes with time ($di/dt$), the collapsing or expanding magnetic field cuts the conductor itself, inducing an electromotive force known as self-induced back-EMF ($e$):

$$e = -L \frac{di}{dt}$$

This formulation combines Faraday's Law with Lenz's Law. The negative sign indicates that the induced back-EMF always acts in direct opposition to the change in current that produced it:

• When circuit current attempts to increase, back-EMF opposes current rise.
• When circuit current drops, back-EMF attempts to sustain current flow.

The SI unit of inductance is the Henry ($\text{H}$). A circuit element has an inductance of one Henry when a current changing at a rate of one Ampere per second induces a back-EMF of one Volt across its terminals ($1\text{ H} = 1\text{ V}\cdot\text{s/A}$).

In radio frequency engineering, fractional sub-units are standard:

• Millihenry ($1\text{ mH} = 10^{-3}\text{ H}$): Audio filters, power supply smoothing chokes.
• Microhenry ($1\ \mu\text{H} = 10^{-6}\text{ H}$): HF band resonant circuits, RF chokes.
• Nanohenry ($1\text{ nH} = 10^{-9}\text{ H}$): VHF, UHF, and microwave inductors, component lead parasitics.

Energy Stored in a Magnetic Field

Energy stored in the magnetic field of an inductor carrying instantaneous current $I$ is:

$$W_L = \frac{1}{2} L \cdot I^2 \quad [\text{Joules}]$$

This stored magnetic energy cannot change discontinuously, meaning the current through an inductor cannot change instantaneously.

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Magnetic Shielding

A current-carrying conductor produces a surrounding magnetic field; coil turns reinforce it. Low-frequency magnetic interference is difficult to shield with thin copper or aluminium. A high-permeability enclosure, such as suitable steel or mu-metal, offers a low-reluctance path that diverts magnetic flux around the protected circuit. At higher frequencies, induced currents in a conductive shield oppose the penetrating field, with effectiveness depending on thickness and skin depth. Shielding electric fields and shielding low-frequency magnetic fields therefore require different material choices.

2. Factors Affecting Inductance and Core Materials

Solenoid Inductance Formula

For a single-layer cylindrical solenoid coil wound on a non-magnetic or magnetic core, the inductance is given by:

$$L \approx \frac{\mu_0 \cdot \mu_r \cdot N^2 \cdot A}{l}$$

where:

• $N$ is the total number of turns.
• $A$ is the cross-sectional core area in square metres ($\pi r^2$).
• $l$ is the physical length of the coil winding in metres.
• $\mu_0$ is the permeability of free space ($4\pi \times 10^{-7}\text{ H/m} \approx 1.2566 \times 10^{-6}\text{ H/m}$).
• $\mu_r$ is the relative magnetic permeability of the core material.

Exam Critical Factor: Inductance is proportional to the square of the turns count ($N^2$). Doubling the number of turns on an identical former quadruples the inductance ($2^2 = 4$). Halving the number of turns reduces inductance to one-quarter ($0.5^2 = 0.25$).

Magnetic Core Materials in Amateur Radio

Toroidal Cores and the Inductance Index ($A_L$)

Toroidal (doughnut-shaped) cores provide a closed magnetic path that greatly reduces external flux and unwanted coupling. Practical windings and their connections still have leakage fields, so a toroid does not guarantee complete isolation between adjacent stages.

Manufacturers publish an Inductance Factor ($A_L$) for each core type, defined in microhenries per 100 turns (iron powder) or nanohenries per turns-squared (ferrite):

$$L = A_L \cdot N^2 \times 10^{-4} \quad (\text{Iron Powder, } A_L \text{ in } \mu\text{H}/100\text{ turns})$$ $$N = 100 \sqrt{\frac{L}{A_L}}$$

For example, to wind a $10\ \mu\text{H}$ inductor on a T50-2 iron powder core ($A_L = 49\ \mu\text{H}/100\text{ turns}$):

$$N = 100 \sqrt{\frac{10}{49}} = 100 \times 0.4518 \approx 45\text{ turns}$$

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3. Inductor Combinations and Mutual Inductance

Inductors in Series (Without Magnetic Coupling)

When inductors are separated so their magnetic fields do not interact:

$$L_{\text{total}} = L_1 + L_2 + L_3 + \dots + L_n$$

Inductors in Parallel (Without Magnetic Coupling)

$$\frac{1}{L_{\text{total}}} = \frac{1}{L_1} + \frac{1}{L_2} + \dots + \frac{1}{L_n}$$

For two uncoupled inductors in parallel:

$$L_{\text{total}} = \frac{L_1 \cdot L_2}{L_1 + L_2}$$

Mutual Inductance ($M$) and Coupling Coefficient ($k$)

When two coils are positioned in close proximity, magnetic flux from one coil cuts the turns of the other, establishing mutual inductance ($M$):

$$M = k \sqrt{L_1 \cdot L_2}$$

where $k$ is the dimensionless coupling coefficient ($0 \le k \le 1$):

• $k = 1.0$: Unity coupling; every line of magnetic flux from coil 1 threads coil 2 (ideal iron/ferrite core transformer).
• $k = 0.0$: Complete magnetic isolation (perpendicular orientation or extensive shielding).

When two inductors with mutual coupling are connected in series:

$$L_{\text{total}} = L_1 + L_2 \pm 2M$$

• Series-Aiding ($+2M$): Coil magnetic flux fields reinforce each other (windings in same direction).
• Series-Opposing ($-2M$): Coil magnetic flux fields oppose and partially cancel each other.

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4. Inductive Reactance ($X_L$)

Inductive Reactance ($X_L$) represents the opposition presented by an inductor to alternating current. It is measured in Ohms ($\Omega$):

$$X_L = 2\pi f L = \omega L$$

Key physical relationships:

• Linear Frequency Dependence: $X_L$ is directly proportional to frequency ($f$). Doubling frequency doubles inductive reactance; halving frequency halves reactance.
• Direct Inductance Dependence: $X_L$ is directly proportional to inductance ($L$).
• DC Behavior: At direct current ($f = 0$), $X_L = 0\ \Omega$. An ideal inductor behaves as a complete short circuit to DC.

Phase Relationship in Pure Inductors: "ELI"

In a purely inductive circuit, current cannot change instantaneously due to induced back-EMF. As an alternating voltage is applied, current rise is delayed by a quarter of a cycle. Consequently, alternating voltage LEADS alternating current by $90^\circ$ (or current lags voltage by $90^\circ$).

Standard Mnemonic: ELI — E (voltage/EMF) Leads I (current) in an L (inductor).

Worked Example 1.7: Inductive Reactance across Amateur Bands

Calculate the reactance of a $4.7\ \mu\text{H}$ RF choke across three operating frequencies:

1. At $1.80\text{ MHz}$ (160 m band):

$$X_L = 2\pi \times (1.80 \times 10^6\text{ Hz}) \times (4.7 \times 10^{-6}\text{ H}) \approx 53.16\ \Omega$$

1. At $14.20\text{ MHz}$ (20 m band):

$$X_L = 2\pi \times (14.20 \times 10^6\text{ Hz}) \times (4.7 \times 10^{-6}\text{ H}) \approx 419.34\ \Omega$$

1. At $146.0\text{ MHz}$ (2 m VHF band):

$$X_L = 2\pi \times (146.0 \times 10^6\text{ Hz}) \times (4.7 \times 10^{-6}\text{ H}) \approx 4311.5\ \Omega$$

Notice that as frequency increases by roughly 80 times from $1.8\text{ MHz}$ to $146\text{ MHz}$, the reactance increases by the identical factor of 80.

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5. Capacitance and Electrostatic Energy Storage

The Farad and Electrostatic Charge

Capacitance ($C$) is the property of an arrangement of electrical conductors separated by a non-conducting dielectric to store potential energy in an electrostatic field. When a potential difference ($V$) appears across two conducting plates, equal and opposite charges ($+Q$ and $-Q$) accumulate on their surfaces:

$$Q = C \cdot V \Longleftrightarrow C = \frac{Q}{V}$$

Differentiating with respect to time yields the fundamental current equation for a capacitor:

$$i(t) = \frac{dQ}{dt} = C \frac{dv}{dt}$$

Physical Consequence: Electric current through a capacitor is proportional to the rate of change of voltage. If voltage is constant (DC), $dv/dt = 0$, so $i = 0$. Hence, a capacitor completely blocks steady direct current, while allowing alternating displacement current to traverse the dielectric.

The SI unit of capacitance is the Farad ($\text{F}$). A capacitor has a capacitance of one Farad when an applied potential of one Volt stores a charge of one Coulomb ($1\text{ F} = 1\text{ C/V}$).

Practical fractional sub-units in radio communications include:

• Microfarad ($1\ \mu\text{F} = 10^{-6}\text{ F}$): Power supply filtering, audio coupling.
• Nanofarad ($1\text{ nF} = 10^{-9}\text{ F} = 1000\text{ pF}$): RF bypassing, IF tuned circuits.
• Picofarad ($1\text{ pF} = 10^{-12}\text{ F}$): HF/VHF resonant circuits, oscillator feedback, neutralisation.

Energy Stored in an Electrostatic Field

Energy stored in the electrostatic field between capacitor plates charged to potential $V$ is:

$$W_C = \frac{1}{2} C \cdot V^2 \quad [\text{Joules}]$$

Because stored electrostatic energy cannot change discontinuously, the voltage across a capacitor cannot change instantaneously.

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6. Physical Factors Governing Capacitance and Dielectric Types

Parallel-Plate Capacitance Formula

For a two-plate parallel capacitor:

$$C = \frac{\varepsilon_0 \cdot \varepsilon_r \cdot A}{d}$$

where:

• $A$ is the overlapping surface area of the conductive plates ($\text{m}^2$).
• $d$ is the dielectric separation thickness (distance between plates in metres).
• $\varepsilon_0$ is the permittivity of free space ($8.8542 \times 10^{-12}\text{ F/m}$).
• $\varepsilon_r$ is the relative permittivity (dielectric constant) of the insulating medium.

To increase capacitance:

1. Increase plate area $A$ (e.g. multi-layer interdigitated construction).
2. Decrease plate separation $d$ (limited by dielectric breakdown voltage).
3. Use a dielectric with high relative permittivity $\varepsilon_r$.

Capacitor Dielectrics and RF Applicability

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7. Capacitor Combinations

Capacitors in Parallel

When capacitors are connected in parallel across common nodes, their plate surface areas effectively combine:

$$C_{\text{total}} = C_1 + C_2 + C_3 + \dots + C_n$$

Total charge is the sum of individual charges ($Q_{\text{total}} = Q_1 + Q_2 + \dots$). The working voltage rating of the parallel group is limited by the lowest voltage rating among the connected capacitors.

Capacitors in Series

When capacitors are connected in series, the effective dielectric separation thickness increases:

$$\frac{1}{C_{\text{total}}} = \frac{1}{C_1} + \frac{1}{C_2} + \dots + \frac{1}{C_n}$$

For two capacitors in series:

$$C_{\text{total}} = \frac{C_1 \cdot C_2}{C_1 + C_2}$$

The total capacitance of any series combination is always strictly less than the smallest capacitor in the chain.

Voltage Distribution across Series Capacitors

Because the same displacement current passes through each capacitor, identical charge $Q$ accumulates on each element ($Q = C_1 V_1 = C_2 V_2$). The voltage drops divide inversely proportional to capacitance:

$$\frac{V_1}{V_2} = \frac{C_2}{C_1}$$

The smaller capacitor drops the larger fraction of total voltage.

Safety Note: Series electrolytics increase voltage capability, but each part must stay within its rating under worst-case DC voltage, ripple and sharing imbalance. Four $450\text{ V}$ parts sum to $1800\text{ V}$ of ratings; the working-bank voltage needs design margin. Equalising resistors across each capacitor require leakage-based resistance selection, tight tolerance and adequate voltage and power ratings.

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8. Capacitive Reactance ($X_C$)

Capacitive Reactance ($X_C$) represents the opposition presented by a capacitor to alternating current. It is measured in Ohms ($\Omega$):

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

Key physical relationships:

• Inverse Frequency Dependence: $X_C$ is inversely proportional to frequency ($f$). Doubling frequency cuts capacitive reactance in half; halving frequency doubles reactance.
• Inverse Capacitance Dependence: $X_C$ is inversely proportional to capacitance ($C$).
• DC Behavior: At direct current ($f = 0$), $X_C \to \infty\ \Omega$. An ideal capacitor represents an open circuit to DC.

Phase Relationship in Pure Capacitors: "ICE"

When an alternating voltage is applied across an uncharged capacitor, maximum charging current flows at the instant voltage crosses zero ($dv/dt$ is maximum). As the capacitor charges to peak voltage, current falls to zero. Consequently, alternating current LEADS alternating voltage by $90^\circ$ (or voltage lags current by $90^\circ$).

Standard Mnemonic: ICE — I (current) Leads C (capacitance) across E (voltage/EMF).

Combining both mnemonics yields the universal engineering rule:

$$\mathbf{ELI} \quad \text{the} \quad \mathbf{ICE} \quad \mathbf{man}$$

• In an Inductor ($L$): Voltage ($E$) leads Current ($I$).
• In a Capacitor ($C$): Current ($I$) leads Voltage ($E$).

Worked Example 1.8: Receiver RF Decoupling Sizing

An RF amplifier in a 20-metre receiver operating at $14.10\text{ MHz}$ requires a decoupling capacitor between the DC supply rail and ground to bypass stray RF currents. The designer evaluates a $100\text{ nF}$ ($0.1\ \mu\text{F}$) ceramic capacitor:

1. Calculate reactance at operating frequency $f = 14.10\text{ MHz}$:

$$X_C = \frac{1}{2\pi \times (14.10 \times 10^6\text{ Hz}) \times (100 \times 10^{-9}\text{ F})} = \frac{1}{8.859} \approx 0.1129\ \Omega$$

1. Calculate reactance of the same capacitor to an unwanted $100\text{ Hz}$ power supply ripple:

$$X_C = \frac{1}{2\pi \times 100\text{ Hz} \times (100 \times 10^{-9}\text{ F})} = \frac{1}{6.283 \times 10^{-5}} \approx 15915\ \Omega \approx 15.92\ \text{k}\Omega$$

The ideal capacitive reactance is only about $0.11\ \Omega$ at $14.1\text{ MHz}$ but $15.92\text{ k}\Omega$ at $100\text{ Hz}$. Effective RF bypassing also requires low ESR/ESL and short, low-inductance connections; the ideal reactance alone does not establish the installed impedance.

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9. Transient Response and Time Constants

When a DC step voltage is applied to an RC or RL network, the voltages and currents do not change instantaneously but transition exponentially toward steady state.

The RC Time Constant

For a series resistor-capacitor circuit, the time constant ($\tau$, Greek letter tau) is:

$$\tau = R \cdot C \quad [\text{seconds}]$$

During capacitor charging from a source $V_{\text{in}}$ through resistor $R$:

$$v_C(t) = V_{\text{in}} \left(1 - e^{-t/\tau}\right), \quad i(t) = \frac{V_{\text{in}}}{R} e^{-t/\tau}$$

During capacitor discharge through resistance $R$ from initial voltage $V_0$:

$$v_C(t) = V_0 \cdot e^{-t/\tau}$$

The RL Time Constant

For a series resistor-inductor circuit, the time constant is:

$$\tau = \frac{L}{R} \quad [\text{seconds}]$$

During current rise across an inductor from source $V_{\text{in}}$ through resistance $R$:

$$i_L(t) = \frac{V_{\text{in}}}{R} \left(1 - e^{-t/\tau}\right), \quad v_L(t) = V_{\text{in}} \cdot e^{-t/\tau}$$

Time Constant Progression Milestones

Amateur Applications of Time Constants

• CW Key-Click Shaping: Abrupt carrier switching produces wide keying sidebands. An RC network rounds the edges; its 10–90% step rise time is about $2.2\tau$, so $\tau=4$–$5\text{ ms}$ gives roughly $9$–$11\text{ ms}$. Shaping reduces key clicks, but bandwidth compliance must be checked for the actual keying speed and transmitted waveform.
• Automatic Gain Control (AGC): AGC circuits in communications receivers use asymmetrical time constants: a fast attack (typically $\tau \approx 1$ to $5\text{ ms}$) to clamp sudden strong incoming signal peaks, combined with a slow decay (typically $\tau \approx 500$ to $1000\text{ ms}$) to maintain stable receiver gain between words during SSB voice communications.

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10. High-Frequency Skin Effect in RF Conductors

At direct current and low audio frequencies, electric current distributes uniformly across the entire cross-sectional area of a conductor. At radio frequencies, this uniform distribution breaks down completely due to the skin effect.

Physical Mechanism of Skin Effect

An alternating current creates an alternating internal magnetic flux within the body of the conductor itself. This internal flux induces tiny internal circulating eddy currents. By Lenz's Law, these eddy currents oppose the primary current at the center of the conductor and reinforce the primary current near the outer conductor surface. Consequently, charge carriers are forced outward into a thin annular layer along the perimeter.

Skin Depth Equation

The skin depth ($\delta$) is defined as the depth beneath the conductor surface at which the RF current density drops to $1/e$ (approximately $36.8\%$) of its surface value:

$$\delta = \sqrt{\frac{\rho}{\pi \cdot f \cdot \mu}}$$

where:

• $\rho$ is the electrical resistivity of the conductor material ($\Omega\cdot\text{m}$; for annealed copper, $\rho \approx 1.724 \times 10^{-8}\ \Omega\cdot\text{m}$).
• $f$ is the operating frequency in Hertz.
• $\mu$ is the absolute magnetic permeability of the material ($\mu = \mu_0 \mu_r$).

For copper at room temperature, this simplifies to:

$$\delta_{\text{copper}} \approx \frac{0.0661}{\sqrt{f}} \quad [\text{metres}]$$

Copper Skin Depth across Radio Frequencies

Practical Engineering Implications in Amateur Radio

1. High AC Resistance ($R_{\text{AC}}$): Because the effective cross-sectional area through which current flows is restricted to a thin surface layer, the AC resistance of an inductor or antenna conductor increases proportionally with the square root of frequency ($R_{\text{AC}} \propto \sqrt{f}$):

$$R_{\text{AC}} \approx R_{\text{DC}} \cdot \frac{r_0}{2\delta}$$

1. Hollow Tubing for Antennas and Coils: Because high-frequency currents do not penetrate the interior of a conductor, hollow copper tubing exhibits the same RF resistance as solid copper rod of the identical outside diameter. High-power transmitter tank coils and Yagi antenna elements are constructed from lightweight, rigid hollow tubing.
2. Silver Plating: Silver has slightly lower electrical resistivity than copper ($\rho = 1.59 \times 10^{-8}\ \Omega\cdot\text{m}$ vs. $1.72 \times 10^{-8}\ \Omega\cdot\text{m}$). In high-Q VHF and UHF cavity resonators, tank circuits, and coaxial relays, copper surfaces are plated with a thin layer of silver (several skin depths thick) to minimize $I^2 R$ heat losses.
3. Litz Wire at Low Frequencies: In the LF and MF bands (e.g. $136\text{ kHz}$ and $472\text{ kHz}$), coils are wound using Litz wire (bundles of individually insulated fine strands woven together). Each strand is engineered to be smaller than the skin depth, forcing uniform current sharing. Above $\sim 2\text{ MHz}$, inter-strand capacitive coupling degrades Litz wire performance, making solid tubing superior.