3.2 Inductors, Capacitors, and RC/RL Time Constants

Inductors, Capacitors, and RC/RL Time Constants

Reactive passive components—inductors and capacitors—store electrical energy temporarily rather than dissipating it as heat. Inductors store energy in magnetic fields, while capacitors store energy in electrostatic fields. Understanding reactive transient behaviors, combination rules, and time constants ($\tau$) is essential for analyzing aircraft power filters, radar pulse-forming networks, and digital bus termination networks.

Fundamentals of Capacitors and Inductors

Capacitance and Electrostatic Energy Storage

A capacitor consists of two conducting plates separated by an insulating dielectric material. When a DC voltage is applied across the plates, electric charge accumulates, creating an electrostatic field. Capacitance ($C$) measures the ability to store charge ($Q$) per unit of applied voltage ($V$):

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

Capacitance is measured in Farads ($\text{F}$), though avionics circuits typically utilize microfarads ($\mu\text{F} = 10^{-6}\text{ F}$), nanofarads ($\text{nF} = 10^{-9}\text{ F}$), or picofarads ($\text{pF} = 10^{-12}\text{ F}$). The physical capacitance depends on plate surface area ($A$), plate separation distance ($d$), and dielectric permittivity ($\epsilon$):

$C = \frac{\epsilon \cdot A}{d} = \frac{\epsilon_r \cdot \epsilon_0 \cdot A}{d}$

Energy stored in a charged capacitor's electrostatic field is given by:

$E_C = \frac{1}{2} C V^2$

Inductance and Magnetic Energy Storage

An inductor consists of a conductive wire coil wrapped around a magnetic or air core. When current flows through the coil, a magnetic field is established. Inductance ($L$) is the property of a circuit that opposes any change in electric current by inducing a counter-electromotive force ($\text{CEMF}$) proportional to the rate of change of current:

$v_L(t) = L \cdot \frac{di}{dt}$

Inductance is measured in Henries ($\text{H}$), millihenries ($\text{mH} = 10^{-3}\text{ H}$), or microhenries ($\mu\text{H} = 10^{-6}\text{ H}$). Physical inductance depends on coil turns ($N$), core magnetic permeability ($\mu$), core cross-sectional area ($A$), and coil length ($l$):

$L = \frac{\mu \cdot N^2 \cdot A}{l}$

Energy stored in an inductor's magnetic field is given by:

$E_L = \frac{1}{2} L I^2$

Series and Parallel Combination Rules

Calculating total capacitance ($C_T$) and total inductance ($L_T$) requires applying rules that reflect their inverse physical relationships relative to resistance.

Combination Rule Matrix

Mathematical Rationale

• Capacitors in Parallel: Connecting capacitors in parallel combines their plate surface areas ($A_{\text{total}} = A_1 + A_2$), directly increasing total capacitance ($C_T = C_1 + C_2$).

• Capacitors in Series: Connecting capacitors in series effectively increases dielectric thickness ($d_{\text{total}} = d_1 + d_2$), reducing total capacitance. For two series capacitors:

  $$C_T = \frac{C_1 \cdot C_2}{C_1 + C_2}$

• Inductors in Series/Parallel: Assuming negligible mutual magnetic coupling, inductors behave identically to resistors because adding series turns increases total magnetic flux linkage.

RC and RL Transient Time Constants

When DC voltage is suddenly applied to or removed from a reactive circuit, current and voltage do not transition instantaneously; they undergo an exponential transient phase.

Time Constant Definitions

• RC Time Constant ($\tau$): The time required for capacitor voltage or current to change by 63.2% of its total initial-to-final voltage differential:

  $\tau = R \cdot C \quad (\text{Seconds} = \Omega \times \text{Farads})$

• RL Time Constant ($\tau$): The time required for inductor current to build up or decay by 63.2% of its maximum value:

  $\tau = \frac{L}{R} \quad \left(\text{Seconds} = \frac{\text{Henries}}{\Omega}\right)$

Universal Charge / Discharge Curve Table

Exponential Voltage and Current Equations

• Capacitor Voltage Charging: $v_C(t) = V_{\text{source}} \left(1 - e^{-t/\tau}\right)$
• Capacitor Voltage Discharging: $v_C(t) = V_{\text{initial}} \cdot e^{-t/\tau}$
• Inductor Current Build-up: $i_L(t) = \frac{V_{\text{source}}}{R} \left(1 - e^{-t/\tau}\right)$
• Inductor Current Decay: $i_L(t) = I_{\text{initial}} \cdot e^{-t/\tau}$

Worked Calculations and Avionics Applications

Example 1: RC Pulse Timing Circuit

An aircraft navigation pulse generator uses an RC timing network with $R = 150\text{ k}\Omega$ and $C = 4.7\ \mu\text{F}$ driven by a 28 VDC step voltage.

1. Calculate Time Constant ($\tau$): $\tau = (150 \times 10^3\ \Omega) \times (4.7 \times 10^{-6}\text{ F}) = 0.705\text{ seconds}$
2. Capacitor Voltage after $1\tau$: $V_C(1\tau) = 28\text{ V} \times 0.632 = 17.70\text{ Volts}$
3. Time to Reach Full Steady-State Charge ($5\tau$): $t_{\text{steady}} = 5 \times 0.705\text{ s} = 3.525\text{ seconds}$

Example 2: Inductive Solenoid Kickback

When an aircraft fuel shutoff solenoid ($L = 2.0\text{ H}, R = 40\ \Omega$) is disconnected from 28 VDC, the sudden collapse of its magnetic field ($di/dt \to -\infty$) induces a massive negative voltage spike ($\text{CEMF} = -L \frac{di}{dt}$) reaching several hundred volts. This inductive kickback destroys control transistors unless suppressed by a parallel flyback diode wired across the solenoid coil.

Avionics Traps & Maintenance Warnings

• Capacitor High-Voltage Shock Risk: High-capacitance filter capacitors inside weather radar transmitters can retain lethal DC voltages for hours after system shutdown. Technicians must verify capacitor discharge using an approved safety grounding stick before touching internal terminals.
• Electrolytic ESR Degradation: Aged aluminum electrolytic filter capacitors exhibit increased Equivalent Series Resistance (ESR), causing high AC ripple voltage on DC power buses without blowing fuses.

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Capacitor Fault Isolation (NCATT Standard)

Beyond ESR aging, the NCATT AET standard requires technicians to isolate capacitors for improper installation, open-plate, and shorted-plate faults using visual and non-visual methods.

Improperly Installed Capacitors

Polarized electrolytics installed with reversed polarity, wrong voltage ratings, or non-compliant mounting (leads stressed, wrong footprint) fail early and can vent or explode. Confirm polarity marks, working voltage (WVDC), and installation standards before applying power.

Open Capacitors

An open capacitor loses continuity to one plate. AC coupling or filter action stops; no charging current flows in that branch. Voltage applied to the circuit (or branch) appears across the open component. On an ohmmeter, an open capacitor does not show the brief charging kick toward a rising resistance—it stays open/OL.

Shorted Capacitors

A shorted capacitor occurs when conductive plates make physical contact. Capacitance disappears electrically; the device behaves like a near-zero-ohm conductor and often trips a fuse or circuit breaker. Ohmmeter readings remain near 0 Ω instead of climbing toward infinite as a healthy dielectric would.

Avionics Trap: Confusing electrolytic ESR rise (high ripple, fuse still intact) with a hard short (breaker trips) delays the correct replacement strategy.