2.3 Capacitive Reactance and Capacitance

2.3 Capacitive Reactance and Capacitance

Capacitance is the electrostatic property of an electrical circuit component that allows it to store electrical energy in an electrostatic field and oppose any changes in voltage across its terminals. In aircraft electronics, capacitors serve vital roles in power supply smoothing, noise filtering, audio/RF coupling, timing circuits, and specialized sensor applications such as aircraft fuel quantity gauging probes.

Principles of Capacitance and Dielectric Energy Storage

A fundamental capacitor consists of two parallel conductive metal plates separated by an insulating material known as the dielectric (such as air, mica, ceramic, tantalum, or oil). When a DC potential is applied across the capacitor terminals, electrons accumulate on the negative plate while leaving the positive plate, creating a concentrated electrostatic field across the dielectric.

Mathematical Definition of Capacitance

Capacitance ($C$) is defined as the quantity of electric charge ($Q$) in Coulombs stored per Volt of applied potential ($V$):

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

Where:

• $C$ = Capacitance in Farads (F)
• $Q$ = Stored charge in Coulombs (C)
• $V$ = Applied potential in Volts (V)

One Farad (F) is an extraordinarily large unit. Practical avionics capacitors are rated in microfarads ($\mu\text{F} = 10^{-6}\text{ F}$), nanofarads ($\text{nF} = 10^{-9}\text{ F}$), or picofarads ($\text{pF} = 10^{-12}\text{ F}$).

Physical Factors Governing Capacitance

The capacitance value of a parallel-plate capacitor is mathematically governed by three physical construction variables:

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

Where:

• $A$ = Overlapping plate surface area ($m^2$)
• $d$ = Distance separating the plates (meters)
• $\epsilon_0$ = Permittivity of free space ($8.854 \times 10^{-12}\text{ F/m}$)
• $\epsilon_r$ = Relative dielectric constant of the insulating material (dimensionless, e.g., Air = 1.0, Vacuum = 1.0, Jet-A Fuel $\approx 2.1$, Glass = 5-10)

Energy Storage in an Electrostatic Field

Like inductors, an ideal capacitor dissipates no active power ($0\text{ Watts}$). Energy is stored in the electrostatic field established within the dielectric during charging and returned completely to the circuit during discharge:

$W_C = \frac{1}{2} \cdot C \cdot V^2$

Where $W_C$ is stored energy in Joules (J), $C$ is in Farads (F), and $V$ is potential in Volts (V).

The ICE Phase Relationship

When alternating current is connected across a capacitor, the capacitor continuously charges, discharges, and recharges in opposite polarities as the AC voltage alternates. Current flows into and out of the capacitor plates at maximum rate when the applied AC voltage is crossing zero (where the rate of voltage change $\frac{dv}{dt}$ is highest).

Conversely, when the applied AC voltage reaches its positive peak ($90^\circ$), the rate of voltage change drops to zero, causing current flow to cease. Thus, in a purely capacitive AC circuit, the AC current leads the applied AC voltage by 90 electrical degrees ($\pi/2$ radians).

This relationship is remembered in avionics using the classic ICE mnemonic:

$\mathbf{I} \quad \mathbf{C} \quad \mathbf{E}$

• I (Current) leads E (Voltage) in a C (Capacitor) by 90 degrees.
• Combined with ELI, avionics technicians memorize the complete phase rule: ELI the ICE man.

On a phasor diagram, the capacitive current vector $\vec{I}_C$ points horizontally along the $+0^\circ$ real axis, while the capacitive voltage vector $\vec{V}_C$ points straight down along the $-j$ axis at $-90^\circ$.

Capacitive Reactance ($X_C$) Formula and Analysis

Capacitive Reactance ($X_C$) is the opposition offered by a capacitor to alternating current flow, measured in ohms ($\Omega$). Unlike resistance, $X_C$ varies inversely with frequency and capacitance value:

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

Where:

• $X_C$ = Capacitive Reactance in ohms ($\Omega$)
• $f$ = Frequency in Hertz (Hz)
• $C$ = Capacitance in Farads (F)

Behavior Across Frequency

• At DC ($f = 0\text{ Hz}$): $X_C = \frac{1}{0} \to \infty\ \Omega$. A capacitor acts as a complete open circuit to DC, blocking steady direct current.
• As Frequency Increases ($f \to \infty$): $X_C$ approaches zero. At high frequencies, a capacitor acts as a short circuit, easily passing high-frequency AC noise signals to ground.

Worked Formula Example: 400 Hz Capacitive Reactance Calculation

Calculate the capacitive reactance ($X_C$) of a $5\ \mu\text{F}$ ($5 \times 10^{-6}\text{ F}$) capacitor operating on a 115 VAC RMS, 400 Hz aircraft bus bar.

Step 1: Apply the $X_C$ Formula $X_C = \frac{1}{2 \cdot \pi \cdot 400\text{ Hz} \cdot (5 \times 10^{-6}\text{ F})} = \frac{1}{0.012566} = 79.58\ \Omega$

Step 2: Calculate Current ($I_C$) $I_C = \frac{V_{rms}}{X_C} = \frac{115\text{ VAC}}{79.58\ \Omega} = 1.445\text{ Amperes}$

Series and Parallel Capacitor Networks

Because connecting capacitors in parallel effectively increases total plate surface area, capacitor combination formulas are inverted relative to resistors and inductors:

Parallel Capacitors

Connecting capacitors in parallel adds plate surface area directly, increasing total capacitance:

$C_T = C_1 + C_2 + C_3 + \dots + C_n$ $X_{C,T} = \frac{1}{\frac{1}{X_{C1}} + \frac{1}{X_{C2}} + \dots}$

Series Capacitors

Connecting capacitors in series effectively increases the overall dielectric spacing distance ($d$), reducing total capacitance below the smallest single component value:

$\frac{1}{C_T} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \dots + \frac{1}{C_n}$

For two series capacitors: $C_T = \frac{C_1 \cdot C_2}{C_1 + C_2}$

Avionics Application: Aircraft Fuel Quantity Capacitance Probes

Modern transport category aircraft measure fuel tank quantity using non-moving capacitance fuel probes. A capacitance fuel probe consists of two concentric aluminum tubes mounted vertically inside the aircraft fuel tank, acting as capacitor plates.

When the fuel tank is empty, the dielectric between the probe tubes is air ($\epsilon_r = 1.0$). As Jet-A fuel fills the tank, fuel displaces air between the concentric tubes. Because Jet-A fuel has a relative dielectric constant of $\epsilon_r \approx 2.1$, total probe capacitance increases in direct linear proportion to fuel level.

$C_{probe} = \frac{\epsilon_0 A}{d} \left[ h_{fuel} \cdot \epsilon_{fuel} + (h_{total} - h_{fuel}) \cdot \epsilon_{air} \right]$

Because fuel density changes with temperature, capacitance probes intrinsically measure fuel mass (pounds or kilograms) rather than mere volume, providing pilots with exact weight-and-balance fuel figures regardless of thermal expansion.