Section 7.2: RLC Circuits & Impedance

Why This Matters for the Exam

In AC circuits, the opposition to current flow is not limited to resistance ($R$). Inductors ($L$) and capacitors ($C$) introduce reactive opposition, which varies with frequency. EASA Part-66 exams test your understanding of how these components behave under alternating current, how to calculate inductive and capacitive reactance, and how to combine them with resistance to find the total impedance ($Z$). A solid grasp of the phase relationships, particularly using the 'CIVIL' mnemonic, is essential for answering questions on AC circuit analysis.

Inductive Reactance ($X_L$)

When alternating current flows through an inductor, the changing magnetic field induces a self-induced electromotive force (EMF) that opposes the change in current. This opposition is called inductive reactance ($X_L$). It is measured in Ohms ($\Omega$) and is calculated as: $X_L = 2\pi f L$ where $f$ is the frequency in Hertz and $L$ is the inductance in Henries.

• Frequency Relationship: Inductive reactance is directly proportional to frequency. At DC ($f = 0$), $X_L$ is zero, and the inductor behaves as a short circuit. At high aircraft frequencies (400 Hz), inductive reactance is significantly higher, creating a large opposition to current.
• Phase Shift: In a purely inductive circuit, the current lags the voltage by exactly 90 degrees ($`\pi/2$ radians). This is because the induced back-EMF is proportional to the rate of change of current, which is greatest when the current wave crosses zero.

Capacitive Reactance ($X_C$)

When alternating voltage is applied to a capacitor, current flows into and out of the plates as they charge and discharge, even though no physical current flows through the dielectric. The opposition to this current flow is called capacitive reactance ($X_C$). It is measured in Ohms ($\Omega$) and is calculated as: $X_C = \frac{1}{2\pi f C}$ where $f$ is the frequency in Hertz and $C$ is the capacitance in Farads.

• Frequency Relationship: Capacitive reactance is inversely proportional to frequency. At DC ($f = 0$), $X_C$ is infinite, meaning the capacitor blocks DC (acts as an open circuit). At high frequencies, $X_C$ decreases, allowing AC current to flow easily.
• Phase Shift: In a purely capacitive circuit, the current leads the voltage by exactly 90 degrees ($`\pi/2$ radians). The maximum current flows when the voltage is changing most rapidly (at the zero-crossing of the voltage wave).

The "CIVIL" Mnemonic

To remember the phase relationships between voltage ($V$) and current ($I$) in reactive components, engineers use the word CIVIL:

• C I V: In a Capacitor, the Current (I) leads the Voltage (V).
• V I L: In an Inductor (L), the Voltage (V) leads the Current (I). This mnemonic is highly tested and is a quick reference tool for EASA exams.

Impedance ($Z$) in Series RLC Circuits

Impedance is the total opposition to alternating current in a circuit, combining resistance and reactance. In a series RLC circuit, the same current ($I$) flows through the resistor, inductor, and capacitor. However, the voltages across these components are out of phase:

• $V_R$ is in-phase with current.
• $V_L$ leads the current by 90 degrees.
• $V_C$ lags the current by 90 degrees. Since $V_L$ and $V_C$ point in opposite directions, they subtract directly. The net reactive voltage is $V_L - V_C$. Because the resistive and reactive voltages are at 90 degrees to each other, they must be combined using the Pythagorean theorem: $V_{total} = \sqrt{V_R^2 + (V_L - V_C)^2}$ Dividing by current $I$, we obtain the total impedance ($Z$): $Z = \sqrt{R^2 + (X_L - X_C)^2}$ The phase angle ($?\theta$) between the total voltage and the current is: $\theta = \arctan\left(\frac{X_L - X_C}{R}\right)$
• If $X_L > X_C$, the circuit is inductive (voltage leads current, $\theta$ is positive).
• If $X_C > X_L$, the circuit is capacitive (current leads voltage, $\theta$ is negative).
• The Power Factor is the cosine of the phase angle ($`\cos\theta = R/Z$).

Impedance ($Z$) in Parallel RLC Circuits

In a parallel RLC circuit, the voltage ($V$) across each branch is identical, but the currents through the branches are out of phase:

• $I_R$ is in-phase with voltage.
• $I_L$ lags the voltage by 90 degrees.
• $I_C$ leads the voltage by 90 degrees. The total current is the vector sum: $I_{total} = \sqrt{I_R^2 + (I_C - I_L)^2}$ The total impedance is the reciprocal of the total admittance ($Y = 1/Z$): $Y = \sqrt{\left(\frac{1}{R}\right)^2 + \left(\frac{1}{X_C} - \frac{1}{X_L}\right)^2}$ $Z = \frac{1}{Y}$ The phase angle in parallel is: $\theta = \arctan\left(\frac{I_C - I_L}{I_R}\right) = \arctan\left(R \left(\frac{1}{X_C} - \frac{1}{X_L}\right)\right)$

Comparison of RLC Configurations

Worked Exam Calculations

Example 1: A series circuit has $R = 30\ \Omega$, an inductor with $L = 15.9\text{ mH}$, and a capacitor with $C = 26.5\ \mu\text{F}$, connected to a 400 Hz supply. Find $X_L$, $X_C$, and $Z$.

1. Inductive Reactance: $X_L = 2\pi f L = 2 \times \pi \times 400 \times 0.0159 \approx 40\ \Omega$.
2. Capacitive Reactance: $X_C = 1 / (2\pi f C) = 1 / (2 \times \pi \times 400 \times 26.5 \times 10^{-6}) \approx 15\ \Omega$.
3. Total Impedance: $Z = \sqrt{30^2 + (40 - 15)^2} = \sqrt{30^2 + 25^2} = \sqrt{900 + 625} = \sqrt{1525} \approx 39.1\ \Omega$.
4. Phase Angle: $\theta = \arctan((40-15)/30) = \arctan(25/30) \approx 39.8^{\circ}$ (voltage leads current).

Exam Traps & Tips

• Vector Addition: Never add resistance and reactance algebraically (e.g., $30 + 40 - 15 = 55\ \Omega$ is incorrect). You must use the vector sum formula ($Z = \sqrt{R^2 + X^2}$).
• Inverse Reactance in Parallel: In parallel circuits, remember that a smaller reactance draws more current. An inductive branch with small $X_L$ draws a large current, making the parallel circuit predominantly inductive.