Source: University Physics: Elec & Mag, UIUC
Tags: AC circuit, reactance, capacitive reactance, inductive reactance, impedance, phasor diagram, phase angle, RLC series, Kirchhoff, voltage amplitude
Difficulty: Intermediate to Advanced Prerequisites: RL circuits, LC/RLC oscillations, basic trigonometry (sine, cosine, phase shifts).
Big picture: Everything up to this point dealt with DC (constant voltage) or free oscillation. Now the circuit is driven by an AC source: ε = V_max sin(ωt). The key new idea is that capacitors and inductors introduce phase shifts between voltage and current, and each has a frequency-dependent "resistance" called reactance. Combining all three (R, L, C) in series leads to impedance and phasor diagrams, which is the framework for analysing any AC circuit. This is the section where earlier concepts (inductor EMF, capacitor charging) all converge.
In an AC circuit, resistors keep voltage and current in phase, capacitors make current lead voltage by 90°, and inductors make current lag voltage by 90°. Each has a reactance (X_C = 1/(ωC), X_L = ωL) that plays the role of resistance. For a series RLC circuit, the total impedance is Z = √[R² + (X_L − X_C)²], and the phase angle between generator voltage and current is tan(φ) = (X_L − X_C)/R.
Reactance (X)
The frequency-dependent opposition to current flow by a capacitor or inductor. Unlike resistance, reactance does not dissipate energy as heat. Think of it as "AC resistance" that depends on how fast the voltage is oscillating.
Capacitive reactance (X_C)
X_C = 1/(ωC). At high frequencies, X_C is small (capacitor passes current easily). At low frequencies, X_C is large (capacitor blocks current). In simple terms, a capacitor becomes more "transparent" to faster-changing signals.
Inductive reactance (X_L)
X_L = ωL. At high frequencies, X_L is large (inductor opposes rapidly changing current). At low frequencies, X_L is small. The opposite behaviour to a capacitor.
Impedance (Z)
The total effective resistance of a series RLC circuit to AC current: Z = √[R² + (X_L − X_C)²]. Measured in ohms. It combines real resistance with the net reactance into a single number. Think of it as the AC version of total resistance.
Phase angle (φ)
The angle by which the current leads or lags the generator voltage. tan(φ) = (X_L − X_C)/R. When X_L > X_C, the circuit is inductive and current lags. When X_C > X_L, the circuit is capacitive and current leads.
Phasor diagram
A vector diagram where voltage amplitudes across R, L, and C are drawn as arrows (phasors) at the correct phase angles relative to the current. Used to find the total generator voltage amplitude and the phase angle graphically.
Source: ε = V_max sin(ωt)
Current: I = (V_max / R) sin(ωt)
I_max = V_max / R
Voltage and current are in phase: they peak and cross zero at the same time.
No phase shift, no reactance. The resistor behaves the same at every frequency.
Source: ε = V_max sin(ωt), with V_C = Q/C.
From KVR: V_max sin(ωt) − Q/C = 0, so Q = C V_max sin(ωt).
Current: I = dQ/dt = ωC V_max cos(ωt).
Current amplitude: I_max = ωC V_max = V_max / X_C, where X_C = 1/(ωC).
Phase relationship: V_C(t) has a sine term, I_C(t) has a cosine term. Current leads voltage by 90°.
Mnemonic: "ICE" (I leads in a C circuit, with E for EMF).
Physical reason: current must flow first to charge the capacitor before voltage builds up across it.
Source: ε = V_max sin(ωt).
Inductor voltage: L(dI/dt) = V_max sin(ωt).
Solve for I: I = −(V_max / ωL) cos(ωt).
Current amplitude: I_max = V_max / X_L, where X_L = ωL.
Phase relationship: voltage leads current by 90° (equivalently, current lags voltage by 90°).
Mnemonic: "ELI" (E leads I in an L circuit).
Physical reason: the inductor opposes changes in current, so current responds sluggishly, lagging behind the driving voltage.
Voltage across R: in phase with I_R.
Voltage across C: lags I_C by 90° (current leads).
Voltage across L: leads I_L by 90° (current lags).
In a series RLC circuit driven by ε = E_max sin(ωt), the same current I flows through all three components.
Voltage amplitudes across each:
V_R(max) = I_max R
V_C(max) = I_max X_C
V_L(max) = I_max X_L
These voltages are not in phase with each other, so they do not add algebraically. Instead, use the phasor diagram.
In the phasor diagram:
V_R points along the current direction (in phase).
V_L points 90° ahead of I (upward).
V_C points 90° behind I (downward).
The net reactive voltage is I_max(X_L − X_C), pointing either up or down depending on which reactance is larger.
The generator voltage amplitude is the vector sum:
E_max = I_max Z, where Z = √[R² + (X_L − X_C)²]
Phase angle: tan(φ) = (X_L − X_C) / R
φ > 0: circuit is inductive (current lags).
φ < 0: circuit is capacitive (current leads).
φ = 0: resonance (X_L = X_C).
Quantity | Formula |
|---|---|
Capacitive reactance | X_C = 1/(ωC) |
Inductive reactance | X_L = ωL |
Impedance (series RLC) | Z = √[R² + (X_L − X_C)²] |
Generator voltage | E_max = I_max Z |
Phase angle | tan(φ) = (X_L − X_C) / R |
Current amplitude | I_max = E_max / Z |
AC circuit analysis is the foundation of electrical engineering. X_C and X_L explain why capacitors are used to filter high-frequency noise (low X_C at high ω lets the noise bypass to ground) and why inductors are used in power supply smoothing (high X_L at high ω blocks ripple). Every audio crossover network, radio tuner, and power filter is a practical RLC circuit.
Students often try to add V_R, V_L, and V_C directly to get E_max. These voltages are out of phase with each other, so you must use the phasor (vector) addition. E_max = V_R + V_L + V_C is wrong; E_max = √[V_R² + (V_L − V_C)²] is correct.
Confusing which component leads and which lags. Current leads voltage in a capacitor (ICE), current lags voltage in an inductor (ELI). Mixing these up inverts the sign of φ.
Thinking reactance is the same as resistance. Reactance is frequency-dependent and does not dissipate power. Only R dissipates power.
Assuming X_C and X_L are always equal. They are equal only at the resonant frequency. At any other frequency, one dominates.
⚠️ Be able to draw the phasor diagram for a series RLC circuit from scratch: V_R along the horizontal, V_L pointing up, V_C pointing down, and the resultant E_max as the hypotenuse.
⚠️ Know the "ELI the ICE man" mnemonic cold. Questions often ask whether current leads or lags for a given component.
⚠️ Expect to calculate Z and φ for a series RLC circuit given numerical values of R, L, C, and ω.
⚠️ Remember that I_max = E_max / Z, not E_max / R (a common slip).
Fill in the blank: Capacitive reactance is X_C = ______.
True or false: In a purely inductive AC circuit, current and voltage are in phase.
Fill in the blank: Impedance of a series RLC circuit is Z = ______.
True or false: At resonance, the phase angle φ is zero.
Fill in the blank: The mnemonic for remembering phase relationships is ______ the ______ man.
Answers: 1. 1/(ωC). 2. False (current lags voltage by 90°). 3. √[R² + (X_L − X_C)²]. 4. True. 5. ELI, ICE.
Q: An AC source at 60 Hz drives a series circuit with R = 100 Ω, L = 0.5 H, and C = 10 μF. Find X_L, X_C, Z, and φ.
A: ω = 2π(60) = 377 rad/s. X_L = ωL = 377 × 0.5 = 188.5 Ω. X_C = 1/(ωC) = 1/(377 × 10⁻⁵) = 265.3 Ω. Z = √[100² + (188.5 − 265.3)²] = √[10000 + 5902] = √15902 ≈ 126.1 Ω. tan(φ) = (188.5 − 265.3)/100 = −0.768, so φ ≈ −37.5° (capacitive, current leads).
Q: In a series RLC circuit, if X_L = X_C, what is the impedance and what is the phase angle?
A: Z = √[R² + 0] = R. The phase angle φ = 0. This is the resonance condition.
Q: A capacitor is connected to an AC source. Does the current peak before or after the voltage peaks?
A: Before. Current leads voltage by 90° in a capacitor. The current peaks a quarter-cycle earlier than the voltage.
Q: Why can you not simply add V_R + V_L + V_C to find the source voltage amplitude in a series RLC circuit?
A: Because the three voltages peak at different times (different phases). They must be added as phasors (vectors), not as plain numbers.
AC circuit analysis builds directly on the inductor and capacitor behaviour studied in the RL and LC/RLC sections. The impedance concept introduced here is the gateway to resonance (covered next), where minimising Z maximises current. The phasor approach also connects to complex number representations of AC circuits used in more advanced electrical engineering courses.
Related Terms / Search Tags: AC circuit, alternating current, reactance, capacitive reactance, inductive reactance, impedance, phasor, phasor diagram, phase angle, ELI the ICE man, series RLC, voltage amplitude, frequency dependent, PHY 212, UIUC, midterm 3