E5B: Time Constants and Phase
E5B covers how reactive components behave over time and with respect to phase in AC circuits. Topics include the RC and RL time constant, the specific percentages associated with one time constant, phase angle calculation for series RLC circuits with arbitrary values, the fundamental voltage-current phase relationships in capacitors and inductors, and the admittance and susceptance representations of circuit impedance.
The Extra exam draws one question from E5B. Questions require calculating time constants for combined parallel circuits, computing phase angles using arctangent, correctly identifying voltage-leading or voltage-lagging behavior, and working with admittance and susceptance definitions.
RC and RL Time Constants
When a voltage is applied to an RC or RL circuit, the voltage or current does not jump instantly to its final value — it rises (or falls) exponentially. The rate of this change is described by the time constant τ (tau).
RL time constant: τ = L / R (in seconds, when L is in henries and R in ohms)
After one time constant:
- A charging capacitor reaches 63.2% of the applied voltage
- A discharging capacitor falls to 36.8% of its initial voltage
These percentages come from the exponential function: 1 − e⁻¹ ≈ 0.632 for charging and e⁻¹ ≈ 0.368 for discharging. The time constant is defined as the time to reach exactly these values — not 50% or 100%.
Time Constants in Parallel Circuits
When resistors and capacitors are combined in parallel, the equivalent R and C must be computed before applying the time constant formula.
Capacitors in parallel add: C_total = 220 + 220 = 440 μF
Resistors in parallel: R_total = (1 MΩ × 1 MΩ) / (1 + 1 MΩ) = 0.5 MΩ
τ = R × C = 0.5 × 10⁶ × 440 × 10⁻⁶ = 220 seconds
Voltage-Current Phase Relationships
In a pure capacitor, the current and voltage are always 90° out of phase — but which one leads? The key rule is:
Inductor: Voltage leads current by 90° (the voltage must build up first before current can flow through the inductor's inductance)
The memory aid "ELI the ICE man" captures these relationships:
- ELI — in an inductor (L), voltage (E) comes before current (I)
- ICE — in a capacitor (C), current (I) comes before voltage (E)
Phase Angle in Series RLC Circuits
In a series RLC circuit with all three components, the net phase angle between the total voltage and the current depends on how much inductive versus capacitive reactance is present relative to the resistance:
Positive θ → inductive net → voltage leads current
Negative θ → capacitive net → voltage lags current
Working through the exam scenarios:
| XC | R | XL | XL − XC | θ = arctan(net/R) | Phase |
|---|---|---|---|---|---|
| 500 Ω | 1000 Ω | 250 Ω | −250 Ω | arctan(−0.25) = −14° | 14°, voltage lags |
| 300 Ω | 100 Ω | 100 Ω | −200 Ω | arctan(−2) = −63° | 63°, voltage lags |
| 25 Ω | 100 Ω | 75 Ω | +50 Ω | arctan(+0.5) = +27° | 27°, voltage leads |
When XC exceeds XL, the net reactance is capacitive and voltage lags current. When XL exceeds XC, the net reactance is inductive and voltage leads current. When they are equal (resonance), the phase angle is zero.
Admittance and Susceptance
Admittance (Y) and susceptance (B) are the reciprocal counterparts to impedance (Z) and reactance (X). Working with admittances simplifies parallel circuit calculations in the same way that working with impedances simplifies series circuits.
Susceptance B = the imaginary part of admittance (symbol B)
For a pure reactance X: B = 1/X (susceptance is the reciprocal of reactance)
Converting polar impedance to admittance: take the reciprocal of the magnitude and change the sign of the angle. If Z = |Z| ∠ θ, then Y = (1/|Z|) ∠ (−θ). The magnitude inverts and the angle negates — this is the complete conversion procedure.
The letter B is used universally to represent susceptance. Admittance, susceptance, and conductance (G, the real part of Y) form a complete parallel to impedance, reactance, and resistance.
E5B Practice Questions
Check Your Knowledge
E5C: Coordinate Systems and Phasors →
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