Time constants
Note where R sits. Multiplied in the RC case, divided in the RL case. More resistance slows an RC circuit and speeds an RL one, and the asymmetry is the thing to remember.
What is the term for the time required for the capacitor in an RC circuit to be charged to 63.2% of the applied voltage or to discharge to 36.8% of its initial voltage?
One time constant
The other figure worth having: after one time constant a discharging capacitor has fallen to 36.8% of its initial voltage. (Both numbers are 1−1/e and 1/e.) Five time constants is conventionally “fully” charged or discharged.
Worked, in the pool’s own shape: two 220 µF capacitors and two 1 MΩ resistors, all in parallel.
- Capacitors in parallel add: 440 µF.
- Resistors in parallel halve: 500 kΩ.
- τ = 500×10³ × 440×10⁻⁶ = 220 seconds.
The trap is the parallel combination, not the time constant. Combine first — using the capacitor rules from General, where parallel capacitors add — and the arithmetic is trivial.
Phase, and getting the sign right
Two relationships to hold absolutely:
What is the relationship between the AC current through a capacitor and the voltage across a capacitor?
Current leads voltage by 90 degrees.
What is the relationship between the AC current through an inductor and the voltage across an inductor?
Voltage leads current by 90 degrees.
The mnemonic ELI the ICE man: in an invductor (L), E leads I; in a capacitor (C), I leads E.
For a series RLC circuit the net reactance decides:
- Net reactance positive (inductive): voltage LEADS current.
- Net reactance negative (capacitive): voltage LAGS current.
Three worked answers, matching the pool exactly:
| X_C | R | X_L | Net X | θ | Direction |
|---|---|---|---|---|---|
| 500 | 1000 | 250 | −250 | arctan(−0.25) = 14° | voltage lagging |
| 300 | 100 | 100 | −200 | arctan(−2) = 63° | voltage lagging |
| 25 | 100 | 75 | +50 | arctan(0.5) = 27° | voltage leading |
The arctangent is the easy part. The sign is the exam question, and the distractors always offer the right magnitude with the wrong direction.
Method, every time:
- Net reactance = X_L − X_C. Inductive minus capacitive, in that order.
- Sign positive → inductive → voltage leads. Negative → capacitive → voltage lags.
- Magnitude = arctan(|X| / R).
Admittance and susceptance
The reciprocal world, which simplifies parallel circuits the way conductance does.
What is admittance?
The inverse of impedance.
What is susceptance?
The imaginary part of admittance.
What letter is commonly used to represent susceptance?
B.
The correspondence, worth laying out because the exam tests it as vocabulary:
| Series world | Parallel world |
|---|---|
| impedance Z | admittance Y = 1/Z |
| resistance R | conductance G |
| reactance X | susceptance B |
And the conversions:
How is impedance in polar form converted to an equivalent admittance?
Take the reciprocal of the magnitude and change the sign of the angle.
In polar form that is all there is to it: Z = 100∠30° becomes Y = 0.01∠−30°. Reciprocal magnitude, negated angle — which is why polar form is worth using for this operation and rectangular form is not.
What is the effect on the magnitude of pure reactance when it is converted to susceptance?
It is replaced by its reciprocal.
Check yourself
- A 100 kΩ resistor and a 10 µF capacitor in series. Time constant?
- X_L = 400, X_C = 100, R = 300. Phase angle and direction?
- Z = 50∠−45°. What is the admittance?
Answers
- τ = 100×10³ × 10×10⁻⁶ = 1 second.
- Net X = 400 − 100 = +300, inductive. θ = arctan(300/300) = 45°, voltage leading.
- Reciprocal magnitude, negated angle: 0.02∠+45°.