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In an AC circuit containing a resistor and capacitor, resistance alone is not enough to predict current. Use complex impedance: an ideal resistor has ZR=R, while an ideal capacitor has ZC=−jXC, where XC=1/(2πfC). Capacitive reactance is measured in ohms, falls as frequency or capacitance increases, and causes current to lead capacitor voltage by 90°.
The circuit must be identified as series RC, parallel RC, or a more complex network before calculating. The formulas below apply to sinusoidal steady-state AC; capacitor charging transients require time-domain analysis.
Resistance, reactance, and impedance
Ideal resistor
An ideal resistor opposes current with resistance R in ohms. Its impedance is purely real:
ZR=R
Resistor voltage and current are in phase, and the RMS relationship is VR=IR. It dissipates real power as heat:
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PR=IRMS2R
The ideal model treats resistance as frequency-independent. Real resistors have parasitic inductance and capacitance that can matter at sufficiently high frequency. See OpenStax’s AC circuit overview.
Capacitive reactance
Capacitive reactance is the magnitude of a capacitor’s frequency-dependent opposition to AC:
XC=1/(2πfC)
- f is frequency in hertz.
- C is capacitance in farads.
- XC is in ohms.
Increasing frequency or capacitance lowers reactance. In the ideal steady-state DC limit, f=0 and reactance tends toward infinity. At very high frequency, ideal reactance tends toward zero, but ESR, ESL, leakage, and self-resonance limit real capacitors. A capacitor stores and returns electric-field energy rather than dissipating it like a resistor. OpenStax explains the frequency dependence of reactance.
Capacitor impedance
Reactance is only a magnitude. The full complex impedance is:
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ZC=1/(jωC)=−j/(ωC)=−jXC, with ω=2πf.
In polar notation, ZC=XC∠−90°. The negative imaginary term carries the phase information: for an ideal capacitor under sinusoidal steady-state conditions, current leads capacitor voltage by 90°. Writing ZC=XC would lose that information.
Series RC circuits
A series RC circuit carries the same current through the resistor and capacitor. Add their impedances directly:
Z=R−jXC
The magnitude and impedance angle are:
|Z|=√(R2+XC2)
θ=−tan−1(XC/R)
The negative angle means the total current leads the source voltage. If current is the reference phasor, resistor voltage is at 0° and capacitor voltage is at −90°. If source voltage is the reference, current has a positive lead angle. Both conventions are valid when used consistently. OpenStax’s series-AC treatment discusses these phasor relationships.
Calculation procedure
- Convert capacitance to farads and identify whether voltage is RMS or peak.
- Calculate XC=1/(2πfC).
- Form Z=R−jXC.
- Calculate |Z| and θ.
- Find current with I=V/|Z| for magnitudes, or Ī=V̲/Z for phasors.
- Calculate VR=IR and VC=IXC.
- Check the source magnitude with the perpendicular phasor sum, VS=√(VR2+VC2).
Worked example
Use R=1.00 kΩ, C=0.100 μF, f=1.00 kHz, and VS=10.0 V RMS.
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- Reactance: XC=1/[2π(1000)(0.100×10−6)]≈1.59 kΩ.
- Impedance: Z=1000−j1592 Ω.
- Magnitude: |Z|≈1.88 kΩ.
- Angle: θ≈−57.9°, so Z≈1.88 kΩ∠−57.9°.
- Current: I=10/1880≈5.32 mA RMS.
- Component voltages: VR≈5.32 V RMS and VC≈8.46 V RMS.
The capacitor voltage exceeds the source-voltage magnitude without violating Kirchhoff’s voltage law because the two component voltages are 90° apart; phasors, not scalar magnitudes, add. Real power is dissipated in the resistor:
P=I2R≈28.3 mW.
Parallel RC circuits
In a parallel RC circuit, both branches have the same voltage. Their currents are different and add as phasors, so admittance is usually the simplest method:
Y=1/R+jωC
Then Z=1/Y. Branch currents are:
IR=V/R and IC=jωCV=V/XC∠+90°.
The admittance magnitude and total-current magnitude are:
|Y|=√[(1/R)2+(ωC)2]
|I|=V|Y|
Total current leads applied voltage because the capacitor-branch current leads by 90°. Do not use √(R2+XC2) for a parallel network; that is the series-RC impedance magnitude.
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| Feature | Series RC | Parallel RC |
|---|---|---|
| Common quantity | Current | Voltage |
| Convenient calculation | Add impedances | Add admittances |
| Total impedance | R−jXC | 1/(1/R+jωC) |
| Low-frequency behavior | Capacitor blocks; current is small | Capacitor branch current is small |
| High-frequency behavior | Current approaches V/R | Capacitor branch draws increasing current |
Phase angle, power factor, and AC power
For a series RC circuit, power factor is PF=cosθ=R/|Z| and is described as leading. The standard AC power quantities are:
- Real power: P=VI cosθ, measured in watts.
- Reactive power: Q=VI sinθ, measured in vars. Under the usual convention, capacitive Q is negative.
- Apparent power: S=VI, measured in VA.
Here, voltage and current must use the same RMS convention. An ideal capacitor can carry substantial RMS current while consuming zero average real power; it exchanges reactive energy with the source. Real capacitors dissipate some power through ESR and dielectric loss. Fluke’s terminology guide covers impedance and apparent-power terms.
How frequency changes an RC circuit
- Low frequency: XC is large, current is small, and most series-circuit source voltage appears across the capacitor.
- High frequency: XC is small, current approaches the resistor-limited value, and more voltage appears across the resistor.
With output across the resistor, a series RC divider is a first-order high-pass filter:
HR(jω)=VR/VS=R/[R+1/(jωC)]
Its cutoff is fc=1/(2πRC). At this frequency, output magnitude is 70.7% of the passband value, or approximately −3.01 dB. Taking output across the capacitor produces the corresponding low-pass response. These frequency-response formulas describe steady-state sine waves; charging and discharging use the time constant τ=RC.
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RMS, peak, and reference conventions
For a sine wave, VRMS=Vpeak/√2 and Vpeak=√2VRMS. Impedance works with either peak phasors or RMS phasors, but every voltage, current, and power value must use a consistent convention. Never add a peak value to an RMS value.
Measuring an RC circuit
Digital multimeter
A DMM can check resistance, AC voltage, and—if supported—capacitance. Readings depend on waveform, frequency, crest factor, signal size, and the meter’s specified bandwidth. At higher frequencies, meter input resistance and capacitance load the circuit; Keysight documents these loading errors. An ammeter must be inserted in series, never placed directly across a voltage source.
Oscilloscope and function generator
- Drive a low-voltage, sinusoidal test circuit with a function generator.
- Place a known series resistor as a current shunt.
- Measure source voltage on one channel and resistor or capacitor voltage on the other.
- Infer current from I=VR/R.
- Measure time offset between corresponding waveform points and calculate φ=360°(Δt/T).
Standard bench-scope grounds are earth-referenced. Do not clip one arbitrarily to a mains or floating node; use correctly rated differential probes, isolation, and procedures for hazardous circuits.
Impedance analyzer
An impedance analyzer can sweep frequency and report magnitude, phase, series resistance, reactance, admittance, capacitance, dissipation factor, or Q. Digilent WaveForms documentation describes reference-resistor impedance measurements and phase reporting. For laboratory-grade work, fixture parasitics and calibration are as important as the instrument itself.
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- ESR: adds series loss and real power dissipation.
- ESL: adds inductive behavior at high frequency.
- Leakage: gives a real parallel-current path, especially important for long DC intervals.
- Dielectric, voltage, and temperature effects: make capacitance and loss dependent on operating conditions.
- Polarity: many electrolytic capacitors must not be reverse-biased.
- Self-resonant frequency: above it, a real capacitor can become inductive rather than an ideal short circuit.
These effects matter in RF, switching supplies, fast digital circuits, and precision impedance measurements. The ideal model remains appropriate when its frequency, voltage, and accuracy assumptions are satisfied.
Quick Recap
Common mistakes and checks
- Identify series, parallel, or mixed topology before choosing a formula.
- Convert μF and nF to farads.
- Include the 2π in 1/(2πfC).
- Use ZC=−jXC, not a positive real reactance.
- Add phase-shifted voltages and currents as phasors, not ordinary scalars.
- Check whether every value is peak or RMS.
- As frequency approaches zero, series-RC current should approach zero; as frequency rises, it should increase toward V/R.
- Remember that “blocks DC” means ideal steady-state behavior; charging current and leakage still exist.
- Do not generalize “passes high frequencies” without specifying the topology, output node, source impedance, and load.
- Do not use a grounded oscilloscope or current meter in an unsafe configuration on mains circuits.
Formula reference
| Quantity | Formula |
|---|---|
| Angular frequency | ω=2πf |
| Capacitive reactance | XC=1/(ωC) |
| Capacitor impedance | ZC=−jXC |
| Resistor impedance | ZR=R |
| Series RC impedance | Z=R−jXC |
| Series magnitude | |Z|=√(R2+XC2) |
| Series angle | θ=−tan−1(XC/R) |
| Parallel admittance | Y=1/R+jωC |
| Filter cutoff | fc=1/(2πRC) |
| AC power | P=VI cosφ, Q=VI sinφ, S=VI |
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