At the very beginning, let me emphasize that inductor acts as short circuit and capacitor acts as open circuit in DC analysis.
RC Circuits
Series RC Circuit
In the configuration above, we analyze the voltage across the capacitor when a step voltage (V=0 at t=0– s, and V=V at t=0 s) is applied.
i = C*dVC/dt, VR = iR = RC*dVC/dt,
VC = V – VR = V – RC*dVC/dt => VC + RC*dVC/dt = V.
Apply Laplace transform and denote L{VC(t)} as F(s):
F(s) + RC*sF(s) = V/s => F(s) = V/[s(1 + RC*s)].
Apply inverse Laplace transform,
VC(t) = V[1 – exp(-t/(RC))]
For a 5V step voltage and RC = 1s, the plot is as follows:
With different step voltages or RC time constants, the plot follows similar pattern. In general, at multiples of RC time constants, the percentage of VC to V is fixed:
| At a*RC s | VC/V% |
| a = 1 | 63.2% |
| a = 2 | 86.5% |
| a = 3 | 95.0% |
| a = 4 | 98.2% |
| a = 5 | 99.3% |
Does this look familiar to confidence interval? They both point to exponential functions after all. We shall discuss the specific numbers in another time.
For a harmonic AC signal (sinusoidal wave) input, the series RC circuit simply acts as a filter. Remember the impedance of capacitor is 1/(jωC), the higher the frequency, the lower the impedance (magnitude). Hence for higher frequency, the signal is more strongly attenuated on the capacitor, and the circuit acts as a low-pass filter when the voltage across the capacitor is measured. On the other hand, if the voltage across the resistor is measured, the circuit acts as a high-pass filter. For complex signals, applying Fourier transform on the signal makes it convenient to analyze the circuit.
Parallel RC Circuit
