RC Circuit Analysis: Series & Parallel (Explained in Plain English)

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Key learnings:

RC Circuit Definition: An RC circuit is an electrical configuration consisting of a resistor and a capacitor used to filter signals or store energy.
Parallel RC Circuit Dynamics: In a parallel RC circuit, the voltage is uniform across all components, while the total current is the sum of individual currents through the resistor and capacitor.
Impedance and Phase Calculation: The impedance in an RC circuit helps determine how the voltage and current are phased, impacting the signal’s overall behavior.
Transfer Function Importance: The transfer function in RC circuits, whether series or parallel, provides a crucial link between input and output, influencing how signals are processed.
Time Constant Significance: The time constant of an RC circuit is fundamental in understanding how quickly the circuit responds to changes in voltage or current, essential for timing applications.

What is an RC Circuit?

An RC circuit, or resistor-capacitor network, is an electrical circuit containing the passive circuit components of a resistor (R) and capacitor (C). Depending on its connection and output point, the network can shape signals, set timing or store and release energy while driven by a voltage source or current source.

When current flows, the resistor dissipates energy as heat, while an ideal capacitor stores energy in its electric field and can return it to the circuit. The same distinction applies to the resistive and reactive parts of an RL circuit or RLC circuit.

An ideal LC circuit has no resistance and therefore no resistive loss. Real inductors, capacitors and connecting conductors have non-zero resistance and other losses, so practical circuits dissipate some energy.

Series RC Circuit

In a series RC circuit, a resistor with resistance R and a capacitor with capacitance C carry the same current. Their voltage drops have different phases under sinusoidal steady-state conditions.

Here $I$ is the RMS value of the current in the circuit.

$V_R$ is the voltage across the resistor R.

$V_C$ is the voltage across the capacitor C.

$V$ is the RMS value of the supply voltage.

The figure shows a vector diagram of the series RC circuit.

R-C Circuit Vector Diagram — **VECTOR DIAGRAM**

Because the current $'I'$ is common to both series components, the phasor diagram uses it as the reference.

$V_R = IR$ is in phase with current $'I'$ because an ideal resistor has voltage and current in phase.

$V_C=I X_C$ lags current $'I'$ by $90^0$ for an ideal capacitor. Their voltage and current are $90^0$ apart: voltage lags current by $90^0$ , or current leads voltage by $90^0$ .

Now $V$ is the vector sum of $V_R$ and $V_C$ .

$\begin{align*} \,\, therefore, \,\, V^2 = {V_R}^2 + {V_C}^2 \end{align*}$

$\begin{align*} \begin{split} V = {\sqrt{{V_R}^2 + {V_C}^2}} \ & = {\sqrt{{IR}^2 + {IX_C}^2}} \ & = I {\sqrt{{R}^2 + {X_C}^2}} \ & = IZ \ \end{split} \end{align*}$

The magnitude of the impedance of an R-C series circuit is

$\begin{align*} Z = {\sqrt{{R}^2 + {X_C}^2}} \end{align*}$

$\begin{align*} \,\, where, \,\, X_C = \frac{1}{{\omega}C} = \frac{1}{2{\pi}fC} \end{align*}$

The figure shows the voltage and impedance triangles. The complex impedance is R minus jXC; the displayed triangle gives its magnitude.

As seen, the vector $V$ lags $I$ by an angle ø where

$\begin{align*} tan{\phi} = \frac{IX_C}{IR} \end{align*}$

$\begin{align*} {\phi} =tan^-^1 \frac{X_C}{R} \end{align*}$

Thus in an R-C series circuit current $'I'$ leads the supply voltage $'V'$ by an angle

$\begin{align*} {\phi} =tan^-^1 \frac{X_C}{R} \end{align*}$

$\begin{align*} \,\, i.e. \,\ if \,\,V = V_m sin{\omega}t \end{align*}$

$\begin{align*} i = I_m sin({\omega}t + {\phi}) \end{align*}$

$\begin{align*} \,\, where, \,\, I_m = \frac{V_m}{Z} \end{align*}$

The figure shows the sinusoidal voltage and current waveforms of the R-C series circuit.

R C Circuit Waveform — **VOLTAGE AND CURRENT WAVEFORM**

Power in an RC Series Circuit

Instantaneous power is the product of instantaneous voltage and current. In the derivation below, the uppercase symbols in the first line represent time-varying values before peak-value expressions are substituted.

$\begin{align*} P = V I \end{align*}$

$\begin{align*} = (V_m sin{\omega}t) [I_m sin({\omega}t + {\phi})] \end{align*}$

$\begin{align*} = \frac{V_m I_m}{2} [2sin{\omega}t * sin({\omega}t + {\phi})] \end{align*}$

$\begin{align*} = \frac{V_m I_m}{2} [cos[{\omega}t-({\omega}t+{\phi})] - cos[{\omega}t+({\omega}t+{\phi})]] \end{align*}$

$\begin{align*} = \frac{V_m I_m}{2} [cos({-\phi}) - cos({2\omega}t+{\phi})] \end{align*}$

$\begin{align*} = \frac{V_m I_m}{2} [cos{\phi} - cos({2\omega}t+{\phi})] \end{align*}$

$\begin{align*} \,\, [where, \,\, cos ({-\phi}) = cos {\phi} \,\, because \,\, cos \,\, curve \,\, is \,\, symmetric] \,\, \end{align*}$

$\begin{align*} = \frac{V_m I_m}{2} cos{\phi} - \frac{V_m I_m}{2} cos({2\omega}t+{\phi}) \end{align*}$

Thus the instantaneous power consists of two parts.

1. A constant part = $\frac{V_m I_m}{2} cos{\phi}$

2. A varying component with the minus sign shown in the derivation = $\frac{V_m I_m}{2} cos({2\omega}t+{\phi})$ , which varies at twice the supply frequency.

The average value of the varying power component over a complete cycle is zero.

Thus the average power consumed in an RC series circuit over one cycle is

$\begin{align*} \begin{split} P = \frac{V_m I_m}{2} cos{\phi} \ & = \frac{V_m}{\sqrt{2}} \frac{I_m}{\sqrt{2}} cos{\phi} \ & = V I cos{\phi} \ \end{split} \end{align*}$

Where $V$ and $I$ are the RMS values of the applied voltage and current in the circuit.

Power Factor in an RC Series Circuit

Consider the figure showing the power and impedance triangles.

$\begin{align*} \begin{split} \,\, (power \,\, factor) \,\, cos{\phi} = \frac{P \,\, (active \,\, power)\,\,} {S \,\, (apparent \,\, power)\,\,} \ & = \frac{R} {Z} \ & = \frac{R} {\sqrt{{R}^2 +{X_C}^2}} \ \end{split} \end{align*}$

Parallel RC Circuit

In a parallel R-C circuit, a resistor and a capacitor are connected across the same two nodes. The resistor has resistance $R$ , and the capacitor has capacitance $C$ .

Each branch of a parallel RC circuit has the same voltage. With