ELC520S: Capacitance and Capacitors Part 2 — Vocabulary (RC Transient)

GROWTH AND DECAY

The charging of an RC circuit (resistor R, capacitor C, source voltage V) involves an instantaneous current given by i = \frac{V - v}{R} , where v is the instantaneous capacitor voltage. Since i = C\frac{dv}{dt} , the differential equation for charging is RC\frac{dv}{dt} = V - v . This leads to the analytical solutions for charging voltage:

v(t) = V\left(1 - e^{-t/\tau}\right)

and charging current:

i(t) = \frac{V}{R} e^{-t/\tau}

where \tau = RC is the time constant of the circuit. The initial rate of rise of the capacitor voltage is \left.\frac{dv}{dt}\right\|_{t=0} = \frac{V}{RC} = \frac{V}{\tau} .

DISCHARGE OF A CAPACITOR THROUGH A RESISTOR

When a capacitor, initially charged to a voltage V, discharges through a resistor R, the voltage and current decay exponentially according to:

v(t) = V e^{-t/\tau}

i(t) = -\frac{V}{R} e^{-t/\tau}

\tau = RC. The negative sign for current indicates an opposite direction compared to charging.

TRANSIENTS WITH THÉVENIN EQUIVALENT

For more complex networks, the transient response can be analyzed by finding the Thévenin equivalent. If the Thévenin equivalent seen by the capacitor has a source voltage V_{th} and series resistance R_{th}, the time constant becomes \tau = R_{th} C. The transient equations are:

v(t) = V_{th}\left(1 - e^{-t/\tau}\right)

i(t) = \frac{V_{th}}{R_{th}} e^{-t/\tau}

More generally, for an initial capacitor voltage V_1 and a final steady-state voltage V_f (reached after roughly five time constants), the capacitor voltage is described by:

v_c = V_f + (V_1 - V_f) e^{-t/\tau}

NOTES ON SCOPE AND CONTEXT

The core concept of RC transient responses is that the time constant \tau = RC governs the charging/discharging speed. Thévenin equivalents are crucial for simplifying complex networks, and the steady-state value is found by treating the capacitor as an open circuit.