AcademyDC Circuits
Academy
RC Transients
Level 1 - Physics topic page in DC Circuits.
Principle
An RC circuit changes exponentially because the capacitor voltage changes the resistor current, and the resistor current changes the capacitor charge.
Notation
\(R\)
resistance in the charging or discharging path
\(\mathrm{\Omega}\)
\(C\)
capacitance
\(\mathrm{F}\)
\(\tau\)
time constant
\(\mathrm{s}\)
\(q(t)\)
capacitor charge
\(\mathrm{C}\)
\(V_C(t)\)
capacitor voltage
\(\mathrm{V}\)
\(I(t)\)
circuit current
\(\mathrm{A}\)
Method
Derivation 1: Charging from an ideal source
For a series resistor and capacitor connected to a DC source, the loop equation relates the source, the resistor drop, and the capacitor voltage.
Loop equation
\[\mathcal E-IR-\frac{q}{C}=0\]
Current is charge rate
\[I=\frac{dq}{dt}\]
Differential equation
\[R\frac{dq}{dt}+\frac{q}{C}=\mathcal E\]
Charging solution
\[q(t)=C\mathcal E\left(1-e^{-t/RC}\right)\]
Derivation 2: Discharging
With the source removed, the capacitor drives current through the resistor. The stored charge decays exponentially.
Discharge loop
\[IR+\frac{q}{C}=0\]
Discharge charge
\[q(t)=q_0e^{-t/RC}\]
Discharge voltage
\[V_C(t)=V_0e^{-t/RC}\]
Derivation 3: Time constant
The time constant is the product of the resistance seen by the capacitor and the capacitance.
Time constant
\[\tau=RC\]
One time constant while charging
\[V_C(\tau)=\mathcal E(1-e^{-1})\approx0.632\mathcal E\]
One time constant while discharging
\[V_C(\tau)=V_0e^{-1}\approx0.368V_0\]
Rules
Time constant
\[\tau=RC\]
Charging voltage
\[V_C(t)=\mathcal E(1-e^{-t/\tau})\]
Charging current
\[I(t)=\frac{\mathcal E}{R}e^{-t/\tau}\]
Discharging voltage
\[V_C(t)=V_0e^{-t/\tau}\]
Capacitor current relation
\[I=C\frac{dV_C}{dt}\]
Examples
Question
A
\[20\,\mathrm{k}\Omega\]
resistor charges a \[50\,\mu\mathrm{F}\]
capacitor. Find \(\tau\).Answer
\[\tau=RC=(2.0\times10^4)(50\times10^{-6})=1.0\,\mathrm{s}\]
Checks
- Capacitor voltage cannot jump discontinuously in a circuit with finite current.
- At the instant an uncharged capacitor is connected, it acts like a zero-voltage element.
- After a long time in a DC circuit, an ideal capacitor branch is open.
- The exponential approaches its final value asymptotically.
- When the capacitor is connected to a larger circuit, use the Thevenin resistance seen by the capacitor for \(R\).