Circuits with a single reactive element (inductor or capacitor).
1. Capacitor
Recall that “capacitor’s law” is
or
Remember that the x in the integral is a dummy variable and disappears during integration. Time shows up in the integral’s upper limit, so we can’t also legally use t as the variable of integration. |
2. Inductor
Recall that “inductor’s law” is
or
Remember that the \(t=0\) in the lower limit and the initial condition is just a placeholder that sets the boundary between “then” and “for any time afterwards”. The initial condition doesn’t need to be at 0 seconds. |
3. Single time-constant solution
a.k.a. a first-order circuit
Single time constant refers to the τ in the \(\exp(-t/\tau)\) term. First-order refers to the circuit solution being a first order differential equation. It is always true that a circuit with N total inductors and capacitors yields an Nth-order differential equation.[1]
A circuit with a reactive element (i.e. a capacitor) has a circuit analysis solution that involves a differential equation. Since there is only one reactance, there is only one time constant. The solution to this first-order differential equation is an exponential: \(A \exp(-t/\tau) + B\)
The R in the τ=R·C is an effective resistance and may NOT be the value of a single physical resistor! Think "Thevenin-equivalent resistance seen by the capacitor" instead. |
The general solution from your differential equations course is correct, but you always then need to solve for the two constants using the initial conditions of the particular circuit. A more practical and easier form is:
If t is in seconds, then this forces τ to be in units of seconds because it makes no sense to raise a number to a power having units, for example “2.7 to the 3 meters power”.
Does it seem weird that resistance × capacitance equals time? See how the units work out once you expand them into their more basic units:
The ampere is an SI base unit while the coulomb is a derived unit (A·s). But it is common to think of the reverse: an ampere is a coulomb per second. |