Solution to the equation and transient behaviour

Solving the differential equation

The equation τ*(duc/dt) + uc = E has as its general solution the sum of a particular solution (the steady state, uc = E) and a homogeneous solution in the form of a decaying exponential: uc(t) = E + A*exp(-t/tau)

The initial condition uc(0) = 0 gives A = -E, hence the complete solution: uc(t) = E*(1 - exp(-t/tau))

The current is given by i(t) = C*(duc/dt) = (E/R)*exp(-t/tau).

Physical interpretation

• At t = 0: uc = 0, i = E/R (maximum).
• As t approaches +∞: uc approaches E, i approaches 0: the capacitor is fully charged and behaves like an open switch in steady-state continuous operation.
• At t = tau: uc(tau) = E*(1 - exp(-1)) ≈ 0.63*E, i.e. approximately 63% of the final charge.

Common reference points

Elapsed time	Value of uc	% of charge
t = tau	E*(1-e^-1)	~63%
t = 3*tau	E*(1-e^-3)	~95%
t = 5*tau	E*(1-e^-5)	~99%

In practice, the capacitor is considered to be fully charged after 5*tau.

Common pitfall

Be careful when using the method of the tangent at the origin to determine tau graphically: the tangent to the curve uc(t) at t=0 intersects the horizontal asymptote uc=E exactly at t = tau. A common mistake is to read tau at the point where uc reaches E/2 (which would give a half-charge time different from tau).