# Circuit Theory/Time Constants

Homogeneous equations are exponential. A homogeneous differential equation is discharging or charging (0 value constant DC source). A non-homogeneous differential equation has a sinusoidal source. It can be solved by splitting into a homogeneous solution plus a particular solution. The steady state solutions earlier were particular solutions.

homogeneous equation ⇒ discharging (DC source) ⇒ homogeneous solution

non-homogeneous equation ⇒ AC source ⇒ particular (steady state) + homogeneous (discharging) solution

The proof of the exponential solution to the the discharging circuit is is hard:

• guess solution
• see if it is possible
• if possible assume it is the solution

In this case it works because there is only one guess:

Capacitor voltage step-response.
Resistor voltage step-response.

that we know of.

A first order differential looks like this:

$g(t) +\tau \frac{d g(t)}{dt} = 0$

A solution is:

$g(t) = e^{- \frac{t}{\tau}}$ (Gödel proved that there are always other truths possible. We can not be certain this is the only solution.)

Tau, τ, or $\tau$ has a name:

$\tau$ = time constant

The Steady State(particular) definition is:

$t \ge 5*\tau$

since for charging and discharging:

$1-e^{-5} = .9933, e^{-5} = 0.0063$

Try to write all answers in the form of

$g(t) = e^{- \frac{t}{\tau}}$

$g(t) = e^{number * t}$