# 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 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:

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

A solution is:

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

Tau, τ, or ${\displaystyle \tau }$ has a name:

${\displaystyle \tau }$ = time constant

${\displaystyle t\geq 5*\tau }$

since for charging and discharging:

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

Try to write all answers in the form of

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

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