# 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.{width="230"}
!Resistor voltage
step-response.{width="230"}

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

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

$$g(t) = e^{number * t}$$ The reason is: looking at a number and taking
the inverse of it is hard. This makes it hard to look at an equation and
\"see\" it\'s time constant.

The units of any time constant are seconds. In circuits with both
inductors and capacitors, there are two time constants that are called
\"zeros\". They are called zero\'s because they are solutions to the
particular solution which sets the differential to zero. They are given
the symbol \"s\" and are identical to the complex frequency concept.

