# Circuit Theory/Initial Conditions

## Both Voltage and Current

Initial conditions describe the energy stored in every capacitor and
every inductor.

Initial conditions are completely specified only when both voltage and
current for all capacitors and all inductors is known.

## Assume Zero

Assume zero for the capacitor voltage and/or inductor current if no
information is given.

## Inductor Initial Conditions

Every inductor has two initial conditions: current and voltage. When a
switch is thrown that eliminates all power supplies, (or connects new
power supplies) the inductor can turn into a power supply itself. The
current through an inductor maintains it\'s direction and magnitude
between $t(0_-)$ and $t(0_+)$. The voltage may instantaneously switch
polarity and/or magnitude.

Energy changes elsewhere in a circuit cause an inductor to block these
changes by appearing as an open. An Inductors will allow any voltage to
appear across it\'s terminals in order to maintain the energy in its
magnetic field. An inductor doesn\'t want the current to increase or
decrease. Any current change will change the energy stored in its
magnetic field.

But gradually, an inductor will accept change. And after a very long
time it acts like a wire or short allowing current only limited by other
devices in the circuit to pass through it.

## Capacitor Initial Conditions

Every capacitor has two initial conditions: voltage and current. When a
switch is thrown that eliminates all power supplies, (or adds power) the
capacitor can turn into a power supply itself. A capacitor maintains
it\'s voltage polarity and value between $t(0_-)$ and $t(0_+)$, but the
current may instantaneously switch directions and may switch magnitude.

A capacitor reacts to energy changes by instantly turning itself into a
short or an ideal wire with no resistance. A capacitor doesn\'t care
about the current flowing through it. A capacitor doesn\'t want the
energy stored in the electric field between its plates to change. So the
voltage is maintained, but the current can switch direction and
magnitude. The current will only be limited by other devices in the
circuit.

But after a very long time, a capacitor gradually adapts to change and
eventually develops a voltage that the rest of the circuit doesn\'t try
to change. At this point the capacitor turns into an open.

Inductors and capacitors both react to energy changes, but in exactly
the opposite ways.

## Particular Solution

The particular solution in response to a step function turns into a set
of conditions that have to be present at t = ∞.

These conditions should be true at all times .. and can be used to help
evaluate constants.

## Progression

Start with an expression with a voltage across the capacitor. This way
the progression through all possible initial conditions requires taking
derivatives:

$$H(s)=\frac{\mathbb{V}_C}{\mathbb{V}_s} \Rightarrow V_C(t) \Rightarrow i(t) = C{d V_C(t) \over dt} \Rightarrow V_L = L{d i(t) \over dt}$$
Starting at V_L means evaluating integrals and their constants:

$$H(s)=\frac{\mathbb{V}_L}{\mathbb{V}_s} \Rightarrow V_L(t) \Rightarrow i(t) = \frac{1}{L}\int V_L dt \Rightarrow V_C = \frac{1}{C}\int i(t) dt$$
Starting at the current means that may have to do both a derivative and
an integral to evaluate all constants.

This detail is displayed by doing one example
problem
four different ways.

## Summary

  t=0~-~ t=0~+~             source                   energy imbalance response   value                               starting point
  ------------------------- ------------------------ --------------------------- ----------------------------------- -----------------------------------------
  V~c~(0~-~) = V~c~(0~+~)   capacitor itself         short                       problem statement/assume 0          particular solution to previous problem
  I~c~(0~-~) ≠ I~c~(0~+~)   circuit cap is in        doesn\'t care               determined by rest of circuit       capacitor terminal relation, node, mesh
  I~L~(0~-~) = I~L~(0~+~)   inductor itself          open                        problem statement/assume 0          particular solution to previous problem
  V~L~(0~-~) ≠ V~L~(0~+~)   circuit inductor is in   doesn\'t care               determined by the rest of circuit   inductor terminal relation, node, mesh

