The math in this article is from Theory and Calculation of Transient Electric Phenomena and Oscillations, written by Charles Steinmetz in 1909.
Consider the following circuit: a battery connected to a resistor, capacitor, inductor, and a switch.
This is an idealized abstraction of any electric circuit; all circuits will contain some level of resistance, inductance, and capacitance. In fact, even individual electrical components will contain all three properties; one property is dominant. For example, all inductors contain some resistance and capacitance, but the inductance is the dominant property, because its ability to store magnetism is enhanced by the geometry of the conductor (a coil).
What happens when the switch closes?
At steady-state, the electrostatic potential (dielectric field strength) between the plates of the capacitor will be equal to the voltage of the battery (12 volts in this case), and there will be no current.
Before this steady state is reached, the circuit has to adjust to the change of conditions (switch closed, voltage applied).
The Steinmetz equations can be used to calculate the voltage and current as functions of time after the switch is closed:
is the quantity of magnetism stored by the inductor, is the voltage across the inductor, is the inductance, is the quantity of dielectricity stored by the capacitor, is the voltage across the capacitor, is the capacitance, and is the current.
The action of the resistor is governed by Ohm's Law:
Notice that a subscript has been given to each voltage, to show that these are referring to the same electromotive force. (Electromotive force, or e.m.f., is another term for, and interchangeable with, voltage.)
The total voltage is the e.m.f. consumed by resistance plus the e.m.f. consumed by the reactance of the inductor plus the voltage consumed by the capacitance of the capacitor. This is Kirchhoff's Law: the sum of the voltages around a circuit must equal zero.
where is the battery voltage.
Let's write this in terms of only the current, i, and the circuit constants.
First, use the Steinmetz equations to solve for the voltage consumed by the reactance of the inductor:
Next, solve for the voltage across the capacitor in terms of current:
Plug these in:
Take the derivative of both sides:
is a constant 12 V, so .
This is a homogeneous linear differential equation. Therefore, i is an exponential function of the form , where e is Euler's number.
If , and .
Plugging these into our equation:
Since this must be true for all values of t,
Solve with the quadratic formula:
There are two roots:
The general solution for i:
Solve for the voltage across the capacitor terminals over time:
If for brevity:
Solve for the constants and . At ,
and
So
and
Substituting those constants back in gives the final equations for the current and capacitor potential as functions of time:
However, these equations are only practical when . When , then is negative, and s is imaginary.
Use since i is already used for current.
Substitute , so that
and
Now
Euler's formula:
With and ,
and
Now
and
so
Finally:
and
Example Problems
The Impulse Case
Suppose the battery is 12 volts, the resistance of the resistor is 4 ohms, the inductance of the inductor is 1 henry, and the capacitance of the capacitor is 1 farad.
Set to the moment the switch closes, so and .
Plug in these values:
so
and
Now we can graph the current over time:
and the voltage across the capacitor over time:
The Oscillating Case
Suppose the battery is 12 volts, the resistance of the resistor is 1 ohm, the inductance of the inductor is 1 henry, and the capacitance of the capacitor is 1 farad.
Set to the moment the switch closes, so and .
Plug in these values:
so
and
Now we can graph the current over time:
and the voltage across the capacitor over time: