Analyzing RLC Circuits

NOTE: This is new material, and there may quite possibly be a few errors which I did not catch. Please report any errors as soon as possible.

The following is a step by step procedure which may be used to determine all voltages and currents in both Parallel and Series RLC circuits. The most significant differences between the parallel circuit and the series circuit are the calculation of the neper frequency, the parameter common to all elements, and the means of calculating the derivative of that common element in order to determine the final constants.

NOTE 1: I do not follow the book's technique for solving RLC circuits, since I believe my method is conceptually simpler, although it may require more steps in some cases. Specifically, I suggest that you always solve for the parameter (voltage or current) that is COMMON to both the inductor and capacitor first, regardless of whether or not that is one of the parameters you are trying to find.

NOTE 2: Unfortunately, I see no way to condense this synopsis into a really compact form (compare the synopsis of basic circuit analysis techniques) and still maintain sufficient detail and clarity. If you have a grasp of the details and simply need a reminder of the basic outline, see the Brief Synopsis of RLC circuits. (NOT YET IMPLEMENTED)

NOTE 3: There are still a few problems with alignment of text and equations (e.g. the integral equations near the end of the page several screens down). It looks bad, but should not be an impediment to understanding. I'll get it fixed one of these centuries.

Step 0 involves the circuit PRIOR TO ANY CHANGES THAT OCCUR.

Step 0: If not specified, determine the initial capacitor voltage V_C(0^-) and initial inductor current I_L(0^-) using circuit analysis.
These are the two parameters that cannot change instantly, thus

V_C(0^-) = V_C(0⁺)

and

I_L(0^-) = I_L(0⁺)

In order to do this analysis, (assuming the circuit has been connected for a long time, i.e. many time constants, prior to any changes that occur), replace the capacitor with an open circuit and the inductor with a short circuit (a wire) and use any circuit analysis techniques which are appropriate to determine the voltage across the capacitor (the open circuit that replaced it) and the current through the inductor (the wire that replaced it).

The rest of the steps involve the circuit AFTER ANY CHANGES THAT OCCUR.

Step 1: Determine whether the circuit is a parallel RLC or a series RLC (after any switches move or other changes occur).
NOTE: Prior to the time the changes occur, the capacitor and inductor may be connected (or not connected) in an entirely different fashion. This step applies to the circuit only AFTER the changes occur (switches move, sources turn on or off, or whatever).

The circuit is a parallel RLC circuit if the capacitor and inductor are in parallel, regardless of how other components are connected thereto.
The circuit is a series RLC circuit if the capacitor and inductor are in series, regardless of how other components are connected thereto.

Step 2: Determine the equivalent resistance of the circuit connected to the capacitor and inductor. This may be as simple as a single resistor, or something considerably more complicated, such as a circuit containing one or more dependent sources. Depending on the circuit, you might be able to use circuit reduction techniques, or a more complex analysis may be required. For more information on determining the equivalent resistance of a circuit, see the handout on Thevenin and Norton Equivalents, remembering that you only need the equivalent resistance, not the entire Thevenin equivalent circuit.

Step 3: Calculate the neper frequency (a) and the resonant frequency w_o.
(Note that for the calculation of a, the resistance R is the equivalent resistance found in Step 2.)

w_o is the same for both parallel and series circuits:
For the PARALLEL circuit, a = 1 / 2RC
For the SERIES circuit, a = R / 2L

Step 4: Using a and w_o, determine whether the circuit is overdamped, underdamped, or critically damped, and calculate any other parameters required for each case.

a > w_o: Overdamped
Calculate:
a = w_o: Critically damped
No further parameters necessary in this step
a < w_o: Underdamped
Calculate: (damped frequency)

Step 5: Determine the response for the parameter that all three elements have in common. Note that the final value for the common parameter will always be zero.

CASE 1: PARALLEL RLC - find V_C(t) first.

In order to complete this step, you will need V_C(0⁺) from Step 0 above, and the initial current I_C(0⁺) through the capacitor at t = 0+. Finding this current may require a little bit of work. The circuit below shows various parameters associated with the parallel circuit at t= 0+. We already know two of the four parameters shown from Step 0 (Shown in Blue). We need to find the initial capacitor current (shown in Green). If we could find the current into the rest of the circuit at t = 0+ (shown in Red), KCL would then give us the initial capacitor current.

If the circuit connected to the LC combination is fairly simple (e.g. a single resistor or a resistor and an independent source), you may be able to calculate the current I_R(0⁺) without much effort since you know the voltage [V_C(0⁺)] across the circuit . (See solutions to problems 8.6 (part b) and 8.7 (part a) for examples.)

If the circuit connected to the LC combination is more complicated, replace the capacitor with a voltage source whose value equals V_C(0⁺) and replace the inductor with a current source whose value is I_L(0⁺). (See circuit below.)

Analyze the resulting circuit by any technique that is appropriate to determine I_R(0⁺), then use KCL to determine I_C(0⁺). (See the solution to problem 8.15 for an example.)

Now that we have the initial capacitor current, the rest is purely cookbook. There are the three forms of damping, each of which is a bit different.

Overdamped:
The form of the solution is

where
V_C(0⁺) = A₁ + A₂ and
I_C(0⁺) / C = s₁A₁ + s₂A₂ (The two s values were obtained in Step 4.)
Solving the above two equations gives the values for A₁ and A₂, and substituting the s and A values into the solution form above gives V_C(t).

Critically Damped:
The form of the solution is

where
V_C(0⁺) = D₂ and
I_C(0⁺) / C = D₁ - a D₂ (a was obtained in Step 3.)
Solving the above two equations gives the values for D₁ and D₂, and substituting the a and D values into the solution form above gives V_C(t).

Underdamped:
The form of the solution is

where
V_C(0⁺) = B₁ and
I_C(0⁺) / C = - a B₁ + w_d B₂ (The two s values were obtained in Step 4.)
Solving the above two equations gives the values for B₁ and B₂, and substituting a and w_d along with these values into the solution form above gives V_C(t).

CASE 2: Series RLC - find I_L(t) first.

In order to complete this step, you will need I_L(0⁺) from Step 0 above, and the initial voltage V_L(0⁺) across the inductor at t = 0+.

NOTE: The general method for finding the initial inductor voltage is conceptually very similar to finding the initial capacitor current in the parallel circuit which was explained in some detail above. For the moment, further elucidation will have to wait since I need to get this posted before the test. If I manage to update it prior to the test, I will inform the class via email.

Now that we have the initial inductor voltage, the rest is purely cookbook. There are the three forms of damping, each of which is a bit different.

Overdamped:
The form of the solution is

where
I_L(0⁺) = A₁ + A₂ and
V_L(0⁺) / L = s₁A₁ + s₂A₂ (The two s values were obtained in Step 4.)
Solving the above two equations gives the values for A₁ and A₂, and substituting the s and A values into the solution form above gives I_L(t).

Critically Damped:
The form of the solution is

where
I_L(0⁺) = D₂ and
V_L(0⁺) / L = D₁ - a D₂ (a was obtained in Step 3.)
Solving the above two equations gives the values for D₁ and D₂, and substituting the a and D values into the solution form above gives I_L(t).

Underdamped:
The form of the solution is

where
I_L(0⁺) = B₁ and
V_L(0⁺) / L = - a B₁ + w_d B₂ (The two s values were obtained in Step 4.)
Solving the above two equations gives the values for B₁ and B₂, and substituting a and w_d along with these values into the solution form above gives I_L(t).

NOTE: The explanation of Step 6 is rather abbreviated due to time constraints. I will inform the class when it is updated.

Step 6: Determine the remaining parameters (voltages and currents) of the circuit as desired using Ohm's law, Kirchoff's Laws, and the V-I relationships for inductors and capacitors. In the simplest case (the circuit attached to the LC combination is a single resistor) proceed as follows:

Parallel:

The voltage across all three elements is the same.
The current through the capacitor and be found from I_C(t) = C dv/dt since we know the capacitor voltage from Step 5.
The current through the resistor can be found using Ohm's Law.
The current through the inductor can now be found from KCL, or from .

Series:

The current through all three elements is the same.
The voltage across the inductor and be found from V_L(t) = L di/dt since we know the inductor current from Step 5.
The voltage across the resistor can be found using Ohm's Law.
The voltage across the capacitor can now be found from KVL, or from .

For slightly more complicated circuits (e.g. two or three resistors, or one resistor and a source, the solution can be accomplished almost as easily in most cases.

For circuits too complicated for this approach, replace the capacitor with a voltage source whose value is V_C(t) and replace the inductor with a current source whose value is I_L(t).

For the parallel circuit, V_C(t) is from Step 5 and

. (Note that V_L(t) = V_C(t) since parallel.)

For the series circuit, I_L(t) is from Step 5 and

. (Note that I_C(t) = I_L(t) since series.)

There is actually a shorter method for the more complicated circuits, but I'm out of time, so further explanation will have to be postponed.

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