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Resistors with Inductors and Capacitors 

In our previous page on the ideal circuit with inductance and capacitance, we deliberately assumed no circuit resistance. This is not a reasonable assumption in real life; there is always some resistance associated with any circuit, and especially with an inductance. Therefore, we will consider here what effects that resistance might have, and how it can change the behavior of the ideal LC circuit.
Our example circuit is shown to the right. Since the resistance is largely inherent in the inductor, we show it in series with the inductance. Of course, we can add more resistance to the circuit. This can have interesting effects on circuit behavior, as we will see shortly.
For the numerical calculations shown below, we assign reasonable component values as follows:
As before, we start with a charged capacitor, which we connect to the resistor and inductor with switch S, at time t = 0. We'll assume no charge leakage from the capacitor before time 0. So what happens in this circuit when the switch is closed?
The figure to the right shows the capacitor voltage of this circuit, for small values of R. The red curve shows the capacitor voltage over time, beginning at the moment switch S is closed. We still get oscillations, but R removes energy from the circuit as current flows through it, so the oscillations gradually decrease in amplitude, according to the exponential decay curves shown in blue.
R has one more effect on this circuit: it actually slows down the oscillations, reducing the sine wave frequency as well as its amplitude. For small values of R, the reduction isn't very much, but it must be recognized and accounted for. Thus, we must account for the effect of R in two different ways.
First, of course, we must account for the amplitude decay. This takes the usual form of:
Here, τ is the lowercase Greek letter tau, and is used to represent the decay time constant in this type of equation. For this circuit,
This is the curve plotted in blue on the graph, and mirrored on the negative half to clearly show the effect.
Without R in the circuit, the frequency of oscillation would be set by L and C, and would be:
However, with R in the circuit, this is no longer the case. With the L and C component values given above, the frequency of the oscillations should be about 1591 Hz. But the actual frequency is about 1583 Hz — a small amount lower. The difference is controlled by the value of τ. The basic equation describing this is:
In this equation, ω is the lowercase Greek letter omega, and is used to represent angular frequency in radians per second. To convert to cycles per second (Hz), we use the relationship ω = 2πf. ω_{osc} is the actual frequency of oscillation, while ω_{o} is the original frequency determined solely by L and C. α is the lowercase Greek letter alpha, and is equal to 1/τ.
As a result, the ratio R/2L affects both the oscillating frequency of the circuit and the amplitude decay of those oscillations. The presence of R in the circuit has a damping effect on this circuit's oscillations. This is a very important concept in electronics.
Now, what happens as R is increased in value? Clearly, the ratio R/2L (α in the equations above) will increase. This will increase the damping effect, making it more pronounced. When R increases enough so that α = ω_{o} above, the actual frequency of oscillation, ω_{osc}, will be zero. This is in fact the case, and for the values of L and C given at the top of this page, R = 2K will accomplish this condition. With this value of R, the circuit is said to be critically damped. With smaller R values, it is underdamped. With larger R values, it is overdamped.
It is interesting to note that an overdamped circuit, while it doesn't oscillate (frequency would be a complex number), it takes longer to dissipate the initial energy than a critically damped circuit. This is very important in some circuits. So much so that we use a damping factor, represented by the lowercase Greek letter zeta (ζ = α/ω_{o}). Sometimes we use the quality factor, or Q, of a circuit instead, where Q = 1/2ζ. There are three possible conditions to consider:
An underdamped circuit can allow the oscillations (sometimes called ringing) to occur. This can mean trouble under certain conditions, so we must be able to determine and adjust the damping factor of any circuit that contains both inductance and capacitance. In other situations, we will want this phenomenon to occur. But we will still want to be able to predict and control it.


 
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