RLC Circuits

An RLC circuit contains resistors, inductors and capacitors. In this section we shall look at the two simplest RLC circuits, the series RLC circuit and the parallel RLC circuit.

Series RLC Circuits

Since the resistor inductor and capacitor are in series, the total impedance of the series RLC circuit is the sum of the individual impedances:

Z = R +X_L + X_C

After substitution of the formulas for X_L and X_C one gets the following expression for Z in terms of f, R, L and C:

This can be simplified by combining the imaginary terms:

This expression in more complex than those derived in earlier sections for the series RL and RC circuits and as a result, the behavior of a series RLC circuit is much more interesting. Let us consider several interesting cases:

Case 1: f <<1.

If the frequency is very close to 0 Hz, the j2pfL term in the original expression for Z is approximately 0. In this case the impedance of the parallel RLC circuit can be approximated by the following equation

At very low frequencies, the series RLC circuit behaves like a series RC circuit.

Case 2: f >> 1

When the frequency is very large, the -j/(2pfC) term in the original expression for Z is approximately 0. In this case the impedance of the series RLC circuit can be approximated by the following equation

At very high frequencies, the series RLC circuit behaves like a series RL circuit.  

Case 3: 4p²f²LC = 1

When this condition occurs, the imaginary part of Z is zero. The impedance of the series RLC circuit is real and is equal to the resistance:

Z = R

The circuit behaves as if the inductor and capacitor were not present. The negative reactance of the capacitor and positive reactance of the inductor add up to 0, creating a condition is known as resonance, which occurs at a specific frequency called the resonant frequency. It is possible to derive an equation for the resonant frequency in terms of L and C from the condition 4p²f²LC = 1:

where:

    f_R = resonant frequency (Hz)

    L = inductance (H)

    C = capacitance (F)

The resonant frequency depends on the product of L and C, not the individual values, and is independent of R. We can use the formula for the resonant frequency to simplify our equation for the impedance of the series RLC circuit as follows:

 Let us examine the behavior of a series RLC circuit in more detail by studying the circuit shown below:

.

We can immediately calculate the resonant frequency:

We can substitute in values for f_R, R, and L to get the total impedance:

The graph below shows how Z varies with frequency:

We can find the current through the series RLC circuit by using Ohm's Law:

rather than simplify this ugly expression further, we will graph the current as a function of frequency:

The magnitude of the current is maximum at the resonant frequency and |I_max |= V_0/R = 12/50 = 240 mA. This type of circuit behavior is called a band-pass response. If a complex signal containing many different frequencies was applied to the series RLC circuit and the output taken across R, the output would contain mostly frequencies near the resonant frequency. 

The next graph shows the phase of the current through the series resonant circuit.

At frequencies below the resonant frequency, the current leads the voltage, which is characteristic of an RC circuit. At frequencies above resonance, the current lags the voltage, and the series RLC circuit looks like a series RL circuit. At resonance, the current and voltage are in phase.

Although there are many combinations of L and C that will give the same resonant frequency, the shape of the curve representing V_R depends on the L/C ratio as shown below.

As the L/C ratio increases, the response becomes sharper.

The value of series resistance, R, also affects the shape of the V_R curve as shown below:

As the series resistance gets larger, the voltage curve gets broader. 

In order to make sense of all these possibilities, it is necessary to introduce some new parameters. This first is Q, the quality factor of the series RLC circuit. Q is defined as:

Another way to look at this is to say that the Q is the ratio of the reactance of the inductor to the resistance, at resonance. High Q RLC circuits have a very sharp response. Low Q RLC circuits have a broad response.

We have talked about the bandwidth of a circuit without defining exactly what it is. Now we will define it.

Referring to the figure above, bandwidth (BW) is defined as the difference in frequency between the upper frequency at which the voltage drops to 70.7% of the maximum and the lower frequency at which the voltage drops to 70.7% of the maximum. The 70.7% voltage points were chosen because they correspond to a power decrease of 50%. Thus:

BW = f_U - f_L

Bandwidth and Q are also related as follows:

In the graph above, Q = 17.5, and f_R = 35 KHz. The bandwidth is:

.

The voltage response of the series RLC circuit can be completely characterized by its resonant frequency, f_R, and Q or the bandwidth, BW.

Parallel RLC Circuits

A parallel RLC circuit, also called a tank circuit,  is one in which R, L, and C are in parallel, as shown below.

Much of what we just learned about series RLC circuits will carry over into parallel RLC circuits. Before we begin we need to define a new concept known as admittance. Admittance is the reciprocal of impedance and its symbol is Y:

The unit of admittance is the Siemens (S). 1 Siemens = 1 ohm ^-1 . Like impedance, admittance is a complex number, consisting of a real part and an imaginary part. The real part is called conductance, G,  and the imaginary part is called susceptance, B:

Y = G+jB

Admittance is very handy when analyzing parallel circuits because the total admittance of a group of components in parallel is simply the sum of their individual admittances. Our formulas for inductive and capacitive reactance can be converted to formulas for susceptance:

Inductive susceptance:   

Capacitive susceptance:

And of course, conductance is defined as follows:

By analyzing the parallel RLC circuit using admittance, we will be able to use nearly all of what we have already learned about series RLC circuits. Admittances in parallel add, just like impedances in series. We can immediately write the equation for the admittance of the parallel RLC circuit by adding the admittances of the three components:

This can be simplified to the following expression:

The imaginary part of the admittance vanishes when:

We define the frequency at which this occurs to be the resonant frequency, given by the following formula:

The equation for the admittance of the parallel RLC circuit can be simplified further:

Let us look at a specific parallel RLC circuit and explore its behavior.

The graph below shows the variation of the admittance of this parallel RLC circuit with frequency.

We can compute the resonant frequency from the component values:

Note that this parallel RLC circuit has the same resonant frequency as the series RLC circuit examined earlier. In general, the resonant frequency depends only on L and C and not whether they are connected in parallel or series.

The next graph shows the variation in the current through the parallel RLC circuit. Notice that the current is a minimum at the resonant frequency. This is an example of a "band-stop" circuit response. A parallel RLC circuit can be used to block frequencies near the resonant frequency, while allowing others to pass. 

The graph below shows the variation in phase shift of the current through the parallel RLC circuit.

Notice that the current has no phase shift at the resonant frequency. At frequencies below the resonant frequency the current lags the applied voltage and the circuit acts like an inductor. At frequencies above resonance, the current leads the applied voltage and the circuit acts like a capacitor.

The shape of the band-stop response of a parallel RLC circuit depends on the value of R and L/C as shown in the next graphs.

Note that the band-stop characteristic becomes narrower as the value of R increases. This the opposite of what we saw in the series RLC circuit.  The response also becomes narrower as the L/C ratio decreases, which again is the opposite of what we saw in the series RLC circuit. We can summarize this behavior by using two additional parameters, Q and the bandwidth, BW. We define bandwidth for parallel RLC circuits, as the difference in frequency between the upper frequency at which the current increases to 141.4% of the maximum and the lower frequency at which the current increases to 141.4% of the minimum. The 141.4% current points were chosen because they correspond to a doubling of the power.

BW = f_U - f_L

Q is defined as:

Bandwidth and Q are also related as follows:

The graph below illustrates the definition of BW.

The bandwidth is 35.8 - 34.3 = 1.5 KHz. Now we can determine Q:

As a check, we can use our other definition to determine Q:

As one would expect, both results agree.

Let us summarize what we have learned about parallel and series RLC circuits: