Frequency Dependent Circuits
Why is Frequency Response Important?
An Example Circuit
Using Frequency Response Concepts
A Frequency Response Lab Problem
Why Worry About Frequency Response?

        Did you ever buy audio equipment and look carefully at how the manufacturer specified how well the equipment would work? (And audio equipment is one of the few consumer items where people actually try to sell things on the basis of how well they work!) If you looked at the specifications for audio equipment you would probably find the following.

Frequency response is an important concept in many areas - within electrical engineering and outside of electrical engineering. Having a good grasp of frequency response is important in many areas, so our objectives in this lesson include the following.
An Example Circuit

        We are going to examine a simple circuit that has frequency dependent behavior, a resistor-capacitor (RC) circuit. It is shown below. To illustrate how this circuit responds to a sinusoidal signal input we can do any of the following.

        We will use the third approach - and we will assume a steady state output and work backwards from the output to compute the input.

        Since the first thing we want to do is just to look at how a circuit can affect sinsusoidal signals, we're going to assume a sinusoidal output and work backwards to calculate the input voltage that produces that output. That's not a very general approach, but it will get us what we want now, and prepare us for other things to come.  We will be able to do that without too much algebraic pain, and we can learn some things from the result.

        So, we will assume that the output voltage is given by:

vout(t) = B sin(wt)

        Be sure that you understand that B is the magnitudeof the output signal

        Now, what does that form for the output voltage imply?

vout(t) = B sin(wt)
i(t) = Cdvout/dt = CBw cos(wt)
vR(t) = Ri(t) = RCB w cos(wt)

Now, we can apply KVL to get the input voltage.

vin(t) = vR(t) +vout(t)
vin(t) = B(RCw cos(wt) + sin(wt))

        At this point step back from this. It may not be obvious, but we can take advantage of a trigonometric identity,

sin(x+y) = sin(x)cos(y) + cos(x)sin(y)

if only we can make the things that multiply the sines and cosines in the second bullet above look like other sines and cosines.

sin(x+y) = sin(x)cos(y) + cos(x)sin(y)
vin(t) = B(RCw cos(wt) + sin(wt))

We need to refer to a little geometrical construction - at the right.  "Clearly" we have the relationships indicated below for cos(f) and sin(f)

So, now we can write:

Which reduces to:

There are two conclusions to draw from the resulting expression for the input voltage.

        You can use the expressions for the gain, B/A, and the phase shift, f, to predict behvarior of circuits like this.  You should note the following in these expressions.

With those thoughts you can think a little more deeply using the simulator we have just below.
A Simulation of the Circuit

Note: - This simulator is real time.  However, to let you see how the circuit behaves, we have made the signals very slow - on the order of a few Hertz, or even a fraction of a Hertz.  The time constant (the R-C product) should be correspondingly long - on the order of a second (from a fraction of a second to a few seconds).  You won't see much if you stray far from these limits - even though these are long time constants and the bandwidths are quite low.  That's just for purposes of illustration.  (However, note that you could get a one second time constant using R = 1.0 MW, and C = 1.0mf.)

        Here is the simulator.

Using this simulator, you can do the following.

Problems & Questions

P1.   Here is an RC filter circuit - the same one discussed above.

In this circuit, the parameters are: Determine the bandwidth of the circuit.

Enter your answer in the box below, then click the button to submit your answer.  You will get a grade on a 0 (completely wrong) to 100 (perfectly accurate answer) scale.

Your grade is:

P2.   In the RC filter circuit determine the resistance value that gives a bandwidth ten times as large as in problem 1.

Your grade is:

P3.   In the RC filter circuit , the parameters are: Determine the phase shift between output and input at 200Hz.  Remember the sign and give your answer in degrees (not radians).

Your grade is:


P4.   Assume that you have an RC filter circuit with a 0.5 second time constant.  Determine the phase shift between output and input at 1.0Hz.  Remember the sign and give your answer in degrees (not radians).

NOTE:  This problem has a long time constant and a low frequency so that you can use a real time simulator to check your answer before you submit it. Click here to get the simulator in a separate window.

Your grade is:


P5.   In an RC filter circuit , the parameters are: Determine the frequency (in Hertz) for which the attenutation is 0.707.  In other words, the output is reduced by 29.3%.

Your grade is:

Reflections

        In this lesson you have been introduced to a simple frequency dependent circuit.  We have used a very brute-force method - assuming a voltage at the output and chasing that back to the input.  That worked here, but it won't work everywhere.  Moreover, this is not the easiest way to make such predictions, and before you attempt more complex circuits you should learn a better way to analyze frequency dependent circuits.  You need to learn about impedance and phasors.  There are links below that will take you to many other topics.


Problem and Labs
Links to Other Lessons on Frequency Topics Send your comments on these lessons.