Frequency Dependent Circuits

Frequency Dependent Circuits

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.

• A frequency response for the unit.

• If the unit is a speaker set, you'll find separate frequency responses for the different speakers like the mid-range, or the woofer and the tweeter.

• If the unit is a microphone, you'll find a frequency response that tells you how the unit responds to different frequencies.

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.

• Given a linear system or circuit described mathematically,

• Be able to compute a frequency response for the system.

• Be able to predict an output signal from a given input sinusoidal signal.

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 can write the differential equation relating the input and output voltages and solve for the output assuming a sinusoidal signal input.

• We can assume a sinusoidal input and use LaPlace transform methods to compute the output voltage.

• Since the input is a sinusoid, we know that the output contains a sinusoid and terms that decay to zero. We can work from there.

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:

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

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

• If the output voltage is given by

• Then, since the output voltage is across a capacitor, we can compute the current flowing through R and C as:

• And then we can compute the voltage across the resistor, R, as:

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

• The input voltage is given by,

• Or:

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

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

• We know: