This course explains how to analyze circuits that have alternating current (AC) voltage or current sources. Circuits with resistors, capacitors, and inductors are covered, both analytically and experimentally. Some practical applications in sensors are demonstrated.
2.6 Bode Plot of RC Circuits
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From the course by Georgia Institute of Technology
Linear Circuits 2: AC Analysis
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From the lesson
Module 2: Frequency Reponse
The description goes here
Meet the Instructors
Dr. Bonnie H. FerriProfessor
Electrical and Computer Engineering
Dr. Joyelle HarrisDirector of the Engineering for Social Innovation Center
School of Electrical and Computer Engineering
[SOUND] Welcome back to Linear Circuits.
This is Dr. Ferri, in this lesson we will be looking at Bode Plots of RC Circuits.
We'll be looking at two different circuits, this one and this one.
And, the difference is that where we're taking the output,
here it's across the resister, and here it's across the capacitor.
And it turns out it makes a big difference in what the Bode plot looks like
of that circuit.
We're going to be building upon the definition of Bode plots and
how we've introduced them in the last lesson.
We'll also be looking at transfer functions of RC circuits.
Now let's start with the Bode plot of the RC circuit,
where we take the output across the capacitor.
In our previous lessons, we've found this to be our transfer function.
And remember how we got that.
We looked at the impedance equivalent circuit of this and
then did a voltage divider law of this impedance over the sum of the two.
And if this is a transfer function, then I can find the magnitude here and
the angle right here.
All I need to do is then plot it on the Bode scale.
And there's some characteristic things that we want to look at
on RC circuits like this.
One is what happens at low frequency, what happens at high frequency?
And what do we determine as being the corner frequency of this?
We're going to look at the plot in a little more detail.
At low frequency, we've got zero decibels.
At high frequency, I've got what looks like a slope here.
And it is a slope with a value of minus 20 decibels per decade.
Remember a decade is the distance between a frequency and ten times its frequency.
So I go over one decade, I go down 20 decibels.
If I continue on here, it's like an asymptotic line right here.
If I continue on right here, I can match up where these two asymptotic lines
are and that's what's called the Corner Frequency.
I'll call it omega sub c.
And in the angle plot we have, we start out at zero degrees
and we end up at -90 at high frequency.
So to summarize this behavior, the magnitude
at low frequency is zero decibels, the angle's zero.
At high frequency the magnitude has a slope -20 dB per decade, angle of -90.
The corner frequency I've defined as a omega sub c according to the plot here.
Now this particular plot was drawn for particular values of RC, but it turns
out no matter what RC is, that corner frequency always occurs at one over RC.
We can also find the Bode plot from experimental data.
The key is to remember how the transfer function relates to input and
output magnitudes.
So the amplitude of the output signal divided by the amplitude of the input
signal, is the magnitude of the transfer function at that frequency.
And similarly, the phase angle is defined as the difference between
the input waveform and the output waveform.
Now let's look at some experiments relating that.
Here is an oscilloscope trace of an input and an output of an RC circuit.
The input is shown in green and the output is shown in blue.
For this particular frequency, the frequency is at 1,000 Hertz.
And you can see that the output amplitude
over the input amplitude is a ratio of 0.8.
So the transfer function at that frequency is 0.8,
magnitude of the transfer function.
If an instrument like a Bode
plot instrument, electronic instrument wants to find a Bode plot automatically,
what it does is it inputs a frequency like this.
Find its ratio of amplitudes,
finds the phase angle between them at that frequency, and then records it.
And then it changes the frequency, and does those recording again,
the amplitude ratio and the phase lag.
Let me show you an example of running a Bode Plot instrument.
So here's with the same circuit hooked up.
And I'm going to hit run on this and you'll see it building it up.
For each of those points, it's finding the amplitude ratio and
taking 20 times the log of it, and plotting it.
This is gain in decibels and down here is the phase angle in degrees.
Again, found numerically or found experimentally by looking at the angle
difference between the input signal and the output signal.
So this again is an automatic generation of the Bode Plot experimentally,
without even having to worry about the explicit formula for H of omega.
Now I want to take a look at our other circuit where I take the output across
this resistor.
Now this is a transfer function and
we can get the transfer function similar to what we did before.
We used the impedance method, where we use the voltage divider law.
So it's R over R plus this impedant, one over j omega c.
And that would be to find V out given Vs.
So this is a transfer function and if I clear my fractions I get this.
Now if I plot the magnitude versus frequency and
the angle versus frequency, I get these two Bode plots.
Similar to what we did before, I can- Find a corner frequency here and
I can look at what happens at low frequencies and
what happens at high frequencies.
So low frequencies, I get a slope of 20 decibels per decade.
Now this is a slope upwards, so it's plus 20 dB per decade and
I go from 90 to zero degrees as I go from low frequency to high frequency.
And at high frequency in terms of the magnitude, I go to 0 dB.
So to summarize, low frequency magnitude has this slope, angle of 90.
High frequency has a magnitude of 0 dB, angle of zero degrees.
And the corner frequency is defined the same way as it was before, 1/RC.
To summarize, we've looked at an RC circuit in two different configurations.
The only difference between them is where we take the output.
Here, it's across C capacitor, and here it's across the resister.
And notice that we get very different Bode Plots between those two configurations.
In this particular lesson, we looked at high frequency and
low frequency characteristics of these Bode Plots.
So for example, in this particular configuration we saw
that the high frequency slope was minus 20 dB per decade.
And we found that this was a corner frequency.
Well this formula for corner frequency holds for
this particular configuration as well.
But in this case,
we have a slope leading upwards of 20 dB per decade at low frequencies.
And then this being the corner frequency.
And in both cases we're going between a zero and 90.
Here we're going zero to actually minus 90, and
here we're going from plus 90 to zero.
All right, thank you very much.
[SOUND]
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