0:30

We're going to be building upon the definition of Bode plots and

Â how we've introduced them in the last lesson.

Â We'll also be looking at transfer functions of RC circuits.

Â Now let's start with the Bode plot of the RC circuit,

Â where we take the output across the capacitor.

Â In our previous lessons, we've found this to be our transfer function.

Â And remember how we got that.

Â We looked at the impedance equivalent circuit of this and

Â then did a voltage divider law of this impedance over the sum of the two.

Â And if this is a transfer function, then I can find the magnitude here and

Â the angle right here.

Â 1:09

All I need to do is then plot it on the Bode scale.

Â And there's some characteristic things that we want to look at

Â on RC circuits like this.

Â One is what happens at low frequency, what happens at high frequency?

Â And what do we determine as being the corner frequency of this?

Â We're going to look at the plot in a little more detail.

Â At low frequency, we've got zero decibels.

Â At high frequency, I've got what looks like a slope here.

Â And it is a slope with a value of minus 20 decibels per decade.

Â Remember a decade is the distance between a frequency and ten times its frequency.

Â So I go over one decade, I go down 20 decibels.

Â If I continue on here, it's like an asymptotic line right here.

Â 2:20

and we end up at -90 at high frequency.

Â So to summarize this behavior, the magnitude

Â at low frequency is zero decibels, the angle's zero.

Â At high frequency the magnitude has a slope -20 dB per decade, angle of -90.

Â The corner frequency I've defined as a omega sub c according to the plot here.

Â Now this particular plot was drawn for particular values of RC, but it turns

Â out no matter what RC is, that corner frequency always occurs at one over RC.

Â 2:54

We can also find the Bode plot from experimental data.

Â The key is to remember how the transfer function relates to input and

Â output magnitudes.

Â So the amplitude of the output signal divided by the amplitude of the input

Â signal, is the magnitude of the transfer function at that frequency.

Â And similarly, the phase angle is defined as the difference between

Â the input waveform and the output waveform.

Â Now let's look at some experiments relating that.

Â Here is an oscilloscope trace of an input and an output of an RC circuit.

Â The input is shown in green and the output is shown in blue.

Â For this particular frequency, the frequency is at 1,000 Hertz.

Â And you can see that the output amplitude

Â over the input amplitude is a ratio of 0.8.

Â So the transfer function at that frequency is 0.8,

Â magnitude of the transfer function.

Â If an instrument like a Bode

Â plot instrument, electronic instrument wants to find a Bode plot automatically,

Â what it does is it inputs a frequency like this.

Â Find its ratio of amplitudes,

Â finds the phase angle between them at that frequency, and then records it.

Â And then it changes the frequency, and does those recording again,

Â the amplitude ratio and the phase lag.

Â Let me show you an example of running a Bode Plot instrument.

Â So here's with the same circuit hooked up.

Â And I'm going to hit run on this and you'll see it building it up.

Â For each of those points, it's finding the amplitude ratio and

Â taking 20 times the log of it, and plotting it.

Â This is gain in decibels and down here is the phase angle in degrees.

Â Again, found numerically or found experimentally by looking at the angle

Â difference between the input signal and the output signal.

Â So this again is an automatic generation of the Bode Plot experimentally,

Â without even having to worry about the explicit formula for H of omega.

Â Now I want to take a look at our other circuit where I take the output across

Â this resistor.

Â Now this is a transfer function and

Â we can get the transfer function similar to what we did before.

Â We used the impedance method, where we use the voltage divider law.

Â So it's R over R plus this impedant, one over j omega c.

Â And that would be to find V out given Vs.

Â So this is a transfer function and if I clear my fractions I get this.

Â Now if I plot the magnitude versus frequency and

Â the angle versus frequency, I get these two Bode plots.

Â 5:38

Similar to what we did before, I can- Find a corner frequency here and

Â I can look at what happens at low frequencies and

Â what happens at high frequencies.

Â So low frequencies, I get a slope of 20 decibels per decade.

Â Now this is a slope upwards, so it's plus 20 dB per decade and

Â I go from 90 to zero degrees as I go from low frequency to high frequency.

Â And at high frequency in terms of the magnitude, I go to 0 dB.

Â So to summarize, low frequency magnitude has this slope, angle of 90.

Â High frequency has a magnitude of 0 dB, angle of zero degrees.

Â And the corner frequency is defined the same way as it was before, 1/RC.

Â To summarize, we've looked at an RC circuit in two different configurations.

Â The only difference between them is where we take the output.

Â Here, it's across C capacitor, and here it's across the resister.

Â And notice that we get very different Bode Plots between those two configurations.

Â 6:53

So for example, in this particular configuration we saw

Â that the high frequency slope was minus 20 dB per decade.

Â And we found that this was a corner frequency.

Â Well this formula for corner frequency holds for

Â this particular configuration as well.

Â But in this case,

Â we have a slope leading upwards of 20 dB per decade at low frequencies.

Â And then this being the corner frequency.

Â