Lecture Notes:- http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-002-circuits-and-electronics-spring-2007/lecture-notes/
We often build special circuits known as filters to eliminate that specific frequency of 50 to 60 Hz from the signal as AC in most countries are at 50 or 60 Hz. Otherwise we end up with a very low hum.
First, we need to plot the graphs of impedance against omega for capacitors, resistors and inductors.
For very low frequencies, the impedance of the capacitor is very, very high. The capacitor looks like an open circuit but for very high frequencies, the capacitor looks like a short circuit. So for the capacitor, we will be able to see the curve that shows its impedance.
For a resistor R, its behavior is frequency independent. So its impedance is simply R. And so it is
going to be a constant.
For the inductor, notice that for very low values of omega, the impedance is very low, and for high values of omega, the impedance is very high. Not surprisingly, the inductor looks like an open circuit to high frequencies and for DC and very low frequencies, the inductor looks very much like a short circuit.
So let's start with a CR circuit. If we plot the magnitude of H of omega of the transfer function for this filter, what does it look like?
So let's start with the high frequencies just for fun. For very high frequencies, note that the capacitor is a short circuit. And the resistor, of course, has a resistance R. So the capacitor is a short circuit and so Vi will appear directly at Vr. So for very high frequencies, my filter will give me Vi at Vr. So Vr divided by Vi will simply be unity.
What about for very low frequencies? At very low frequencies, the capacitor behaves like an open circuit. And if that behaves like an open circuit, then most of the voltage drop will be across the capacitor and very little will drop across the resistor and so I'm going to get a very low value. So low value for low frequencies and high values for high frequencies. So what we will get is a high pass filter.
These same principles apply for an RC circuit. So for an RC circuit, we will get a low pass filter.
For an RL circuit, at low frequencies, the inductor behaves like a short circuit so H of omega will have a low value. At high frequencies, the inductor behaves like an open circuit, so H of omega will have a high value. The RL circuit will be a high pass filter.
For an RLC circuit with the output voltage measured across the resistor, a bandpass filter is produced. At resonance, omega equals omega0 which is 1 by square root of LC and H of omega will be equal to 1.
For an RLC circuit with the output voltage measured across the capacitor and inductor, a bandstop filter is produced. At resonance, the output voltage is zero.
For an RLC circuit with the output voltage measured across the capacitor and inductor in parallel, a bandpass filter is produced. At resonance, the output voltage is zero. A major application of bandpass filters is in AM radio. The capacitor is made variable so that the bandpass filter can tune to various frequencies.
How does the radio work?
On the x axis of the graph is the frequency of the signal. On the y-axis, is the signal strength.
So the way radio system work is that different radio stations would be transmitting at different frequencies. So, for example, in the Boston area, there are a bunch of radio stations that transmit the following frequency. So for example, one of the radio stations is 1030 in the Amplitude Modulation band, And then,there may be other signals being transmitted by other radio stations. So for example, there may be a station transmitting at 1020, another at 1010 KHz and it goes on and on.
So when stations transmit, they try to maximize its signal strength, say around the 1030 frequency, ,it tries to maximize it there. And then it tries to make sure that the signal strengthdoesn't encroach too much into neighboring bands.So notice that each of these stations gets a roughly 10 kilohertz band.
So let's say, if I want to listen to WBZ News Radio, then I have to tune my capacitor here such that
the band-pass filter have its passing band focused where I care the most.
So when this is passed through, this band-pass filter will allow the 1030 range to mostly get through, and it will attenuate everything else that's further away but it is not perfect and so it will let through a little bit of the neighboring bands. That's why with AM radio you always get some interference from the sides of the two neighboring bands.
Fundamentally, that is a small value. Mostly the 1030 will come through. So this was the filter, the band-pass filter
Notice here that selectivity is important.
What is selectivity? Selectivity has to do with trying to capture the signal in a given range.
In the next video, we will see that the selectivity relates to something that you've seen before.
It will be shown how a RLC circuit, gives voltage values that are much higher than voltage values that are input. We have to look at Vc divide by Vi.. If the graph of Vc divide by Vi against omega is plotted, we will get a low pass filter.
We can do the maths with Vc over Vi using the voltage divider relation. We can write the impedance of the capacitor and divide it by the sum of all of the other impedance.
And for the magnitude, we can simply get the square of the real part plus the square of the
imaginary part square rooted.
You've taken it on faith from me that between these two points the function looks like the graph but that is not true.
Next we need to find out mathematically what happens at resonance. After some mathematical analysis, it can be seen that V_c is equal to Q times V_i. If Q is very large, V_c is more than V_i.
Next, we look at RLC circuit with V_r divided by V_i. We will get a bandpass filter.
So selectivity, we saw this in the context of radios. Selectivity has to do with how selective my filter is. Recall this is a band pass filter And selectivity says, how sharp is this band? How selective am I?
So one way of figuring out selectivity is to look at this ratio omega 0 by delta 0. Where delta 0 is called the bandwidth, it's measured at the point where the various parts of the curve reaches 1
by square root 2 of its highest value.
So higher omega 0 by delta omega, higher the selectivity. So the reason is that as delta omega becomes narrower, my filter becomes more selective.
For the next part, we need to find out what is delta omega. Absolute magnitude of V_r divided by V_i is one by square root 2. Eventually we will get a result of delta omega equal to R/L.
As 2 alpha is equal to R divided by L so omega 0 by delta omega is equal to Q. The lower the R, the sharper the peak, the more selective the filter. So high q implies high selectivity. From Q, you can tell all kinds of things about that circuit.
Q can also be defined as 2 pi times the energy stored in the circuit divided by energy lost per cycle in the circuit.
