Welcome back to Electronics, this is Dr. Robinson. In this lesson we'll cover Second-Order Transfer Functions. In the previous lesson we talked bout cascaded first-order op-amp filters, and our objectives for today's lesson are to introduce second-order filter transfer functions and to examine features of these transfer functions. Now remember, a filter transfer function is just the ratio of the output voltage to the input voltage for a circuit. So once the frequency is chosen, this transfer function is simply a complex number with a magnitude and a phase, or a real part and an imaginary part. That tells us how the input voltage is modified to produce the output voltage. Let's do a quick review of the behavior of a first-order low-pass filter transfer function. Here I've drawn the transfer function in standard form. And we can see that it's a first-order transfer function, because the highest power to which f is raised in the denominator is 1, f to the first power. It's a low pass transfer function because we've formed the transfer function by moving the lowest order term of the denominator, the 1, to the numerator. There are two parameters in this transfer function, K the DC gain, and F not the resonate frequency. On this plot, I've plotted the Bode magnitude plot, of this transfer function, or the magnitude of the transfer function versus frequency. We can see that it's a low-pass filter at low frequencies. I've set the DC gain or K equal to 1 and I've set the resonance frequency or F not to 100 hertz. Now remember, the slope of this asymptote for first order filter in the stop band is equal to minus 20 DB. Per decade or minus 1 decade per decade. A decade per decade means for any increase in decades along the frequency access, we decrease by 1 decade in the magnitude of the transfer function where a decade is a factor of 10. So for example in moving from 100 hertz to 1,000 hertz in frequency, that's one factor of 10. We would expect to decrease in magnitude by one factor of 10. And you can see that we, we change in magnitude from 1 to .1 over that frequency range from 100 to 1,000. Now let's compare a first order. Low pass transfer function to a second order low pass transfer function. I've drawn the two transfer functions here so you can see the differences. I've put the second order transfer function in standard form. You can see that the denominator polynomial is second order, the highest power to which f is raised is 2. And again it's first order, because we've taken the lowest, lowest power term of the denominator or lowest order term and moved it to the numerator to form the transfer function. K in both cases is the DC gain. And in this plot, I've plotted the bode magnitude plot of both transfer functions so you can compare them f naught is the same for both transfer functions equal to 100 hertz. And K is the same for both transfer functions equal to 1. The primary difference you can see here is, in the slope of the transfer function in this top band. The second order filter. Has a slope in the stop band of minus 2 decades per decade. While the slope of the first order filter in blue, has a slope of minus 1 decades per decade. Now remember, an ideal low pass filter, within this cut-off frequency would look like this. We have a gain of one in the pass band and then we would have an infinite slope here at f nod equals 100 so frequencies on this side are completely attenuated or eliminated and frequencies on this side are completely passed. You can see that increasing the order of the filter makes the filter more ideal, in that this slope is steeper or closer to an infinite slope. A thing to note about the second order transfer function, is that we introduced an additional parameter, the parameter Q or quality factor. Let's examine how this third parameter, the quality factor or Q, affects the behavior of a second order low pass, transfer function. I've plotted the Bode magnitude plot of the second or low pass transfer function for three different values of Q, I've kept K equal to 1 for all three cases and F not equal to 100 hertz. The blue curve here is a high Q of, 5, 5. The green curve is a low pass filter, with a Q of 0.2. And the red curve is a second order, low pass filter transfer function, with a Q equal to 0.707 or, one over the square root of two. You can see that the quality factor affects the behavior of the transfer function given the frequency of 100 hertz. Far from the resident frequency, all three of these transfer functions approach an asymptote. Slope 0 decades/decade in the pass band of the filter. And far from the resident frequency on the high side, all the filters approach this slope of minus 2 decades per decade, because it's a second order filter. But near the resident frequency, the q effects the response. A low Q. Filter or a low Q second order transfer function, has this gradual transition from pass band to stop band, where as the high Q transfer function has peaking in the pass band. And here, we add the Q in between the two where there is no peaking. You can see that the high Q filter has a steeper transition between pass band and stop band. But at the expense of this large ripple in the pass band. Let's examine the behavior of second order high pass transfer functions. And compare that behavior to the behavior of a first order high pass transfer function. So I've written here both the first order, low pass and high pass transfer functions, and the second order, low pass and high pass transfer functions. Informing the second-order high-pass transfer function, you can see that the denominator is exactly the same as it was for the second-order low-pass transfer function. But, the numerator is now equal to the highest order term, in the denominator. Whereas for the low-pass filter, the numerator was equal to the lowest order term in the denominator. And similarly for the first order filters, you form a first order high-pass filter by moving the highest ordered term in the denominator to the numerator and you form a low-pass filter by moving the lowest ordered term in the denominator to the numerator. So to visually see the behavior of these high-pass filters. I've plotted three bode magnitude plots on this graph. Two second order high pass filter transfer functions, and one first order for comparison. Both of the second order high pass transfer functions have these same, F not and K but I varied the Q. This is a high-Q, high-pass filter. A Q equal to 2. And the red curve has a quality factor equal to 1 over the square root of 2. The slope in this region for the second order high-pass. Transfer functions is plus two decades per decade because it's a second order filter. The slope here, first order filter, so it has a slope of plus 1 decades per decade. And again you can see the quality factor affects the behavior near the resident frequency of 100 hertz. Here we're going to take a look at band-pass filter transfer functions. I've drawn the transfer function for a Band-Pass Filter here and again the denominator is the same as it was for the high pass filter and the low pass filter. But to form the overall Band-Pass Filter transfer function. I've taken the middle term and moved it to the numerator, to form, the transfer function. Now you can see that, I've plotted on this graph again three plots so we can compare how Q is changing the behavior of the filter transfer function. All three of the Band-Pass Filters have resident frequency or F naught of 100 hertz and they all have a k of one. But the blue curve has a quality factor of five. The green curve has a quality factor of 0.2 and the red curve has a quality factor of whatever the square root of two. You can see as the quality factor increases for a fixed f naught. The filter is becoming more selective. This high Q filter has a narrow bandwidth and quickly attenuate, attenuates frequencies outside of that narrow pass band. The low q band pass filter is a broadband filter. Where it passes frequencies within this wide range. Before they're attenuated in the stop bands on either side of F naught. Now, we can relate the quality factor to the center frequency bandwidth of the filter through this equation. So, for a given F naught,. As we increase the Q, the bandwidth must decrease. Where the bandwidth is defined as the difference between the upper cutoff frequency and the lower cutoff frequency. The frequencies at which the magnitude of the gain is down by a 3db. So for this wide band band pass filter, I've drawn a line here at 1 over root 2. Where this line intersects the magnitude of the transfer function here and here, those two intersections define the upper cut off frequency and the lower cut off frequency. And the distance and frequency between these two is the bandwidth. Let's talk about Butterworth and Chebyshev transfer functions. These are types of transfer functions and for second-order filters, the type is determined by the value of Q. If Q is equal to exactly 1 over root 2, the transfer function is known as a Butterworth transfer function, or a Maximally Flat transfer function. If the quality factor is greater than 1 over root 2, it's known as a Chebyshev transfer function. Function. The type of transfer function indicates its behavior. Let's examine the behavior of a Chebyshev transfer function by examining the Bode magnitude plots for three different Chebyshev filters or three different Chebyshev transfer functions. And while this lesson is primarily concerned with second-order transfer functions, I want to show you increasing the order, affects the Bode magnitude plot. So here I've plotted three different filters of increasing order. The red curve is a second order Chebyshev filter. The green curve is a third order. And the blue is a fourth order Chebyshev filter transfer function. I can see that increasing the order as you would expect increases the steepness of the slope in the stop end. Increasing the order also increases the amount of ripple in the pause band. The third order filter, has three bumps in its pause bands one, two, three and then down. The second order filter has 1, 2, and then down into the stop band. While the fourth ord, order has 1, 2, 3, 4, and then into the stop band. So we get a steeper slope in this region by increasing the order of the filter at the expense of more ripple in the pass band. Let's compare the behavior of three different Butterworth Filters, each of a higher order. Here's a Bode magnitude plot of the three different filters. I've plotted a second order, a third order, and a fourth order, Butterworth transfer function. The reason a Butterworth filter is called a maximally flat filter is because the passband of the filter is flat for as long a distance and frequency as possible for a given F not. There is rippled in the passband for a Butterworth filter. As we increase the order, the steepness of the asymptote in the stop band increases. A slope of -2 decades per decade, a slope of -3 decades per decade, and a slope of -4 decades per decade. So as we increase the order of this filter, the filter is becoming more like an ideal brickwall filter. Let's contrast the behavior of a Fourth-Order Butterworth transfer function with that of a Fourth-Order Chebyshev transfer function. The Butterworth filter in blue, you can see that the transition between the Pass band and the Stop band is not as steep, as it is with the Chebyshev, but the Butterworth, transfer function has no ripple in the pass band. You get this steepness in the transition for the chebyshev at the expense of this ripple in the pass band. So which filter you choose depends on your application. If you need the steepness in the, in the transition and ripple in the pass band is not important, you choose the chebyshev filter. But, if you need a, a flat passband, you are forced to choose a Butterworth filter, at the expense of a more gradual transition here, from passband to stopband. So in summary, during this lesson we introduce second-order filter transfer functions and examined the features of these transfer functions. By looking at bode magnitude plots. In the next lesson, we will look at op-amp second order filter circuits that can be used to implement these transfer functions. So thank you, and until next time.
