This course introduces students to the basic components of electronics: diodes, transistors, and op amps. It covers the basic operation and some common applications.

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From the course by Georgia Institute of Technology

Introduction to Electronics

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This course introduces students to the basic components of electronics: diodes, transistors, and op amps. It covers the basic operation and some common applications.

From the lesson

Op Amps Part 2

Learning Objectives: 1. Examine additional operational amplifier applications. 2. Examine filter transfer functions.

- Dr. Bonnie H. FerriProfessor

Electrical and Computer Engineering - Dr. Robert Allen Robinson, Jr.Academic Professional

School of Electrical and Computer Engineering

Welcome back to Electronics, this is Dr. Robinson.

Â In this lesson we'll cover Second-Order Transfer Functions.

Â In the previous lesson we talked bout cascaded first-order op-amp filters, and

Â our objectives for today's lesson are to introduce second-order filter

Â transfer functions and to examine features of these transfer functions.

Â Now remember, a filter transfer function is just the ratio of

Â the output voltage to the input voltage for a circuit.

Â So once the frequency is chosen, this transfer function is simply a complex

Â number with a magnitude and a phase, or a real part and an imaginary part.

Â That tells us how the input voltage is modified to produce the output voltage.

Â Let's do a quick review of the behavior of a first-order low-pass filter

Â transfer function.

Â Here I've drawn the transfer function in standard form.

Â And we can see that it's a first-order transfer function, because the highest

Â power to which f is raised in the denominator is 1, f to the first power.

Â It's a low pass transfer function because we've formed the transfer function

Â by moving the lowest order term of the denominator, the 1, to the numerator.

Â There are two parameters in this transfer function,

Â K the DC gain, and F not the resonate frequency.

Â On this plot, I've plotted the Bode magnitude plot, of this transfer function,

Â or the magnitude of the transfer function versus frequency.

Â We can see that it's a low-pass filter at low frequencies.

Â I've set the DC gain or

Â K equal to 1 and I've set the resonance frequency or F not to 100 hertz.

Â Now remember, the slope of this asymptote for

Â first order filter in the stop band is equal to minus 20 DB.

Â Per decade or minus 1 decade per decade.

Â A decade per decade means for any increase in decades along the frequency access,

Â we decrease by 1 decade in the magnitude of

Â the transfer function where a decade is a factor of 10.

Â So for example in moving from 100 hertz to 1,000 hertz in frequency,

Â that's one factor of 10.

Â We would expect to decrease in magnitude by one factor of 10.

Â And you can see that we,

Â we change in magnitude from 1 to .1 over that frequency range from 100 to 1,000.

Â Now let's compare a first order.

Â Low pass transfer function to a second order low pass transfer function.

Â I've drawn the two transfer functions here so you can see the differences.

Â I've put the second order transfer function in standard form.

Â You can see that the denominator polynomial is second order,

Â the highest power to which f is raised is 2.

Â And again it's first order,

Â because we've taken the lowest, lowest power term of the denominator or

Â lowest order term and moved it to the numerator to form the transfer function.

Â K in both cases is the DC gain.

Â And in this plot, I've plotted the bode magnitude plot of both transfer

Â functions so you can compare them f naught is the same for

Â both transfer functions equal to 100 hertz.

Â And K is the same for both transfer functions equal to 1.

Â The primary difference you can see here is,

Â in the slope of the transfer function in this top band.

Â The second order filter.

Â Has a slope in the stop band of minus 2 decades per decade.

Â While the slope of the first order filter in blue,

Â has a slope of minus 1 decades per decade.

Â Now remember, an ideal low pass filter,

Â within this cut-off frequency would look like this.

Â We have a gain of one in the pass band and then we would have an infinite slope here

Â at f nod equals 100 so frequencies on this side are completely attenuated or

Â eliminated and frequencies on this side are completely passed.

Â You can see that increasing the order of the filter makes the filter more ideal,

Â in that this slope is steeper or closer to an infinite slope.

Â A thing to note about the second order transfer function, is that we

Â introduced an additional parameter, the parameter Q or quality factor.

Â Let's examine how this third parameter, the quality factor or Q,

Â affects the behavior of a second order low pass, transfer function.

Â I've plotted the Bode magnitude plot of the second or

Â low pass transfer function for three different values of Q,

Â I've kept K equal to 1 for all three cases and F not equal to 100 hertz.

Â The blue curve here is a high Q of, 5, 5.

Â The green curve is a low pass filter, with a Q of 0.2.

Â And the red curve is a second order, low pass filter transfer function,

Â with a Q equal to 0.707 or, one over the square root of two.

Â You can see that the quality factor

Â affects the behavior of the transfer function given the frequency of 100 hertz.

Â Far from the resident frequency,

Â all three of these transfer functions approach an asymptote.

Â Slope 0 decades/decade in the pass band of the filter.

Â And far from the resident frequency on the high side, all the filters approach this

Â slope of minus 2 decades per decade, because it's a second order filter.

Â But near the resident frequency, the q effects the response.

Â A low Q.

Â Filter or a low Q second order transfer function,

Â has this gradual transition from pass band to stop band,

Â where as the high Q transfer function has peaking in the pass band.

Â And here, we add the Q in between the two where there is no peaking.

Â You can see that the high Q filter has a steeper transition between pass band and

Â stop band.

Â But at the expense of this large ripple in the pass band.

Â Let's examine the behavior of second order high pass transfer functions.

Â And compare that behavior to the behavior of

Â a first order high pass transfer function.

Â So I've written here both the first order, low pass and high pass transfer functions,

Â and the second order, low pass and high pass transfer functions.

Â Informing the second-order high-pass transfer function,

Â you can see that the denominator is exactly the same as it was for

Â the second-order low-pass transfer function.

Â But, the numerator is now equal to the highest order term, in the denominator.

Â Whereas for the low-pass filter,

Â the numerator was equal to the lowest order term in the denominator.

Â And similarly for the first order filters,

Â you form a first order high-pass filter by moving the highest ordered term in

Â the denominator to the numerator and you form a low-pass filter by

Â moving the lowest ordered term in the denominator to the numerator.

Â So to visually see the behavior of these high-pass filters.

Â I've plotted three bode magnitude plots on this graph.

Â Two second order high pass filter transfer functions, and

Â one first order for comparison.

Â Both of the second order high pass transfer functions have these same,

Â F not and K but I varied the Q.

Â This is a high-Q, high-pass filter.

Â A Q equal to 2.

Â And the red curve has a quality factor equal to 1 over the square root of 2.

Â The slope in this region for the second order high-pass.

Â Transfer functions is plus two decades per decade because it's a second order filter.

Â The slope here, first order filter, so it has a slope of plus 1 decades per decade.

Â And again you can see the quality factor affects the behavior near

Â the resident frequency of 100 hertz.

Â Here we're going to take a look at band-pass filter transfer functions.

Â I've drawn the transfer function for a Band-Pass Filter here and

Â again the denominator is the same as it was for the high pass filter and

Â the low pass filter.

Â But to form the overall Band-Pass Filter transfer function.

Â I've taken the middle term and

Â moved it to the numerator, to form, the transfer function.

Â Now you can see that, I've plotted on this graph again three plots so

Â we can compare how Q is changing the behavior of the filter transfer function.

Â All three of the Band-Pass Filters have resident frequency or

Â F naught of 100 hertz and they all have a k of one.

Â But the blue curve has a quality factor of five.

Â The green curve has a quality factor of 0.2 and

Â the red curve has a quality factor of whatever the square root of two.

Â You can see as the quality factor increases for a fixed f naught.

Â The filter is becoming more selective.

Â This high Q filter has a narrow bandwidth and

Â quickly attenuate, attenuates frequencies outside of that narrow pass band.

Â The low q band pass filter is a broadband filter.

Â Where it passes frequencies within this wide range.

Â Before they're attenuated in the stop bands on either side of F naught.

Â Now, we can relate the quality factor to the center frequency bandwidth of

Â the filter through this equation.

Â So, for a given F naught,.

Â As we increase the Q, the bandwidth must decrease.

Â Where the bandwidth is defined as the difference between the upper cutoff

Â frequency and the lower cutoff frequency.

Â The frequencies at which the magnitude of the gain is down by a 3db.

Â So for this wide band band pass filter, I've drawn a line here at 1 over root 2.

Â Where this line intersects the magnitude of the transfer function here and

Â here, those two intersections define the upper cut off frequency and

Â the lower cut off frequency.

Â And the distance and frequency between these two is the bandwidth.

Â Let's talk about Butterworth and Chebyshev transfer functions.

Â These are types of transfer functions and

Â for second-order filters, the type is determined by the value of Q.

Â If Q is equal to exactly 1 over root 2, the transfer function is known as

Â a Butterworth transfer function, or a Maximally Flat transfer function.

Â If the quality factor is greater than 1 over root 2,

Â it's known as a Chebyshev transfer function.

Â Function.

Â The type of transfer function indicates its behavior.

Â Let's examine the behavior of a Chebyshev transfer function by examining the Bode

Â magnitude plots for three different Chebyshev filters or

Â three different Chebyshev transfer functions.

Â And while this lesson is primarily concerned with second-order transfer

Â functions, I want to show you increasing the order,

Â affects the Bode magnitude plot.

Â So here I've plotted three different filters of increasing order.

Â The red curve is a second order Chebyshev filter.

Â The green curve is a third order.

Â And the blue is a fourth order Chebyshev filter transfer function.

Â I can see that increasing the order as you would expect increases the steepness of

Â the slope in the stop end.

Â Increasing the order also increases the amount of ripple in the pause band.

Â The third order filter, has three bumps in its pause bands one,

Â two, three and then down.

Â The second order filter has 1, 2, and then down into the stop band.

Â While the fourth ord, order has 1, 2, 3, 4, and then into the stop band.

Â So we get a steeper slope in this region by increasing the order of

Â the filter at the expense of more ripple in the pass band.

Â Let's compare the behavior of three different Butterworth Filters,

Â each of a higher order.

Â Here's a Bode magnitude plot of the three different filters.

Â I've plotted a second order, a third order, and

Â a fourth order, Butterworth transfer function.

Â The reason a Butterworth filter is called a maximally flat filter is because

Â the passband of the filter is flat for as long a distance and

Â frequency as possible for a given F not.

Â There is rippled in the passband for a Butterworth filter.

Â As we increase the order, the steepness of the asymptote in the stop band increases.

Â A slope of -2 decades per decade, a slope of -3 decades per decade,

Â and a slope of -4 decades per decade.

Â So as we increase the order of this filter,

Â the filter is becoming more like an ideal brickwall filter.

Â Let's contrast the behavior of a Fourth-Order Butterworth transfer

Â function with that of a Fourth-Order Chebyshev transfer function.

Â The Butterworth filter in blue, you can see that the transition between

Â the Pass band and the Stop band is not as steep, as it is with the Chebyshev, but

Â the Butterworth, transfer function has no ripple in the pass band.

Â You get this steepness in the transition for

Â the chebyshev at the expense of this ripple in the pass band.

Â So which filter you choose depends on your application.

Â If you need the steepness in the, in the transition and

Â ripple in the pass band is not important, you choose the chebyshev filter.

Â But, if you need a, a flat passband, you are forced to

Â choose a Butterworth filter, at the expense of a more gradual transition here,

Â from passband to stopband.

Â So in summary, during this lesson we introduce second-order filter

Â transfer functions and examined the features of these transfer functions.

Â By looking at bode magnitude plots.

Â In the next lesson, we will look at op-amp second order filter circuits that can

Â be used to implement these transfer functions.

Â So thank you, and until next time.

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