hapter 3: Passive Filters and Transer Functions In this chapter we will look at the behavior o certain circuits by examining their transer unctions. One important class o circuits is ilters. A good example is trying to tune in a radio station. I you want to listen to an FM station broadcasting at 89.3Mhz, you want to process the signal coming rom your antenna and allow any signal very close to 89.3Mz to pass on while blocking, or attenuating, all other signals. ircuits that allow voltages at some requencies to pass while attenuating those at other requencies are called ilters. A: FIST ODE LOW PASS FILTES The simplest ilters, and crudest, are irst order high pass ilters and irst order low pass ilters. These can be made o a resistor and a capacitor or made o a resistor and an inductor. (esistors and capacitors are usually used at low requencies.)we looked at one in the previous chapter and it is shown in ig..3 which I ve duplicated at the right. This is oten called a low pass ilter. We are primarily interested in the complex transer unction, (j) and particularly E(t) Input Output in its polar representation, (j) exp(jφ). The magnitude o (j), or its amplitude, is (rom eqn..40) ( j) + + τ 3. ( ) ( ) where I ve used τ. The phase o (j), φ, is just φ tan - (τ) 3. We usually plot these as a unction o or a given τ, but irst let s look at the general behavior o these unctions. (Note that all circuits with the same product o and yield the same curve or behavior.) ase : I τ <<, the magnitude o (j) and φ 0. In this case the output is approximately the same as the input, i.e. the same amplitude and little phase shit. ase : I τ >>, the magnitude o (j) /τ << and the phase shit φ π/ or 90 o. ase 3: I τ, the magnitude o (j) / 0.707 and the phase shit φ π/4 or 45 o. This suggests that the circuit attenuates and shits the phase o signals whose requency > /τ. /τ is usually called the cuto requency, c or the circuit, since that is where the behavior changes, even though the change is gradual or this type o circuit. (The cuto requency is usually deined as the requency where the output amplitude is 3dB rom the input amplitude.) One could guess this result by noting that the impedance o the capacitor decreases as the requency increases. At low requency, the capacitors impedance is large and almost all the voltage drop is across the capacitor. At τ the magnitude o the capacitors impedance equals the resistors, and the voltage is split between them. (Note that each can have 70.7% o the voltage drop instead o 50% because the maximum voltages drops are out o phase with each other by π/4 at that point, i.e. they do not occur at the SAME TIME. At the same time, the sum o 0
the two drops must equal the input voltage.) At high requencies, the impedance o the capacitor is small and almost all the voltage drop is across the resistor. Since we are oten interested in the behavior over a wide range o requencies and attenuations, we use a logarithmic scale. We plot 0Log 0 ( (j) ) versus the Log 0 (/ c ). Note that is dimensionless, so it is ok to take the log o. (I ve dropped the [j]. Also, i I use Log or log that will be base 0 and ln will be the natural logarithm, or base e.) When we use 0Log 0 ( (j) ), this is (j) in db or decibels. The table at the right shows several useul relations or db. Note that i < it s log is negative. The at the right is not one or any particular circuit; I ve just made up some values or an example. (j) (j) in db 0 0dB 0dB 0.707 3dB 0.5 6dB 0. 0dB 0.0 40dB Note that is the ratio o the magnitudes, or amplitudes, o the output voltage to the input voltage. Thus the db scale here uses the input amplitude as a reerence and measures the output amplitude as a raction o the input s amplitude. (We also use a db scale or other quantities like sound intensity, where the reerence sound intensity is 0 - W/m.) For the low pass ilter example, i τ, the magnitude o 3dB, or the output is 3dB below the input level or 0.707 times the input level. The two plots below show as a unction o /π with the in db and as Log 0 (/ c ). (Note that / c / c.) The second plot shows the phase vs. Log 0 (/ c ). We usually plot these vs Log 0 (/ c ) because they scale as (/ c ). Every irst order low pass ilter will have the same shape when plotted this way. vs Log(/c) Phase vs Log(/c) 0.0-3.00 -.00 -.00 0.00.00.00 3.00 4.00 0.00-3.0 -.0 -.0 0.0.0.0 3.0 4.0 db -0.0-40.0-60.0-80.0 Phase (radians) -0.50 -.00 -.50 -.00 Log(/c) Log(/c) Fig. 3. Fig. 3. The ilter is called a irst order ilter because in the region where there is attenuation the output amplitude is approximately proportional to (/) to the irst power. A second order low pass ilter would be approximately proportional to (/) in that region. (ow would a 3 rd order low pass ilter depend on (/) when >> c?) Try to answer the ollowing 3 questions rom the irst graph. a) I c 000z, where will the output amplitude be /0 the input amplitude? b) I c 0z, where will the output amplitude be /00 o the input amplitude? c) I I want the output amplitude at 60 z to be /0 o the input amplitude, what should the cuto requency c be?
I τ >>, or >> c, (j) /(jτ), or the complex V out equals the integral o E in times /τ. (You should veriy this statement.) This circuit approximately integrates the input voltage i τ >>. B: FIST ODE IG PASS FILTES A irst order high pass ilter will be similar to the low pass ilter, but the capacitor and resistor will be interchanged, i.e. the output voltage will be the voltage across the resistor. The circuit is shown at the right. Again the input is a sinusoidal voltage and we will use its complex representation. This circuit is just a divider circuit, but with the impedances Z and Z reversed in position rom the low pass example. The complex output voltage will be given by V out V in Z Z + Z V in + j V in j + j The transer unction (j) is just the coeicient o V in or, using τ, ( j) jτ τ π ( ) exp j tan τ + jτ + ( τ) V in V o e j t V out ig 3.3 igh Pass Filter The last term has been converted into polar coordinates. (You should veriy that this is the correct orm.) The magnitude and phase o are given by τ π ( j) and φ phase tan ( τ) + τ 3.5 ( ) Again we will consider three cases: ase : I τ <<, (j) τ and φ π/ or 90 o. The output is smaller than the input, and the phase shit approaches 90 o. ase : I τ >>, (j) and the phase shit φ 0. ase 3: I τ, (j) / 0.707 and the phase shit φ π/4 or 45 o. This is called a high pass ilter because requencies above c /τ tend to be passed with little attenuation or phase shit while those below c tend to be attenuated. Notice the phase shit here is positive while or the low pass it was negative. The magnitude o (in db) and the phase o (in radians) are plotted below versus Log(/ c ). It is called a st order ilter because goes 3.3 3.4
(jw ) vs log(/c) Phase o vs Log(/c) in db 0.0-0.0-40.0-60.0-80.0-4.00 -.00 0.00.00 4.00 Phase (radians).00.50.00 0.50 0.00-4.00 -.00 0.00.00 4.00 Log(/c) Log(/c) Fig. 3.4 Fig. 3.5 like (/ c ) to the irst power when << c. (an you guess how a nd order high pass ilter s transer unction would behave when << c?) All st order high pass ilters have the same shape when plotted this way. The transition rom the region o little attenuation, >> c, to the region o strong attenuation is not very sharp with this type o ilter, the transition region being roughly rom (/ c ) /3 to (/ c ) 3. That is the P 0.95 at (/ c ) 3, within 5% o the high requency limit and P 0.36 at (/ c ) /3, where the approximate expression yields 0.333, again a deviation o about 5%. For many applications we can approximate P i (/ c ) > 3 and P (/ c ) i (/ c ) < /3. The irst order high pass ilter blocks the D or constant part o a signal, and only passes the part that depends on time. For example, i the input is 5V + Acos(t) and >> c, the output will be just Acos(t). The 5V will be blocked and disappear. The inputs to some devices, e.g. oscilloscopes, have a choice or A or D coupling. D coupling passes all parts o a signal. A coupling puts the input through a high pass ilter, which blocks the lower requencies. The time constant or an oscilloscope is usually around 0.s, producing a cuto requency o about.6 z. When a signal goes through a high pass ilter, it is shited so that or times >> τ, the average o the output voltage is 0 volts. At requencies below the cuto requency, this circuit approximately dierentiates the input and multiplies it by τ or / c, i.e. (j) τ or / c. (You should veriy this.) Finally, it is oten helpul to write the transer unctions o these ilters in terms o the cuto requency c π c /τ. Then the irst order low pass ilter has a transer unction given by + c, and φ tan and the irst order high pass ilter has a transer unction given by c π P, and φ P tan c + c c 3.6 3.7 3
I one deines x / c / c, they take on a somewhat neater orm., and φ tan ( x) 3.8 + x P x π, and φ P tan ( x ) + x 3.9 A useul approximation is that or x >>, and P x Similarly, or x << and P x : SEOND ODE -LIKE FILTES The simplest second order ilter is two irst order ilters that are 0 cascaded, i.e. one ollows the other in the circuit. owever, or it to behave E(t) Input nicely, the stages should be noninteracting. This will be /0 Output approximately the case when the Thevenin impedance o the irst stage (its output impedance) is much larger Fig. 3.6 than the series impedance o the two elements o the second stage. Note that in the circuit in ig. 3.6, a second order low pass ilter, I ve chosen 0 and /0 which ensures that the series impedance o the second stage is at least 0 times the output impedance o the irst stage. That is usually suicient. I that is the case, the transer unction or the entire circuit is, to a good approximation, the product o the two separate transer unctions, or total (j) (j) (j). This makes it much easier to analyze the circuit. In this example both circuits accidentally have the same time constant τ. (They don t have to have the same τ, but I ll leave some o those cases or you to igure out.) As a result, ( jtan ( τ) ) total ( j) exp jτ 3.0 + + ( τ) where the last expression is in polar orm. Again we will look at three cases. ase : I τ <<, the magnitude o (j) and φ 0. In this case the output is approximately the same as the input, i.e. the same amplitude and no phase shit. ase : I τ >>, (j) (/τ) << and the phase shit φ π or 80 o. ase 3: I τ, the magnitude o (j) / and the phase shit φ π/ or 90 o. 4
The reason it is called a second order ilter is that the amplitude alls o like (/τ) to the second power i τ >>. You should also note that the amplitude o total is the product o the two individual amplitudes and the phase is the sum o the two individual phases. At high requencies, the transer unction (/τ). In this region a doubling o the requency results in a reduction o 4 in the amplitude, or the amplitude changes by db or every doubling o the requency. (In audio systems they usually say db per octave.) db nd Order Low Pas s Filter 0.00-0.00-40.00-60.00-80.00-00.00-4.00-3.00 -.00 -.00 0.00.00.00 3.00 Log(/c) Fig. 3.7 ascading two non-interacting ilters means that the net transer unction is the product o the two individual transer unctions. When I use the db or logarithmic scale, 0 Log( net ) 0 Log( ) 0 Log( ) + 0 Log( ), or in db, net +. In the db scale they add. This is a useul shortcut when plotting the net on a db scale. It is interesting to note that the phase shits also add. This is not a very sharp ilter, i.e. the transition rom to the region above τ where it is decreasing rapidly occurs over a range o requencies. For instance i you want to pass a signal o requency s and not attenuate it by more than 0%, i.e. 0.90, you need to set c o the individual ilters such that 0.90 + ( s / c ) 3. or ( s / c ) /3. This means that c 3 s, the cuto requency o each ilter stage has to be 3 times the signal requency. A noise signal around c would only be reduced in amplitude to ½ its original amplitude. The noise signal would have to be at 3 c 9 s to be reduced in amplitude by a actor o 0 ( 0dB.) There are three dierent versions o cascading two irst order type ilters to get a second order ilter. The irst is cascading two low pass ilters. The second is cascading two high pass ilters and the third is the cascade a low pass and a high pass ilter to produce a band pass ilter. The band pass ilter would like the one below, where I ve again made the two cuto requencies the same. I ve also show a plot o. Note that has a maximum o 6dB or /. (Oten you want the low pass cuto requency higher than the high pass cuto requency.) I will leave these or you to investigate. 5
/0 Band Pass Filter 0.0 E(t) Input 0 Output -0.0-40.0-60.0 Fig. 3.8 D: SEOND ODE L FILTES -80.0-4.00 -.00 0.00.00 4.00 Log(/c) Fig. 3.9 A better second order ilter can be made rom an inductor, capacitor and a resistor in series. You can get all three behaviors by looking at the voltage drop across dierent elements. I your output is the voltage across the capacitor, you will have a second L order low pass ilter. I the output is across the inductor, you will have a second order high pass ilter. I the output is across the resistor, you will have a band pass ilter. onsider the low pass ilter version. It is shown at the right. This looks just like another divider circuit. I the input voltage is V o exp(jt), the output voltage E(t) Fig. 3.0 V out will be Z V out Z + Z + Z Voexp( j t ) 3. L and the transer unction (j) will be given by V out V in. Thereore Z + Z Output ( j) 3.3 Z L + Z + j j jl + L + j Normally one sets L (/ o ) since L has units o over requency squared and this o is the angular requency o ree oscillations or an L circuit, sometimes called the natural requency or undamped resonant requency or the circuit. One can write o j o + o ( j) + j o The magnitude o is just ( j) The phase o, φ, is o + ( ) + ( ) o o L 3.4 3.5 6
φ tan L 3.6 o Again we will consider three cases: ase : I << ο, the magnitude o (j) and φ 0. In this case the output is equal to the input. ase : I >> o, the magnitude o (j) ( o /) and the phase shit φ π or 80 o. The output s magnitude is much less than the input s. ase 3: I o, the phase shit φ π/ or 90 o and the magnitude o (j) is ( j ) L 3.7 o o where I ve put o /(L) / in eqn. 3.. The interesting thing is that the output can be larger than the input or certain choices o, L and. For instance i L m, nf, then (L/) / 000. I < kω, the output is larger than the input at this requency. There are two terms associated with this circuit, the Q o the circuit and the damping actor, d. Most people deine d /(Q), but some deine d as d /Q. (It is d that changes rom author to author, not the Q.) I ll use d /(Q). L Q 3.8 d Q and d are dimensionless. At o, Q. I d / 0.707, the output amplitude is never larger than the input s. I d < 0.707, then or some requency, the output amplitude will be larger than the input s. I d, the circuit is said to be critically damped, and i d >, the circuit is said to be overdamped. These terms are also used to describe mechanical oscillations where the symbol or the damping actor is usually γ. I one lets x (/ o ), one can rewrite as ( j ) 3.9 x + 4x d ( ) Usually 0< d <, so or x << and x>> the magnitude o is not sensitive to d. It is in the region around x that the value o d is important, typically or (/3) < x < 3. You should also note that the maximum value o is not necessarily at x, although it goes to x as d gets smaller. The maximum o occurs when the denominator in 3.5 is a minimum. I you take the denominators derivative and set it 0, you ll ind that the maximum or occurs at x ( d ) i d < /. I d > /, the maximum is at x 0, or 0. {Note that I can substitute / o or / o in 3.5 above. This is because the π s cancel. It is oten convenient to use x/ o instead o x/ o when doing calculations.) I have plotted in the region 0. < x < 5 or several d s below. The d case is the same as two cascaded ilters with the same τ. 7
5 or various d's vs. log (/o) db 0-5 -0-5 -0-5 -30 - -0.5 0 0.5 log(/o) d0.4 d 0.6 d0.7 d Fig. 3. The d s range rom 0.4 to. Once you get away rom the x region, or log(/ o ) 0, they are all similar. For d less than 0.7, the all o is sharper but you get some overshoot, i.e. > near x. I have not shown the phase shit, or phase o, but you can get a rough idea rom eqn. 3.6. The high pass version looks at the voltage across the inductor. I recommend you try to write the magnitude o the transer unction as a unction o x / o / o and d. You simply replace Z in the numerator o irst expression in eqn. 3.9 by Z L and do the algebra. The magnitude o that transer unction should look like the mirror image o the one above, i.e. relected let to right through log(/ o ) 0. For this case P and φ P are x ( ) xd P j and φ P π tan + 4x d x 3.0 ( x ) The band pass output would be the voltage drop across the resistor. ere you replace Z in the numerator o irst expression in eqn. 3.9 by Z and do the algebra. You should get BP is xd π xd ( ) BP j and φbp tan + 4x d x 3. ( x ) The maximum o BP occurs at x, or o and it doesn t depend on d. BP max. At that requency, the phase shit is zero, and the output is the same as the input. owever, the sharpness or narrowness o the band pass does depend on the damping actor d. The width o the band pass is deined in terms o the two requencies where BP /. I the lower requency is and the upper is, the width is. The narrowness o the band pass ilter is given by / o and the Q o the circuit is related to the narrowness by Δ 3. Q o 8
I ve show a plot o BP or a couple o dierent d s. Note that as d gets smaller, the range o requencies that pass with little attenuation becomes smaller and smaller. We say the ilter is sharper or narrower. All o the ones shown are narrower than the two-stage band pass ilter, whose d. or band pass L ilter 0 db -0-0 -30-40 d0.707 d 0.3 d 0. d 0.03-50 - -0.5 0 0.5 Log(/o) Fig. 3. On old radios, the way you selected a station was by turning a dial that changed the capacitance o a variable capacitor that was part o a band pass circuit. This would change o to match the requency o the radio station you wanted to listen to. Then the signal o that station was passed and the signals rom other stations were attenuated. In that case you needed a very narrow band pass ilter, i.e. a large Q or a small d. One last note about L ilters. The voltage across the capacitor and inductor are 80 o out o phase, i.e. they have the opposite phase. By arranging the circuit to have the inductor and capacitor together and measure the voltage across both or the output, one gets a notch ilter. This attenuates the input close to o, but does not attenuate it ar away rom o, just the opposite o a band pass ilter. You use notch ilters to get rid o intererence or noise that occurs at a speciic requency. A common example is a 60z notch ilter to remove noise picked up rom our 60z power mains. {Note that the series impedance o an ideal inductor and capacitor, Z L + Z, goes to zero at o /(L) /.} I should warn you that these analyses assume ideal behavior or the resistor, the capacitor and the inductor. Larger inductors, L > 00µ are seldom ideal. They oten have 5-0Ω o resistance and some capacitance between the windings. I you take a 0m inductor and look at the magnitude o its impedance, you will probably ind that at low requencies the 0-0 Ω o resistance it is likely to have will mean that Z does not 0. Similarly at high requencies the interwinding capacitance means that its impedance will not continue to increase as the requency increases, instead at some point it will decrease. apacitors also exhibit some inductance, but you can usually minimize its eect at requencies below Mz. (This is done by choosing the right type o capacitor. Electrolytic capacitors oten have more inductance.) Because it is harder to get good inductors or ilters below kz, people oten use active ilters in the lower requency ranges. Active ilters use operational ampliiers, or op amps or short, resistors and capacitors to simulate L circuits. We will look at them when we discuss op amps. 9
I have not discussed parallel L circuits; there just isn t time to discuss everything. You might want to try them on your own. E: FEQUENY DOMAIN AND TIME DOMAIN When we describe the response o a circuit to a sinusoidal input, we reer to the description as the requency domain description. The response is the transer unction and is a unction o the requency o the input signal. The alternative is a time domain description. For instance, how would the system respond to a step unction input? ere the response unction is a unction o time, in particular the time ater the application o the step change in the input. They are two dierent ways o looking at the same system, and knowledge o one response unction will allow us to calculate the other, at least in principle. It is useul to have an idea o how low and high pass ilters respond to a step unction input or to a square wave input. For a low pass ilter, i the requency o the square wave is much less than c, then it will be passed with little change. You will merely see a rounding o the corners. I the requency is much greater than c, it will be integrated to look like a triangle wave and it amplitude will be attenuated. See ig. 3.3. V V In In t t Out << c >> c >> c Low pass ilter Fig. 3.3 << c igh pass ilter Fig. 3.4 I you send a square wave into a high pass ilter and its requency is much greater than the cuto requency, it will be passed with little change. owever, the average voltage will be zero, so the swings o the output voltage will be symmetric about 0 volts or ground. I the input signal varied rom 0V to +5V at the input, it will vary rom.5v to +.5V at the output. I the requency is much less than c, the output will look like little spikes that occur when the input changes and the spikes quickly decay back to 0V. The peaks o the spikes would be +5V or the positive one and 5V or the negative one. See ig. 3.4. 30