<latexit sha_base64"lw4qvmxz6lhq39hnu2kq/f65xgq">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</latexit> <latexit sha_base64"lw4qvmxz6lhq39hnu2kq/f65xgq">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</latexit> <latexit sha_base64"lw4qvmxz6lhq39hnu2kq/f65xgq">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</latexit> <latexit sha_base64"f3om9jilwrz9l9gprolwmw6bu8">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</latexit> second-order active filters Building up higher-order active filters from first-order filters is OK, but limiting, because we can never have Q P > 0.5 by using first-order building blocks. To get more flexibility we need slightly different approaches. Here we describe three types of second-order active filter circuits that can achieve higher Q P. Two-integrator loop biquad Start with the transfer function for a second-order high-pass T hp V hp V i G o and re-arrange: s 2 s 2 ω o Q P s ω 2 o V hp ω o Q P s ω2 o s 2 G o V i The left-hand side is essentially a summation of V o, a scaled version of the integral of V o, and a scaled version of the second integral of V o. This way of writing may seem a bit odd, but this is the key observation. EE 230 active second-order filters
<latexit sha_base64"z6rkagnfaukyyxzminpsw9tf4">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</latexit> V hp G o V i ω o V hp Q P s ω 2 o V hp s 2 V o /s integral of V o V o /s 2 integral of V o /s second integral of V o We can create the high-pass output by summing the input, the integral of the output and the second integral of the output, with appropriate scaling. We have seen summing-type amps and integrating amps, so it appears that we might be able make this work. C v i (t) V i R v o (t) V o v i (t) dt V i s EE 230 active second-order filters 2
<latexit sha_base64"scj3qczzw7vki0xi5jmrivn7s">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</latexit> <latexit sha_base64"omy62eirdfyfrzjqjq9vtafpuje">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</latexit> V o G o V i ω o V o Q p s ω 2 o V o s 2 Try building up a circuit. Start with V o applied to two successive integrating circuits. C C R V R V 2 V hp V V 2 V hp s V s V hp () 2 s 2 We see a slight issue with signs in the equation the first and second integral terms have the same sign (negative), but V and V 2 from the integrator circuits have opposite signs. EE 230 active second-order filters 3
We could try to fix the sign problem using an extra inverter, which require one more op amp. Or we can be clever with the implementation for the summing amp. Consider the following circuit (which may have seen this before.): R c R f v c v b R b v o v a R a It is straight-forward to show that v o R f R c R v a a R b R f R c R f R v b b R R c b v c G a v a G b v b G c v c This is exactly what is needed, a three-input circuit that adds two of inputs and subtracts a third. EE 230 active second-order filters 4
Putting it all together: Rc Rf Rb Vi C R Vhp V R V2 Ra Vhp Gb Vi V where the G s are defined on the previous page. V2 <latexit sha_base64"hmhewxsiwun4wiqipgbunfog">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</latexit> Vhp Gb Vi C Vhp s (s) 2 Vhp <latexit sha_base64"oa5w0spvv0xd8z65zscsoaytjo">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</latexit> Vhp Thp Gb Vi s2 <latexit sha_base64"wqc6j5hlhjilfecaedvn97lh9u">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</latexit> EE 230 s2 s ω2o ()2 () ωo Qp 2 active second-order filters 5
But, wait There s more. V is the integral of Vhp V <latexit sha_base64"scj3qczzw7vki0xi5jmrivn7s">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</latexit> Vhp s Gb s s2 s Gb s2 ()2 <latexit sha_base64"ntioqtacrbrh5ejccpuafglpjaq">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</latexit> s s ()2 <latexit sha_base64"w7uskgp/igjur/wp6g3chd6d2pm">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</latexit> sha_base64"f6dckzioewfpv5guzso2loyamga">aaacxnicbzflswmxemft9vflxhsd4tfujcy9alebau9ekhgrvblmcoa2g2wzjzavn6bbz66fwsfgaztuifdgt/gyyzzcrwliqfbw8pewvbxjelmvnf7z7e09wxajg2ujbavirguqqfdrik8suxcheosrn2b3j/8x2nf90tdbdgw9jbqcazlu75qqqtuym78oahnryrprlavn0jznezfxikrvqqudsdq4jlhbvfu4sj8d70sowkghhtoxv3ofkphyn8rjbukflhdpphbrgwzh0kthqm5335fa/xyul7mu7eypjcrx/ldrnpu/az4f2i2gqkxw6adwi6vp38aaj7eamsrj3anymumyqaoexhouukdmucgrypbqhyqra7s7vbppzy5fzkvvqe2boh3hl2zxd7poxrxej2vzef0xjvhpvdr4dts4o2be7ytv2wa7zpauzbumsyh/ss/dt7xn7vwfzcpplldmmeyc/waw8zw</latexit> sha_base64"bxiwq0y/xffhukxpss9tu/qmw90">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</latexit> sha_base64"ijlgdiglx8j7zoztgoba8jhswcu">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</latexit> V is simply a band-pass function!! ωo Qp ω2o Go <latexit sha_base64"ifelqncyhy/ooclcmovybe4vi6o">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</latexit> Gb () 2 And V2 is the integral of V. V2 <latexit sha_base64"gk8ln3sb5hacsziwvcdhx2b0">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</latexit> V s Gb G a s2 ()2 s ()2 <latexit sha_base64"ydv7tdgcijm/konvgafnsbca">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</latexit> V2 is simply a low-pass function!! Gb s2 ()2 s ()2 <latexit sha_base64"n6ahzx5tzxgmosprvwli4mpkmdm">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</latexit> ωo 2 ωo 2 Qp () Gb Go <latexit sha_base64"kiweafgxcxwothdtzedrpvigxem">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</latexit> All three filters achieved with one circuit. Pretty nifty. EE 230 active second-order filters 6
Interesting observation: An network must always have poles on the negative real axis no complex conjugates, so no Q P > 0.5. However, it can have complex conjugate zeroes. If we use the network in the feedback path of a closed-loop circuit, the zeroes of the bare network become the poles of the closed-loop transfer function.!!! To see this: V A T V B T V B V A N D G A A T V B T V A ( network turned around.) If A G T D N As claimed! EE 230 active second-order filters 7
So we need networks that have complex-conjugate zeroes. Two that do are the bridged-t networks. R 2 C 2 C C 2 V x R R 2 R C Note that because these are symmetric, it doesn t matter which side is input and which is output it s the same either way. EE 230 active second-order filters 8
Transfer function for a bridged-t circuit. R 2 As usual, start with node-voltage: sc (V i V x ) V x R sc 2 (V x V o ) sc 2 (V x V o ) V o V i R 2 V i C C 2 V x R V o Re-arranging each (sr C ) V i ( sr C sr C 2 ) V x (sr C 2 ) V o V x sr 2C 2 sr 2 C 2 V o sr 2 C 2 V i Substitute the second into the first to eliminating V x. Then carefully rearrange to get the transfer function in standard form. V o V i s 2 s R 2 C 2 R 2 C R C R 2 C 2 s 2 s R C R 2 C 2 R 2 C R C R 2 C 2 EE 230 active second-order filters 9
V o V i s 2 s R 2 C 2 R 2 C R C R 2 C 2 s 2 s R C R 2 C 2 R 2 C R C R 2 C 2 Again, since we will use this in the feedback loop of a closed-loop amp, we are concerned with the numerator, as it will become the denominator of the closed-loop gain. s 2 s R 2 C 2 R 2 C R C R 2 C 2 s 2 s ω o Q P ω 2 o ω o R C R 2 C 2 ω o Q P R 2 C 2 R 2 C Q P > 0.5 is possible for this function. Picking a few numbers to illustrate: with R kω, R 2 00 kω, C 2 C 2 0 nf, the parameters are ω o 0,000 rad/s and ω o / Q P 2000 rad/s. From these, we find that Q P 5. EE 230 active second-order filters 0
Delyiannis-Friend Insert the circuit into the feedback loop of an op amp circuit. But where is the input? We need to insert the input in a way that does not change the poles of the circuit. Because an ideal voltage source has no resistance, we can connect the source anywhere between ground and a component connected to ground without affecting the transfer function. So we can connect an input source between R and ground. Re-drawing the circuit to make it slightly simpler. R R 2 C C 2 C 2 V i R C R 2 V o EE 230 active second-order filters
Delyiannis-Friend C2 Calculating the transfer function is straight-forward. Once again, start with a couple of node-voltage equations. Vi Vx R sc Vx sc2 (Vx sc Vx R2 ) Vi R Vx Re-arranging each: Vi ( sr C sr C2 ) Vx Vx sr2 C Substitute in for Vx: Vi ( sr C sr C2 ) R2 C sr C2 sr2 C sr C2 Multiply by sr2c, collect terms, and re-arrange to get the transfer function. Vi EE 230 sr2 C s2 (R C R2 C2 ) s (R C R C2 ) active second-order filters 2
Vi sr2 C s2 (R C R2 C2 ) s (R C R C2 ) R C2 s ωo s2 s R2 C2 R C R2 C2 R2 C R C R2 C2 Go ωo QP R2 C2 R2 C s ωo QP s2 s ωo QP ω2o ωo Go QP R C2 This is a bandpass filter. Plug in some typical numbers: R k!, R2 00 k!, C C2 0 nf. Of course, we already did this calculation. (Slide 5.) ωo 0,000 rad/s (fo 59 Hz.) ωo / QP 2000 rad/s QP 5 ωo Go 05 rad/s Go 50 QP EE 230 active second-order filters 3
Design example Design a second-order bandpass filter with f o 3200 Hz (ω o 20,000 rad/s), Q P 0, and G o > 50. Since there are 4 components, it would seem that there are more enough degrees for freedom to specify all three parameters. However, because the resistors and capacitors are always combined as time constants, there isn t enough flexibility to specify all three parameters arbitrarily. You can design for 2 of the 3 independently, and then take what you get for the third. Re-writing each of the parameters directly in terms R s and C s: ω o R C R 2 C 2 Q P R 2 R C 2 C C C 2 G o R 2 R C 2 C (You might want to derive these for yourself.) EE 230 active second-order filters 4
It is typical to choose C C2 C. In that case, the previous set of equations reduces to ωo R R2 C QP 0.5 R2 R Go R2 2R 2Q2P We can choose a value of C and then calculate the R and R2 needed to obtain the desired ωo and QP. Pick C 0 nf. R 250 Ω 2ωo QC 2Q R2 ωo C 00 kω Then, Go 200. (You get what you get.) EE 230 active second-order filters 5
<latexit sha_base64"8rjscjy0hwgxksxeo0anxhd4a">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</latexit> <latexit sha_base64"g9wsxidqrdymxiupzn2qs8a2nma">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</latexit> Sallen-Key Another single-amp bi-quad is the Sallen-Key configuration. It is fundamentally different than the DF circuit. The amp is configured in a non-inverting fashion in fact, it usually has unity gain. The general form of the circuit is shown below. To find the transfer function, we start by noting that the unity-gain feedback means that V i Z Z 2 V x V V o Z 3 Z 4 V o V V V o Writing node equations: V i V x Z V x V o Z 2 V x V o Z 4 (effectively Z 2 and Z 4 are in parallel.) V x V o Z 2 V o Z 3 EE 230 active second-order filters 6
Re-arranging the equations: Vi <latexit sha_base64"uyzfn2edcvcqkxxt0d6mqrrvf8i">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</latexit> Vx <latexit sha_base64"uwyzcnjble9jfqy50mvd764qvms">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</latexit> Z Z Z4 Z3 Z Z Z4 Vx Substitute the second into the first to eliminate Vx Vi <latexit sha_base64"8md0rql85/zgiufka3nxba5ta">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</latexit> Z Z Z4 Z3 Z Z Z4 Multiply everything out Vi <latexit sha_base64"zjpmshl6sa50gvbmht9d7heb9u">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</latexit> Z Z Z Z Z4 Z3 Z3 Z4 Z3 Z Z Z4 And wrangle it into a transfer function T Vi <latexit sha_base64"zwhscm44qry8shhqgfq6mkqhv4a">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</latexit> EE 230 Z3 Z Z3 Z Z4 Z3 active second-order filters 7
T Vi <latexit sha_base64"zwhscm44qry8shhqgfq6mkqhv4a">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</latexit> Z3 Z Z3 Z Z4 Z3 It doesn t mean much until specific impedances are inserted. C4 Z R R2 Z3 sc3 Z4 sc4 <latexit sha_base64"oqnsy5we3fr3wie90ptcifkfiu">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</latexit> Vi R R2 C3 <latexit sha_base64"bo4t3oogvb2umyyf6kznngtgxc">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</latexit> T sr2 C3 sr C3 s2 R R2 C3 C4 Make it high-pass by swapping resistors and capacitors. <latexit sha_base64"/agsb4h03wzrdnpetxokyrujln8">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</latexit> R R2 C3 C4 s2 s R C4 R2 C4 <latexit sha_base64"hzmf8nc8leo0bxmwmch7sbjqqiw">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</latexit> Clearly low-pass with: ω2o <latexit sha_base64"rovmixteivluqwz5yuzygaa6je">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</latexit> EE 230 R R2 C3 C4 <latexit sha_base64"x2nft/3z43vj/9xjadq8/mqly">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</latexit> ωo QP R C4 R2 C4 active second-order filters 8