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PHYS 352

Signal Conditioning: Analog Filters

Analog Filters   reasons to use analog filters

  restrict bandwidth, improving signal-to-noise   accomplish some impedance matching

  along with amplifiers

  integration and differentiation

 note: this course is systems-level   not so concerned about noise characteristics of

individual transistors, for example   we want to examine general ways to condition

signals in measurement systems   so, coming up, a qualitative overview look at

analog filters

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Filters: General Issues   average out noise   eliminate high frequencies   but, in general

  1/f noise may also be present   signal may have high frequency content

you don’t want to filter out

  in general, need to consider bandpass

Want a Fast Roll-off   e.g. RC low-pass filter   what’s important is the

order of the polynomial in the denominator   determines how fast the

function drops off   20*n dB/decade, where n

is the number of “poles”

  cascade one low-pass filter after another (identical or different RC) for faster roll-off?

one pole, n=1

two poles, n=2

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Passive Analog Filters   simply cascading RC low-pass filters

does not work well because   though it does produce steeper

falloff, going as 20*n dB/decade, where n is the number of RC stages

  each successive stage loads the previous   filter response not just based

upon ω = 1/RC in each stage   the “knee” remains soft

  bandwidth   what cutoff frequency?

  roll-off: how fast?   sharp transition   stopband attenuation

  how deep?   passband/stopband ripple

  how flat?   phase shift

  what happens to output pulse shape if the phase response is not constant with frequency?

input output

desire output signal to resemble input; even if two signals at ω1 and ω2 are passed with A(ω1)=A(ω2), the output looks different if they have different phase shift

Key Filter Design Criteria

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  rise time   overshoot   ringing   settling time

  keep these in mind also; however, we'll focus on frequency domain analysis of performance

  both amplitude and phase response impact time-domain performance

Filter Performance in Time Domain

rise time

Bandpass Filters  RC circuits

  they can work; far from ideal though

 passive RLC filters   can achieve virtually any desired flatness

of the passband combined with sharpness of transition and steepness of falloff

  however, inductors are bulky, “expensive”, not lossless (finite resistance), prone to magnetic pickup of interference, winding-induced non-linearity

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Filter Responses   three common design types:

  Butterworth (maximally flat passband)   Chebyshev (steepest transition)   Bessel (maximally flat time delay)

  flattest passband response   at the expense of steepness in the transition   poor phase characteristics

  where n is the order of the filter (number of poles)   e.g. “3rd-order Butterworth filter…”

Butterworth Filter

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  in most applications, some variation allowed in the passband response, say less than 1 dB

  then, Chebyshev has steepest transition   but, phase characteristics still poor

  where Cn is the Chebyshev polynomial of the first kind of degree n and ε is a constant that sets the passband ripple

Chebyshev Filter

Chebyshev Polynomials   e.g. Cn(f/fc) is a function evaluated at f/fc

  n = 2: 2x2 – 1   n = 3: 4x3 – 3x   n = 4: 8x4 – 8x2 + 1   n = 5: 16x5 – 20x3 + 5x   n = 6: 32x6 – 48x4 + 18x2 – 1

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Butterworth versus Chebyshev   Butterworth flat passband looks good, but is

already rolling off near the cutoff frequency, unlike Chebyshev

  Chebyshev amplitude response variations are spread throughout passband

  RLC components of finite tolerance will cause filter to deviate from the predicted response   real-life Butterworth filter has some passband

ripple anyway

Bessel Filter   is a linear-phase filter  amplitude response is less steep than

Butterworth   time-domain properties are (naturally)

better than Butterworth or Chebyshev   it takes a higher order Bessel filter to

give the same steepness of the frequency response; but, linear phase (pulse-shape fidelity) may be worth it

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Bessel Filter Transfer Function

 where θn(s) is a reverse Bessel polynomial function, for example:   n=1; s+1   n=2; s2+3s+3   n=3; s3+6s2+15s+15

  so, again for example, a 3rd order Bessel low-pass transfer function is (in frequency units of ω0):

Example: 3rd Order Bessel Filter

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Linear-Phase Filter   re-examine input x(t) and output y(t)

  so if θ(ω) is a linear function of ω or is a constant (independent of ω), it just produces a time shift that’s the same for all frequencies

  no pulse-shape distortion in the output (provided amplitude response is flat in the passband)

y t( )= A ω1( )cos ω1t +φ1+θ ω1( )( ) + A ω2( )cos ω2t +φ2 +θ ω2( )( )y(t) = A(ω1)cos(ω1(t +

θ(ω1)ω1

) + φ1) + A ω2( )cos ω2 (t +θ ω2( )ω2

)+φ2⎛⎝⎜

⎞⎠⎟

Bessel: Step Response

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Active Filters   purpose:

  to get rid of inductors   to provide gain between sections (buffer)   to avoid cascading impedance mismatch

  amplifier circuits with feedback accomplish this   op-amps are involved in the design

  output attempts to make the voltage difference between the inputs zero (i.e. it's not a fixed gain device...rather, the gain adjusts in order to zero the voltage difference)

  inputs draw no current (i.e. infinite impedance)

Aside: Op-Amps  what's inside an op-amp...

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Aside: Op-Amp Analysis   inverting amplifier

  point B is at ground so therefore point A is also   voltage across R2 is Vout; across R1 is Vin

  no current flows into op-amp, so Vout/R2 = −Vin/R1

  thus, gain: Vout/Vin = −R2/R1

  input impedance: Zin = R1

  for the whole circuit, not just the op-amp

  I = Vin/Z (since V− is at Vin)   Vout = I(R+Z)   Vin – IinR = Vout   Vin – IinR = IZ – IinR = I(R+Z)   thus, Iin = –I and Zin = Vin/Iin = –Z

  this turns a capacitor into an “inverted” inductor

  ZC = –j/ωC → Zin = j/ωC

Negative-Impedance Converter

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things to the right of NIC serve as the Z of the NIC (see previous circuit)

Gyrator   easy to show that this has Zin = R2/Z

  turns ZC = 1/jωC → Zin = jωCR2

  it's a simulated inductor with L = CR2

  establishes that inductorless filters are possible   limitation: one end of inductor is grounded

 2-pole filter, roll-off 40 db/decade  advantage: knee is sharper than

cascaded RC

Sallen-Key Filter

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VCVS Filter   in general these are known as voltage-

controlled, voltage-source filters (VCVS filters); but, Sallen-Key filter is used interchangeably   which itself is a generalization of the original Sallen-

Key filter which is strictly a unity gain filter

  these are 2-pole filters   cascade any number of 2-pole VCVS to

generate higher-order filters (individual filter sections not necessarily identical)

VCVS Filter Table   suitable choice of components generates desired

response

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VCVS Filter Table cont'd  design table is for equal R, equal C

  adjust the gain and RC = 1/2πfnfc for desired response

  if multiple stages, each have different gain and fn

  for high pass, interchange R and C, gain is the same, but inverted fn (“high” fn = 1/fn from “low” table)

  for band pass, put a low-pass and high-pass together

  in general, R's and C's don't have to be equal

  gain K of the op-amp determines response and is set by the voltage divider resistors at the output

  can re-write (equal R's and C's):

  gain linked to Q-factor of response   high Q-factor as G → 3

VCVS Filter Circuits

damping term in forced harmonic motion

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Filter Response: Q-Factor

Feedback: Sallen-Key Filter   C2 provides positive feedback   at low f, C's are open, output

is just the input amplified (gain set by R3 and R4)   Vout = G

  at high f, C's are short to ground   Vout = −G (ω0/ω)2

  around cutoff, it's the positive feedback “enhancing” the signal, sharpening the transition   Vout = -jGQ

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Sallen-Key Limitation   gain must be less than 3   otherwise, the circuit becomes unstable

  resultant high Q with gain = 3 exhibits large sensitivity to variations in R3 and R4   imagine Q = 10, G = 2.9; if the “gain” resistors

change by 1%, the new Q = 16...unstable   imagine Q = 1, G = 2; if “gain” resistors change by

1%, the new Q = 1.02...much better behaved

  higher gain (negative Q) leads to oscillations

Concluding Remarks: Active Filters

  there are books and books on active filters

  many ways to achieve desired response (Bessel, Chebyshev, etc.)

  some designs attempt to compensate for non-ideal behaviour of the op-amp