Active Filters – an Introduction - Penn Engineeringese319/Lecture_Notes/Lec_23...ESE319...

ESE319 Introduction to Microelectronics

12008 Kenneth R. Laker updated 07Dec09 KRL

Active Filters – an Introduction

Active Filters1. Continuous-time or Sampled-data2. Employ active elements (e.g. transistors, amplifiers, op-amps)

a. inductor-less (continuous-time)b. inductor-less & resistor-less (sample-data)c. gain ≥ 1in passband

Vin(s) V

out(s)

Filter circuitG(s)

+

--

+



Active Filters – an Introduction

Vin(s) V

out(s)

Filter circuitG(s)

+

--

+

G s=aM s

MaM−1 sM−1....a1 sa0

sNbN−1 sN−1....b1 sb0

G s=aM sz1sz2........szM s p1s p2........s pN

M ≤ N Filter Order = N



Ideal Filter Response Characteristics

∣G∣=∣G j∣=∣V out jV in j ∣

Stop-bandPassband Stop-band Passband

Passband

Lower Stop-band

Upper Stop-band

Stop-band

Lower Passband

Upper Passband

|G| |G|

|G| |G|

1 1

1 1

0 0

00

P P

PL PH SL SH

Low-pass (LP) High-pass (HP)

Bandpass (BP) Bandstop (BS)



Practical Lowpass Filter SpecificationKey specs:1.2. A

max

3.4. A

min

f B=P /2

f S=S /2

Filter cost increases!1. A

max -> lower

2. Amin

-> larger3. -> larger4. -> 1 S /P

Passband Stop-band

Amax

Amin

Transition band

0

0 P Sz1 z2

selectivity factor = S

P

P

|G| (dB)



G s=aM sz1sz2........szM s p1s p2........s pN

=>

MatLab is a good tool for this task.

Filter Approximation – Design G(s)

|G|



Practical Bandpass Filter Specification

SL

PL=SU

PU

SL

PL≠SU

PU

Symmetric bandpass filter

Transition bands

Passband

0 Amax

Amin

Upper Stop-band

Lower Stop-band

SL PL PU SU

SLPL0

Q=PU−PL

0

|G| (dB) Selectivity factors



Cascade Filter DesignIf N = odd

(N- 1)/2

If N = even

G s=aM s



=a10 sa00sb10

∏ a2i s2a1i sa0i

s2b1i sb0i=∏G i s

i = 1i = 1 i = 0

(N- 1)/2

G s=aM s



=∏ a2i s2a1i sa0i

s2b1i sb0i=∏G i s

i = 1

N/2

i = 1

N/2

....G1(s) G2(s) G3(s) GN/2(s)Vin

VoutVo1

Vo2Vo3 Vo(N-1)/2

N = odd => G1(s) 1st order

N = even => G1(s) 2nd order



2nd order low-pass (LP)

G s=a0

s2s0

Q0

2

∣G j0 ∣=a002

G s=a2 s

2

s2s0

Q0

2

∣G j∞∣=a22nd order bandpass (LP)

2nd order high-pass (LP)

G s=a1 s

s2s0

Q0

2

∣G j0∣=a1Q0

j

j

0

0

0

2Q

0

2Q

0

0

X

X

X

X

O

Oo

j

0

0

2Q

0

X

X

z1=∞z 2=∞

z1=0z 2=0

z 2=∞z1=0

max

|G|a002

0max

max=01− 12Q2

∣a0∣Q

021− 1

4Q2

|G|

0

∣a2∣

max=0/1− 12Q2

∣a2∣Q /1− 14Q2

00

|G|

Gmax

Gmax

0.707 Gmax

a1Q /0

a1Q /20

0 00/Q

1 2

12=02

Gmax

Filter Type s-plane zeros/poles |G|



2nd order Notch (N)

|G|

|G|

|G|

0

0

0

2nd order LP Notch (LPN)

2nd order HP Notch (HPN)

G s=a2s2N

2

s2s0

Q0

2

N0

∣G j0 ∣=∣a2∣N2

02

∣G j∞∣=∣a2∣

G s=a2s2N

2

s2s0

Q0

2

N0

∣G j0 ∣=∣a2∣N2

02

∣G j∞∣=∣a2∣

X

X

X

X

X

X

O

O

O

O

O

O

j

j

j

0

0

0

0

0

0

0

0

0

0

0

N

N

N

N

N

N

max

max

0

2Q

0

2Q

0

2Q

Gmax

Gmax

∣a2∣

∣a2∣

∣a2∣

∣a2∣N2

02

∣a2∣N2

02

∣a2∣N2

02

∣a2∣2

1 2

12=02

Filter Type s-plane zeros/poles |G|



vO t =K v I t−td ⇒T j=∣T j∣e j tdIdeal transmission: ∣T j∣=K

j=− t d

=−d jd

=t dGroup Delay

2nd order All-Pass (AP)

G s=a2s2−s

0

Q0

2

s2s0

Q0

2

∣G j0 ∣=∣G j∞∣=∣a2∣

|G|

∣a2∣

0

j

00

0

X

X

O

O0

2Q0

2Q

0 0

−

−2



Delay Equalization Concept

DelayEqualizerDE

Total Equalized Delay tot =CDE

Cable or Filter tot =CDE

DelayEqualizer

Cable or Filter

equalized data

delay distorted

data



OP Amp Building Blocks

vO t =−1CR∫v I t dt0

t

V o sV i s

=−1sCR

=−int

s

int=1CR

Inverting Integrator Summer

V =−R fR1

V 1R3

R2R31

R fR1

V 2.

R2R2R3

1R fR1

V 3

v¿

v−¿



Two-Integrator-Feedback-Loop Active FilterV hp s=

K s2

s2s0

Q 02V is=

K

1 1Q

0

s02

s2

V is

V hp s=−V hp s1Q

0

s −V hp s02

s2K V i s

V i

V hp−0

sV hp=V bp

02

s2V hp=V lp

V hp −0

sV hp

02

s2V hp

−0

s−0

s

−0

s −0

s

-1

K

1Q

=>

02

s2V hp

−0

s V hp

V iV hp

1 /Q

−1

Kint=1CR

=0

V hp sV hps 1Q

0

s V hp s02

s2=K V i s=>

02

s2V hp 0

2

s2V hp−0

sV hp

−0

sV hp



Feedback Equations

V hp=− 1Q 0

s 02

s2 V hpK V i

Ghp s=V hp

V i=

K

1 1Q

0

s02

s2

=K s2

s20

Qs0

2



Feedback Equations IIV hp

V i=

1

K 1Q

0

s02

s2

=K s2

s20

Qs0

2High Pass Output:

Bandpass Output:V bp

V i=−0

sV hp

V i=−

K0 s

s20

Qs0

2

Lowpass Output:V lp

V i=−0

sV bp

V i=

K02

s20

Q s02



Implementation

Inverting Integrators

Summing Amp

V hp=−0

2

s2V hp−

1Q

0

sV hpK V i

R3

R2

R f

R1

R RC C

V hp

V bp

V lpV i

−0

sV hp=V bp

02

s2V hp=V lp

V hp=−R fR1 0

2

s2 V hpR2

R2R3 1 R fR1 −0

s V hpR3

R2R3 1 R fR1 V i



Implementation II

V hp=−02

s2 V hp−2R2R2R3 0

s V hp2R3R2R3

V i

R f=R1Set: And compare terms:

V hp=−02

s2 V hp−1Q 0

s V hpKV i

R3R2

=2Q−1Q=R2R32R2

⇒Q=12 1 R3R2 =>

circuit symbolic Eq.

spec/numerical Eq.

V hp=−R fR1 0

2

s2 V hpR2

R2R3 1 R fR1 −0

s V hpR3

R2R3 1 R fR1 V i



K – Q DependenceR3R2

=2Q−1

K=2R3R3R2

=2R3R2

1R3R2

= 2 2Q−112Q−1

=2− 1Q

Only Q or K can be the independent variable!

From previous slide: K=2 R3R2R3



Design Equations

R f=R1

R3R2

=2Q−1

KQ=R3R2

=2Q−1⇒K=2− 1Q

RC= 10

Given , choose C, calculate R

Choose Rf, Calculate R1 or vice-versa.

Given Q, choose R2, calculate R3 or vice-versa.

K is fixed by choice of Q.

We have two independent parameters ( and Q, or K) and threeindependent components (C, Rf (or R

1), and R2(or R

3)).

0=2 f 0

0



RestrictionsSince

When Q = 1/2:

V hp

V i=

K s2

s22002=

K s2

s0 2

We have 2 real and equal poles.

For Q > 1/2, we are restricted to complex conjugate poles.

K=2− 1Q 0⇒Q1/2



Adding Finite Zeros – (Notches)To be able to create notches in the response, we need anothersumming amplifier:

Where the weighted inputs come from the highpass,bandpass, and lowpass outputs of the feedback circuit.

V hp

V bp

V lp

V o



Notch CreationAll the output point transfer functions contain the samedenominator, so only the numerator terms will be affected:

G s=− RFRH V hpRFRBV bp

RFRLV lp

G s=−KRF /RH s

2−RF /RB0 sRF /RL02

s20 /Q s02

For a notch at , no connection is made to Vbp, i.e. =N

RL

RFRHRB

V hp

V bp

V lp

V o

RB=∞



“Big Picture” Filter Design Tasks1. Design G(s) from filter specs.2. Determine filter structure (block diagram) to realize G(s).3. Determine filter circuit(s) to implement structure.4. Determine component values.

Filter Design CAD Tools on the Market1. MatLab - Mathworks2. FILTER PRO – Texas Instruments3. Aktiv Filter – New Wave Instruments4. Filter Lab – Microchip5. Filter Wiz Pro – Schematica6. FilterCAD – Linear Technology



R2n=R fn=R1n=10



(MFM)

Active Filters – an Introduction - Penn Engineeringese319/Lecture_Notes/Lec_23...ESE319...

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