FEEDBACK, STABILITY and OSCILLATORSeelinux.ee.usm.maine.edu/courses/ele343/343 Class...

FEEDBACK, STABILITY and OSCILLATORS

à FEEDBACK, STABILITY and OSCILLATORS

- STABILITY OF FEEDBACK SYSTEMS- Example 1: ANALYSIS and DESIGN OF PHASE-SHIFT-OSCILLATORS- Example 2: ANALYSIS and DESIGN OF WIEN BRIDGE OSCILLATORS

à FEEDBACK, STABILITY OF FEEDBACK SYSTEM, OSCILLATORS

ü STABILITY OF FEEDBACK SYSTEM

Vout = Avo.HVin - b.VoutLAv f =

VoutÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄVin

=Avo.HVin - b.VoutLÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ

Vinfi Av f =

AvoÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ1 + b.Avo

The negative feedback effectively reduce the gain.

Note that if » b.Avo » >> 1 then, Av f >Avo

ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄAvo.b

>1

ÄÄÄÄÄÄÄb

For non - inverting opamp amplifier Av f =1

ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄR1ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄR1+RF

Univ. of Southern Maine 1 Prof. M. G. Guvench

Phase at w Ø ¶ (number of zeros -number of poles) . 90°

Even if Avo (0) may be very large , at high frequencies » AvoHjwL » can become very small; making the approximation aboveinvalid.

Av f HjwL =Avo HjwL

ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ1 + Avo HjwL.b HjwL

For DC, the negative feedback results in a (+) sign in the denominator. Therefore, » Av f HjwL » will always be smaller than» AvoHjwL |.But, if the inputs are reversed or somehow AvoHjwL.bHjwL changes its sign, the feedback becomes positive, the denomina-tor can become < |1|, therefore, Av f with feedback becomes even greater than Avo of the operational amplifier.

Interesting Point: Denominator ª 0 fl "Instability" fl Oscillations

Conditions for Instability HOscillationsL 1 + Av HjwL.b HjwL ∫ 0

Equivalently J Re @1 + Avo HjwL.b HjwLD = 0Im @1 + Avo HjwL.b HjwLD = 0

a. 1 + Re @Avo HjwL.b HjwLD = 0 or Re @Avo HjwL.b HjwLD = -1b. Im @HAvo HjwL.b HjwLD = 0 fi Phase of HAvo.bL = 0, ¡180, ¡360

Note that, it is impossible to make Re @1 + Avo HjwL.b HjwLD = 0 for phase = 0 or ¡ 360

ELE 343 FeedbackStabilityOscillatorsActivefilters-Y.


For instability :J [email protected] = ¡180 °[email protected] = -1 N or, J [email protected] = ¡180 °» Avo.b » = 1 N



Stable system since » Avo.b | < 1 at the critical frequency(2) which satisfies the phase condition.

Definitions/Conditions for Stability:

Loop Gain = Avo.b Worst case: b=1 for passive circuits



fM (Phase Margin) > 30 ~ 60° - 70° for a good internally compensated opamp

Gain Margin > 10 dB

xxxxxxxxxxxxxxxxxxxxx

OSCILLATOR ª Unstable but at a unique frequency so that it generates a sine wave with minimal distortion and minimaldrift of oscillation frequency.



à Example 1: ANALYSIS and DESIGN OF PHASE-SHIFT-OSCILLATORS

Note that for an oscillator to oscillate it does not need a signal, therefore, the signal input shown in the block diagrams can be eliminated.

Amplifier Gain = Av Beta HjωL = β Hjω )

b (jw) circuit

Beta HjwL = b HjwL =V2 HjwLÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄV1 HjwL



Educated conclusion: use higher number of RC's than 3 to achieve higher slope for better frequency stability.

@VD = @ZD * @ID V is known, I is unknown stimuli

ikjjjjjjj V1

00

y{zzzzzzz =

ikjjjjjjjjjjjjjjj

R + 1ÄÄÄÄÄÄÄÄÄÄÄÄj wC -R 0

-R 2 R + 1ÄÄÄÄÄÄÄÄÄÄÄÄj wC -R

0 -R 2 R + 1ÄÄÄÄÄÄÄÄÄÄÄÄj wC

y{zzzzzzzzzzzzzzz ikjjjjjjj I1

I2

I3

y{zzzzzzzand output voltage, V2 = R.I3

One can apply Cramer's Rule to solve for I3 and the output voltage V2= R. I3 . In Mathematica the same calculation canbe done by using linear solver which yields { I1 , I2 , I3} and pick out I3 , the third term from the list, and multiply it byR, as done below.

V2 = ExpandAllAR ∗TakeALinearSolveAikjjjjjjjjjjjjjjR + 1

j ω C−R 0

−R 2 R + 1j ω C

−R

0 −R 2 R + 1j ω C

y{zzzzzzzzzzzzzz, ikjjjjjjj V100 y{zzzzzzzE, 83<E ê.

ω −>1

j R C yE

99 V11 + 6 y + 5 y2 + y3

==V1

1 + 6 y + 5 y2 + y3ê. y −>

1

j ω R C

V11 + 1

C3 j3 R3 ω3 + 5C2 j2 R2 ω2 + 6

C j R ω

Rearrange V2 = V1 1H1 − 5

ω2 R2 C2L + 1

jω R C H6 − 1

ω2 R2 C2L

For the phase to reach 0 or ¡180° imaginary term should disappear,



Therefore, 6 −1

ω2 R2 C2= 0 ⇒ ω 0 = $%%%%%%%%%%%%%%%%%%1

6 R2 C2

Then,V2

V1=ikjjjjjj 1

1 − 5. 1ω2 R2 C2

y{zzzzzzω2 = 16 R2 C2

=1

1 − 5.6=

1

1 − 30=

1

−29

If Avo ≥ ikjjj 1

βy{zzzω 0

2= 16 R2 C2

then oscillations will start and grow at ω0.

Summary:

If » Avo » ¥ 29, If Avo < 0 (i.e. inverting)

Oscillations will start and grow up at ω0 =1è!!!!6 RC

Av f =Avo

1 + β HjωL Avo=

Avo

1 − 1= ∞



à Example 2: ANALYSIS and DESIGN OF WIEN BRIDGE OSCILLATORS



Beta = β HjωL =Zb

Za + Zb=

11Rb

+jω CbIRa + 1jω Ca

M + J 11Rb

+jω CbN

=1IRa + 1

jω CaM I 1

Rb+ jω CbM + 1

=1A Ra

Rb+ j Iω Ra Cb − 1

ω Rb CaM + Cb

Ca+ 1E

=1I Ra

Rb+ Cb

Ca+ 1M + j Iω Ra Rb − 1

ω Rb CaM

When ω2 = ω02 =1

Ra Rb Ca Cbthe imaginary term disappears

β Hj ω0L =1

RaRb

+ CbCa

+ 1

If H+L feedback is provided Av f =Avo

1 − β HjωL Avo

at ω = ω0 Av f Hjω0 L Avo

1 − β Hjω0L Avo

or Av f Hjω0 L ∞ if Avo ≥1

β Hjω0L =Ra

Rb+Cb

Ca+ 1

For Ca = Cb = C , Ra = Rb = R Avo ≥ 3



Wien - Bridge Oscillator

ikjj Rf

R1+ 1y{zz ≥

Ra

Rb+Cb

Ca+ 1

Amplifier Design :

Condition 1 : Zin >> Zab where Zab = HZa êê ZbLω = ω0

Condition 2 : Avo > ikjjj 1

β

y{zzzω = ω0

In order to control the amplitude(1) Small Signal Gain > minimum needed for oscillation, (2) Average gain reduces and becomes less than ( 1

βM

ω= ω0

at the amplitude of oscillations it is required to stabilize.

Amplifer Chs. with Saturating Gain



Inverting Amplifier with Gain Saturation

Low amplitudes RF = RF1Mid amplitudes RF ≡ RF1 êê RF2High amplitudes RF = RF1 êê RF2 êê RF3Av f =

ikjjjjjjj 1β

1 + 1β Av

y{zzzzzzz > 1

β



H b.Av L ¥ 1 b.Av > 1 b.Av ª 1 HAvLLarge

Signal

≤ HAvLSmallsignal



ACTIVE FILTERS

à ACTIVE FILTERS

- PASSIVE FILTER + AMPLIFIER- FILTER IS INTEGRATED IN THE FEEDBACK LOOP- TRUE ACTIVE FILTERS IMPLEMENTED WITH DEPENDENT SOURCES

a. ACTIVE TWO-POLE SALLEN-KEY LOW-PASS FILTERb. ACTIVE TWO-POLE SALLEN-KEY HIGH-PASS FILTERc. ACTIVE TWO-POLE SALLEN-KEY BAND-PASS FILTER

à ACTIVE FILTERS

"Active" Filter ∫ Passive Filter Components + OpAmp as amplifier + OpAmp as dependent source

à Example 1: PASSIVE FILTER + AMPLIFIER

Advantage: 1. RL does not load the filter; therefore, it does not affect the frequency response.2. The filter can actually amplify the signal in its path. (Insertion gain vs insertion loss of a passive filter)



observe that ω− 3 dB = 1R Thevenin. C

≠ 1R C

and depends on loading resistance RL.

At higher frequencies, frequency response of "real opamp" will alter the frequency response of the active filter.* Note that the danger of instability at the frequency where phase angle ª 180 °* If the opamp is internally frequency compensated it can introduce no more than 90° phase while its gain is still greaterthan "0 dB". This implies that:Overall filter will never have a gain greater than 0 dB when phase = -180°ï stable.



à Example 2: FILTER IS INTEGRATED IN THE FEEDBACK LOOP

If the opamp has wide enough GBW and low frequency gain;

Avf HjωL =∆ Vout

Vin=

Avo

1 + β Avo>

1

β HjωLThis has the effect of Low Pass ö Low Reject (High Pass)

High Pass ö High RejectBand Pass ö Band Reject

A Good "Band Reject" Filter

Remark: A good Low / High / Band Reject actually reduces the signal in the reject band significantly, i.e. gain << 1.



à Example 3: TRUE ACTIVE FILTERS IMPLEMENTED WITH DEPENDENT SOURCESa. ACTIVE TWO-POLE SALLEN-KEY LOW-PASS FILTERb. ACTIVE TWO-POLE SALLEN-KEY HIGH-PASS FILTERc. ACTIVE TWO-POLE SALLEN-KEY BAND-PASS FILTER

a. ACTIVE TWO-POLE SALLEN-KEY LOW-PASS FILTER

Equivalent circuit:



Node Equations:



ikjjjj VinR1

+ VO s C1

0

y{zzzz = JHG1 + G2 + sC1L −G2−G2 HG2 + sC2LN JV1

V0N

ikjjjj VinR1

0

y{zzzz = JHG1 + G2 + sC1L −HG2 + sC1LH−G2L HG2 + sC2L N JV1V0N

Using Cramer' s Rule to getVOVinR1

V0VinR1

=1I VinR1

M .

DetAikjjjjHG1 + G2 + sC1L VinR1H−G2L 0

y{zzzz EDetAJHG1 + G2 + sC1L −HG2 + sC1LH−G2L HG2 + sC2L N E

=1I VinR1

M .−I Vin

R1M H−G2LHG1 + G2 + sC1L HG2 + sC2L − G2 HG2 + sC1L

VO

Vin= H HsL =

G1.G2

s2 C1 C2 + sC1 G2 + sC2 HG1 + G2L + G2 HG1 + G2L − G22 − G2 sC1

=G1.G2

s2 C1 C2 + sC2 HG1 + G2L + G1.G2

=

1HR1 C1L HR2 C2Ls2 + s I 1

R1 C1+ 1

R2 C1M + I 1

R1 C1M I 1

R2 C2M

Rename :1

R1 C1 R2 C2= ω0

2 ikjjj 1

R1 C1+

1

R2 C1

y{zzz = 2 ζ ω0

H HsL =ω02

s2 + 2 ζω0 s + ω02 No inductor!

The response of this circuit is similar to the response of an RLC - Low Pass passive filter shown below. Except that thereis no need to employ bulky, lossy and expensive inductors to implement it .



Vout ê Vin =

1êsCêHR + sL + 1êsCL = 1êHsRC + s2 LC + 1L = H1êLCLêHs2 + R êLs + 1êLCLwhere ω02 ≡ 1êLC is the "resonant frequency" and

2 ζ ω0 ≡ R êL is the "damping factor"

H =ω02

s2 + 2 ζ ω0 s + ω02a = 1, b = 2 ζ ω0, c = ω02

=constantHs − p1L Hs − p2L

p1, p2 =−b ±

è!!!!!!!!!!!!!!!!!!!b2 − 4 ac

2 a= −

b

2± $%%%%%%%%%%%%%%%%%%%%ikjj b

2

y{zz2 − c

=

−ζ ω0 ±"###########################Hζ ω0L2 − ω02 = −ζ ω0 ± jω0

"##############1 − ζ 2 complex conjugate

H =» constant »» s − p1 »» s − p2 »



b. ACTIVE TWO-POLE SALLEN-KEY HIGH-PASS FILTER

Insertion Loss



c. ACTIVE TWO-POLE SALLEN-KEY BAND-PASS FILTER



Node Equations :

ikjjjj VINR

+ KV2R

0

y{zzzz =ikjjjjjj 1

R+ 1

R+ sC + sC −sC

−sC 1R

+ sC

y{zzzzzz JV1V2N or

ikjjjj VINR

0

y{zzzz =ikjjjjjj2 H 1

R+ sCL −HsC + K

RL

−sC H 1R

+ sCL y{zzzzzz JV1V2N

Using Cramer' s rule,

KV2 = VOUT = K

DetAikjjjj2 H 1R

+ sCL VINR

−sC 0

y{zzzzEDetAikjjjjjj2 H 1

R+ sCL −HsC + K

R

−sC H 1R

+ sCLy{zzzzzzE= K

+ s C VINR

2 H 1R

+ sCL H 1R

+ sCL − sC HsC + KRL



VOUT

VIN= K

sCR

2 HsCL2 + 4 H sCRL + 2 H 1

RL2 − HsCL2 − sC

R K

after dividing by C2,

H HsL = K

sCRHsCL2 + H4 − KL sC

R+ 2 H 1

RL2

H HsL = K

sR C

s2 + H4 − KL 1R C´̈ ¨¨¨̈ ¨¨¨̈ ≠ Æ¨¨¨¨̈ ¨̈ ¨

s

2 γ = 2 ζ ω n

+ J è!!!!2

R CN2

´̈ ¨¨¨̈¨≠ Æ¨¨̈ ¨̈ω n

2

ωn =

è!!!!2

R Cω n = ω 0

2 + γ2

The damping factor : ζ =H4 − KL 1

R C

2 è!!!!2

R C

=H4 − KL2 è!!!!2

Poles = −γ ± jω 0 Resonance is at ω = ω 0

BandWidth = BW = ω 2 − ω 1 = ω 0 + γ − Hω 0 − γL = 2 γ

Quality Factor of a Resonant Circuit : Q =ω peak

BW>

ω 0

2 ζ=

è!!!!2

4 − K



» Hmax » =K jω

RCH+jωL2 + H 4−KR C

L jω + Hω0L2 = K 1

4 − K=

Kè!!!!2

è!!!!2

4 − K= K

Qè!!!!2

K 4 » H »max ∞ unstable! Oscillator



FEEDBACK, STABILITY and OSCILLATORSeelinux.ee.usm.maine.edu/courses/ele343/343 Class...

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