Announcements

• mid-term Thursday (Oct 27th)

• Project ideas to me by Nov 1st latest

• Assignment 4 due tomorrow (or now)

• Assignment 5 posted, due Friday Oct 21st

Lecture 13 Overview

• Active Filters

• Positive feedback

• Schmitt trigger/oscillator

• Analysing a more complex opamp circuit

Recap: Opamps

• DC coupled, very high gain, differential amplifier.• Feed part of the output back into the inverting

input to get stable operation in the linear amplification region

• Golden rules under negative feedback:• The voltage at the inputs is the same (v+=v-)• No current flows into the opamp (i+=i-=0)

What about complex impedances?

S

F

S

out

Z

Z

V

V

S

F

S

out

Z

Z

V

V1

Active low-pass filter

FF

SF

FF

FF

FFF

S

F

S

out

CRj

RRjA

CRj

RZ

CjRZ

Z

Z

V

VjA

1

/)(

1

1111

)(

e.g. RF/RS=10; 1/RFCF=1

Max Amplification: RF/RS

Low pass factor: 1/(1+ jωRFCF)Cut-off frequency (-3dB = 1/√2)when ωRFCF=1, ie ω0=1/RFCF

Active high-pass filter

SS

S

F

SS

F

SSS

S

F

S

out

CRjR

R

CjR

RjA

CjRZ

Z

Z

V

VjA

11

1

1)(

1

)(

e.g. RF/RS=10; 1/RFCF=1

Max Amplification: RF/RS

High pass factor: 1/(1+ 1/jωRSCS)Cut-off frequency: ωRSCS=1

Active band-pass filter

Combine the two:

)1)(1()(

SSFF

SF

CRjCRj

CRjjA

Advantages of active filters: 1)no inductors (large, expensive, pick-up)2)buffered (high input impedance, low output impedance) – so filter performance independent of source and load; can cascade filters

Spot the Difference!

Positive feedback

• Consider what happens when there is a perturbation:

• Negative feedback cancels out the difference between the inputs, providing stable amplification• Positive feedback drives opamp into saturation (at an exponential rate)

SOUT Vv

So what's the use of positive feedback?Comparator:

Simple version - no feedback

Amplifier saturates when v+-v- >10μV

Set v- = vref =0, input signal vsignal on v+:

Comparator compares two input voltages, vref and vsignal.if vsignal> vref the output voltage is highif vsignal< vref the output voltage is low

Real world problem: noisy signal

vref

Small noise fluctuations generate spurious additional pulses before/after the main pulse

(inverted output)

The Schmitt Trigger:Comparator with positive feedback

21

1

RR

RVv S

V15 So Vv

V15 So Vv21

1

RR

RVv S

5.7when

so

v

vvvi

5.7when

so

v

vvvi

Try R1=R2, VS=+/-15V

• The circuit has 2 thresholds, depending on the output state • Gives a clean transition.• Known as hysteresis• Choose resistors to set required difference between the two voltage levels

in this state:

in this state:

Oscillator: We can create a clock

This sets the threshold levels

This sets the clock period ( RC)

v+=vo/2v-=vC

vo sets the voltage at v+ and charges the capacitor

What does this circuit do?

• Break it down into elements

What does this circuit do?

• Two buffers (voltage followers): vo=vi

What does this circuit do?

• Inverting amplifier

1Gain S

F

R

R

• Voltage drop from point X to point Y = 2Vi , so:

What does this circuit do?

C

i

ZR

vi

2

• Also, at point Z,C

iiio ZR

RvviRvv

2

R

XY Z

VXZ

C

i

ZR

vi

2

• Now, gain

• Multiply top and bottom by (1-jωRC):

C

iiio ZR

RvviRvv

2

RZ

RZ

ZR

RZR

ZR

R

v

vg

C

C

C

C

Cin

o

221

RCj

RCj

RCj

RCj

g

1

11

1

1Ag

jAeCR

RCjCR

RCjRCj

RCjRCjg

222

222

1

21

)1)(1(

)1)(1(• None of the amplifiers change the amplitude:

222

222

222

222

1

2

11

12

tanCR

RC

CRCR

CRRC

a

b

What does this circuit do?

R

2221

2tan

CR

RC

• The circuit is a phase shifter!• Output voltage is a phase shifted version of the input• Vary R to vary the degree of phase shift. Nice audio effect – but also…• Very useful for communications applications (e.g Electronically steerable microwave antenna arrays: PATRIOT= "Phased Array TRack to Intercept Of Target" )

1g

What does this circuit do?

• The circuit is a phase shifter!• Output voltage is a phase shifted version of the input• Vary R to vary the degree of phase shift• Very useful for communications applications (e.g Electronically steerable microwave antenna arrays: PATRIOT= "Phased Array TRack to Intercept Of Target" )

Non-Ideal Opamps: Basic Cautions

1) Avoid Saturation• Voltage limits: VS

-< vOUT < VS+

• In the saturation state, Golden Rules of opamp are not valid

Basic Cautions for opamp circuits

2) Feedback must be negative (inverting) for linear behaviour3) There must always be negative feedback at DC (i.e. when ω=0).

• Otherwise any small DC offset will send the opamp into saturation• Recall the integrator: In practice, a high-resistance resistor should be added in parallel with the capacitor to ensure feedback under DC, when the capacitive impedance is high

4) Don't exceed the maximum differential voltage limit on the inputs: this can destroy the opamp

Frequency response limits

• An ideal opamp has open-loop (no feedback) gain A=• More realistically, it is typically ~105-106 at DC, dropping to 1 at a frequency, fT=1-10 MHz• Above the roll-off point, the opamp acts like a low-pass filter - and introduces a 90º phase shift between input and output• At higher frequencies, as the open-loop gain approaches 1, the phase shift increases• If it reaches >180º degrees, and the open loop gain is >1, this results in positive feedback and high frequency oscillations• The term "phase margin" refers to the difference between the phase shift at the frequency where the gain=1 (fT) and 180º

Frequency response limits

• Open loop cut-off frequency, f0 (also known as open loop bandwidth) is usually small (typically 100Hz) to ensure that the gain is <1 at a phase shift of 180º• Closed-loop gain (gain of amplifier with feedback) begins dropping when open loop gain approaches RF/RS (in the case of the inverting amp)• Cut off frequency will be higher for lower closed-loop gain circuits

Inverting amplifier

Slew rate (or rise time)• The maximum rate of change of the output of an opamp is known as the slew rate (in units of V/s)

• The slew rate affects all signals - not just square waves• For example, at high enough frequencies, a sine wave input is converted to a triangular wave output due to limited slew rate

max0 dt

dvS o

square wave input

Slew rate example• Consider an inverting amplifier, gain=10, built using an opamp with a slew rate of S0=1V/μs.• Input a sinusoid with an amplitude of Vi=1V and a frequency, ω.

• For a sinusoid, the slew rate limit is of the form AViω<S0.• We can therefore avoid this non-linear behaviour by

• decreasing the frequency (ω)• lowering the Amplifier gain (A)• lower the input signal amplitude (Vi)

• Typical values: 741C: 0.5V/μs, LF356: 50V/ μs, LH0063C: 6000V/ μs,

)cos()cos( tAVvtVv ioii

0max

0)sin( SAVdt

dvtAV

dt

dvii

o

56

max

101010 dt

dvo