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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 No current flows into the opamp (i + =i + =0) =v )**

What about complex impedances?

*V out V S*

*Z F Z S V out V S*

1

*Z F Z S*

Active low-pass filter

*A*

(

*j*

)

*V out V S*

*Z F Z S*

1

*Z F*

1

*R F*

1 1

*j*

*C F Z F*

1

*R F j*

*R F C F A*

(

*j*

) 1

*R j F*

/

*R R S F C F*

Max Amplification: R F /R S Low pass factor: 1/(1+ j ωR F C F ) Cut-off frequency ( 3dB = 1/√2) when ωR F C F =1, ie ω 0 =1/R F C F e.g. R F /R S =10; 1/R F C F =1

Active high-pass filter

*A*

(

*j*

)

*V out V S*

*Z F Z S Z S*

*R S*

1

*j*

*C S A*

(

*j*

)

*R S*

*R F*

1

*j*

*C S*

Max Amplification: R F /R S High pass factor: 1/(1+ 1/j ωR S C S ) Cut-off frequency: ωR S C S =1

*R F R S*

1 1 1

*j*

*R S C S*

e.g. R F /R S =10; 1/R F C F =1

Active band-pass filter Combine the two:

*A*

(

*j*

) ( 1

*j*

*R F j*

*C F R F*

)( 1

*C S*

*j*

*R S C S*

) 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

• 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)

*v OUT*

*V S*

So what's the use of positive feedback?

Comparator: Comparator compares two input voltages,

*v ref*

if

*v signal*

>

*v ref*

the output voltage is high if

*v signal*

<

*v ref*

the output voltage is low and v

*signal*

.

Amplifier saturates when

*v + -v -*

>10 μV Simple version - no feedback Set

*v = v ref*

=0, input signal v signal on

*v +*

:

Real world problem: noisy signal (inverted output)

*v ref*

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

The Schmitt Trigger: Comparator with positive feedback

*in this state: v o*

*V S*

15 V

*v*

so

*v i*

*R*

1

*V S*

*v*

*R*

1

*R*

2

*v*

when

*v*

7 .

5

**Try R 1 =R 2 , V S =+/-15V**

*v o*

*V in this state: S*

15 V

*v*

so

*v i*

*R*

1

*v V S*

*R*

1

*R*

2

*v*

when

*v*

7 .

5 • 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

Oscillator: We can create a clock This sets the clock period ( RC) v + =v o /2 v =v C This sets the threshold levels v o sets the voltage at v +

**and**

charges the capacitor

• Break it down into elements

• Two buffers (voltage followers): v o =v i

• Inverting amplifier Gain

*R F R S*

1

**X Y R**

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

*i*

*R*

2

*v i*

*Z C*

• Also, at point Z,

*v o*

*v i*

*iR*

*v i*

*R*

2

*v i*

*R Z C*

V XZ

**Z**

*i*

*R*

2

*v i*

*Z C v o*

*v i*

*iR*

*v i*

*R*

2

*v i*

*R Z C*

• Now, gain

*g*

*v o v in*

1 2

*R R*

*Z C*

*R*

*R Z C*

*Z C*

2

*R*

• Multiply top and

*g*

bottom by (1-j ωRC):

*g*

( 1 ( 1

*j*

1

*j*

*C*

*R*

1

*j*

*C j*

*RC*

*R*

)( 1

*RC*

)( 1 1 1

*j*

*RC j*

*RC j*

*RC j*

*RC*

1 2

*R*

2

*C*

2 2 1 2

*R*

2

*C*

2

*j*

*RC*

) )

*Ae j*

*Z C Z C*

*R*

*R*

• None of the amplifiers change the amplitude:

*g*

*A*

1 tan

*b a*

1 2

*RC*

2

*R*

2

*C*

2 1 1 2

*R*

2

*C*

2 2

*R*

2

*C*

2 1 2 2

*RC R*

2

*C*

2

tan

*g*

1 1 2 2

*RC R*

2

*C*

2 tan • The circuit is a phase shifter!

j ¥ ¥ ¥ 0 if -¥ if ¥

**R**

if w w w RC RC RC = ® • Output voltage is a phase shifted version of the input 1 0 ® ¥ ¥ j = • Vary R to vary the degree of phase shift. Nice audio effect – but also… ¥ ¥ ¥¥ ¥ ¥ ¥ 0 + p 2 p 2 • Very useful for communications applications (e.g Electronically steerable microwave antenna arrays: PATRIOT= "

**P**

hased

**A**

rray

**TR**

ack to

**I**

ntercept

**O**

f

**T**

arget" )

• 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= "

**P**

hased

**A**

rray

**TR**

ack to

**I**

ntercept

**O**

f

**T**

arget" )

Non-Ideal Opamps: Basic Cautions 1) Avoid Saturation • Voltage limits: V S < v OUT < V S + • In the saturation state, Golden Rules of opamp are not valid

Basic Cautions for opamp circuits 2) Feedback must be negative (inverting) for linear behaviour 3) 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

• An ideal opamp has open-loop (no feedback) gain A= • More realistically, it is typically ~10 5 -10 6 at DC, dropping to 1 at a frequency, f T =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 (f T ) and 180º

• Open loop cut-off frequency, f 0 (also known as open loop bandwidth) is usually small (typically 100Hz) to ensure that the gain is <1 at a phase shift of 180 º • Closed-loop gain (gain of amplifier with feedback) begins dropping when open loop gain approaches R F /R S (in the case of the inverting amp) • Cut off frequency will be higher for lower closed-loop gain circuits Inverting amplifier

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

*S*

0

*dv o dt*

max square wave input • 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

• Consider an inverting amplifier, gain=10, built using an opamp with a slew rate of S 0 =1V/ μs.

• Input a sinusoid with an amplitude of V i =1V and a frequency, ω.

*v i*

*V i*

cos(

*t*

)

*v o*

*AV i*

cos(

*t*

)

*dv o dt*

*AV i*

sin(

*t*

)

*dv*

0

*dt*

max

*AV i*

*S*

0

*dv o dt*

max 10 10 6 10 5 • For a sinusoid, the slew rate limit is of the form AV i ω~~
~~

~~• We can therefore avoid this non-linear behaviour by • decreasing the frequency (ω) • lowering the Amplifier gain (A) • lower the input signal amplitude (V i ) • Typical values: 741C: 0.5V/μs, LF356: 50V/ μs, LH0063C: 6000V/ μs, ~~