a “difference” amp Operational Amplifiers · Operational Amplifiers Op Amps! Differential Amp Input bias current canceled, since inputs look symmetric. v + v - Remember, v + =

Operational Amplifiers Op Amps!

Differential Amp

Input bias current canceled, since inputs look symmetric.

v+

v-

Remember, v+ = 𝑅2

𝑅1+𝑅

2

V1

v- =v+

i- i+

So Vout = 𝑅1+ 𝑅

2

𝑅1

v- -

𝑅2

𝑅1 V2

Becomes Vout = 𝑅1+ 𝑅

2

𝑅1

v+ -

𝑅2

𝑅1 V2

Vout = 𝑅1+ 𝑅

2

𝑅1

𝑅2

𝑅1+𝑅

2

V1 - 𝑅2

𝑅1 V2

Vout = 𝑅2

𝑅1

(V1 – V2)

Subtraction

We did this circuit two weeks ago – a “difference” amp

There’s a whole lot of “other” signal processing math functions we can use op-amps for.

2017-10-16

Operational Amplifiers

2017-10-16 PHYS351001 L8 Michael Burns

Math Functions – Inverting Summing Amp

v+ = v- = 0, So Vout = ifRf & V1 = i1R1 & V2 = i2R2

We also know if + i1 + i2 = 0

So 0 = Vout

Rf

+ V1

R1

+ V2

R2

Vout = - Rf V1

R1

+ V2

R2

Vout

v-

V2

if

i1

i2

v+

V1

Weighted (inverted) sum of V1 & V2



Math Functions – Logarithmic Amp

Actual Silicon Diode

I ~ Io𝑒V

VT

For Silicon Diodes, VT ~ 0.6-0.7 volts



Math Functions – Logarithmic Amp

v+ = v- = 0,

i =io 𝑒Vd

VT

i = Vin

R1

Only one on at a time

Vout

Vin At input, i = Vin

R1

At output i =io 𝑒Vd

VT

So Vin

R1

= io 𝑒Vd

VT

Or, since Vout = - Vd, Vout = - VT lnVin

R1io



Math Functions – Logarithmic Amp (more generally)

Vout

Vin

R’s limit current. R2 allows it to function when neither diode is on.

Max compression is R1

R

Example: R2 ~ 10R1



Math Functions – Exponential Amp

v+ = v- = 0,

i =io 𝑒Vd

VT

i = - Vout

R2

Only one on at a time Vout

Vin At output, i = Vout

R2

At input i =io 𝑒Vd

VT

So Vout

R2

= io 𝑒Vd

VT

Or, since Vin = Vd Vout = - R2io 𝑒Vin

VT



Math Functions – Exponential Amp (more generally)

Vout

Vin

R’s limit current. R1 allows it to function when neither diode is on.

Max expansion is R2

R

Example R2 ~ R1/10

Operational Amplifiers Math Functions – Log & Exponential Amps

An example where log (& exponential) amps are used is with sensors with larger dynamic range than the data collection system can handle directly.

E1

E2

E3

Light Intensity E1 < E2 < E3 Avalanche photodiodes dynamic range can be 109

(Nuclear Instruments and Methods in Physics Research A 350, 595 (1994))

Dynamic range is the ratio between the largest and smallest possible values of a changeable quantity

Suppose sensor has a dynamic range of 109.

Suppose you want to record this digitally. 16-bit analog-to-digital converter has a dynamic range of 216 , which is 65536. 24-bit analog-to-digital converter has a dynamic range of 224 , which is ~1.7 x 107. 109 directly would need a 30-bit analog-to-digital converter.

http://www.sciencedirect.com/science/article/pii/0168900294912629

http://www.sciencedirect.com/science/article/pii/0168900294912629


The dynamic range of magnetic tape is approximately 55 dB. (decibels)

dB = 10 log10𝑃2

𝑃1

defined in terms of power (H & H, page 15)

Power ~ V2

dB = 20 log10𝑉2

𝑉1

defined in terms of voltage (or other signal)

So a dB is a dB regardless of whether one is talking power or signal.

So magnetic tape has a dynamic range of 1055

20 ~ 102.75 ~ 562

Dynamic range is the ratio between the largest and smallest possible values of a changeable quantity

2017-10-16


The dynamic range of magnetic tape is approximately 55 dB. (decibels)

So magnetic tape has a dynamic range of 1055

20 ~ 102.75 ~ 562 “Dolby A” adds approximately 10 dB to the dynamic range that will fit on magnetic tape (to ~ 1800), DBX adds 30 dB (to ~ 18,000). Vinyl records also about 55 dB, but no compression tricks. CD has a dynamic range of 96dB in theory (~63,000), but practically ~90dB (~31,600). Some digital audio recording could theoretically have 120dB (20-bit) or 144 dB (24-bit) dynamic range, but microphones are not that good & file formats (e.g. MP3) discard data. Human hearing ~120 dB ( ~ 1,000,000).

2017-10-16



Math Functions – Integrator

Vout Vin

i = Vin

R

i = −C 𝑑V

out

𝑑𝑡

So Vin

R = −C

𝑑Vout

𝑑𝑡

Vout = −1

RC Vin 𝑡 𝑑𝑡

Vin

Vout

Generally works better than just a C & R, usually limited by the op-amp input bias current.

V = Q

C & i =

𝑑Q

𝑑𝑡 i = C

𝑑V

𝑑𝑡



Math Functions – Differentiator

Vout Vin

i = - Vout

R

i = C 𝑑V

in

𝑑𝑡 So

Vout

R = −C

𝑑Vin

𝑑𝑡

Vout = −RC 𝑑V

in

𝑑𝑡

Same as integrator, just interchange R & C



Math Functions – Integrators & Differentiator

Using op-amp Integrators & Differentiators, you can simulate ANY differential equation. You are not limited by order or linearity constraints. These analog computers have the advantage over digital computers of computing continuous variables, rather than discretized variables. i.e. Analog computers can compute using real numbers, digital computers are limited to rational numbers. Until just a couple of decades ago, analog computers were the only way to simulate complex fluid dynamics for aircraft/missile/rocket designs.



Math Functions – Analog Computers

1970 Compumedic Analog Computer Model 6F13 Module

1960 EC-1 Module



Math Functions – Analog Computers

X-15 simulator analog computer

X-15 – Still holds highest speed ever recorded by a manned, powered aircraft (1967) @Mach 6.72 at 102,100 feet [4,520 miles per hour (7,274 km/h)]

More on X-15 analog computer stuff: https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19680019932.pdf

https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19680019932.pdf



Simulate other components

Humongous inductor – Simulate making a “Gyrator”

Where L = R2C

Use this when you need a really really big inductor. e.g. R= 100, C = 100f, results in L = 1 Henry

http://en.wikipedia.org/wiki/Gyrator





Vin How does this work?

VA

iin

i1

i2

iin = i1 + i2

iin = Vin −VA

R

i1 = VA

R

What’s going on here?

v+ = v- & v+ = Vin v- = Vin

So… i2 = VA −Vin

Zc

= iC (Vin – VA)

ZC = 𝑖

𝜔𝐶

Remember, Zc = −1

iωC






Vin

VA

iin

i1

i2

iin = i1 + i2

iin = VA

R + iC (Vin – VA)

Since iin = Vin −VA

R

iin = Vin

R + iCR iin – iin

So Vin = iin(2R - iCR2)

Now Zin = Vin

iin

So Zin = (2R - iCR2)






Vin

VA

iin

i1

i2

Zin = (2R - iCR2)

For a resistor, Z = R (No dependence)

For a capacitor of capacitance C, Z = 𝑖

𝜔𝐶

Proportional to 1/ For an inductor of inductance L, Z = -iL Proportional to

Proportional to No dependence “L” = CR2

Resistor-like Inductor-like And since they add, they are in series.




Other Functions

Comparators

An ideal comparator is just an op-amp with an enormous gain so that Vout = if v+ > v-

Vout = - if v+ < v-

Even if |v+ - v-| ~



Other Functions

Comparators

There are special op-amps for this purpose which either short the output to ground or have to ground.

+5

Vin

This way you bias the output to suit your needs separate from the input.

Vout



Other Functions

Comparators

As you will probably noticed from the lab where you will make a comparator, noise an such can make the circuit switch if |V+ - V-| is small.

V+ - V-

t

Vout

We usually don’t want this “jitter”.

t We can eliminate the jitter by adding a small amount of hysteresis to the threshold.



Other Functions

Comparators

Suppose we want to compare Vin to find when it is 3 0.1 volts?

Vin

Vout

+10 +5

+5

0 2.9 3.1

Vout

Notice we are putting the signal into the – input and feedback into the + input.

v+



Other Functions

Comparators

Vin

Vout

+10 +5

Suppose at t=0, we assume Vin > 2.9 volts. Vout =0 So the resistor network looks like:

+5

0 2.9 3.1

Vout

+10

v+



Other Functions

Comparators

+10

v+on

So v+on = 10 R2||R

3

R1+(R

2||R

3)

Set v+on = 2.9 volts

R1 + (R2||R3) = 10

2.9 (R2||R3)



Other Functions

Comparators

Now we look at the off threshold. There, Vout 5 volts.

So the resistor network looks like:

+10 5

i1

i2

i3

V+off

We want V+off to be 3.1 volts. So i1R1 = (10 - V+off) = 6.9 volts i2R2 = 6.9 volts & (R3 + 1K) i3 = (5 - V+off) = 1.9 volts

We know i2 = i1 + i3 from current conservation.

So 3.1 R1 (R3 + 1K) = 6.9 (R3 + 1K)R2 + R1R21.9



Other Functions

Comparators

3.1 R1 (R3 + 1K) = 6.9 (R3 + 1K)R2 + R1R21.9

R1 + (R2||R3) = 10

2.9 (R2||R3)

Two equations, three unknowns.

What do we do?

Make a design decision: Pick R2 to be 10K Then R1 22K & R3 183K



Other Functions

Constant current source

Vi

i

i = Vi

Rin

Note that RL can vary all over, and i through RL

stays at Vi

Rin

.

Of course, this is only as good as Vi & Rin. If they drift, so does i.



Other Functions

Better constant current source

i

Vz

i

Vo

So i = Vz

Rset

Why?

But V- = V+

But V- = Vo - Vz

i = Vo −V

−Rset

+

So i = Vo −V

−Rset

-

i = Vz

Rset



Other Functions

Constant voltage source

Vin Vout

Start with a non-inverting amp.

Vin = V-

V- = R2

R1+R

2

Vout

Vout = V- 1 + R1

R2

The key is how stable you can make Vin.



Other Functions


Vz

Vsupply You already know how to make a fixed voltage using zener diodes.

Vout = Vz 1 + R1

R2

Is there an advantage to using the zener + op-amp instead of just the zener?



Other Functions


Is there an advantage to using the zener + op-amp instead of just the zener?

Yes!

1. We do not want the load (which would be in parallel with the zener) to shift us on the zener I-V curve.

2. We can scale the voltage to anything we want from the zenor (even use a pot so the scaling is adjustable).

Slope

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a “difference” amp Operational Amplifiers · Operational Amplifiers Op Amps! Differential Amp Input bias current canceled, since inputs look symmetric. v + v - Remember, v + =