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Transcript of a “difference” amp Operational Amplifiers · Operational Amplifiers Op Amps! Differential Amp...
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
Operational Amplifiers
2017-10-16 PHYS351001 L8 Michael Burns
Math Functions – Logarithmic Amp
Actual Silicon Diode
I ~ Io𝑒V
VT
For Silicon Diodes, VT ~ 0.6-0.7 volts
Operational Amplifiers
2017-10-16 PHYS351001 L8 Michael Burns
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
Operational Amplifiers
2017-10-16 PHYS351001 L8 Michael Burns
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
Operational Amplifiers
2017-10-16 PHYS351001 L8 Michael Burns
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
Operational Amplifiers
2017-10-16 PHYS351001 L8 Michael Burns
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.
Operational Amplifiers Math Functions – Log & Exponential Amps
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
Operational Amplifiers Math Functions – Log & Exponential Amps
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
Operational Amplifiers
2017-10-16 PHYS351001 L8 Michael Burns
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
Operational Amplifiers
2017-10-16 PHYS351001 L8 Michael Burns
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
Operational Amplifiers
2017-10-16 PHYS351001 L8 Michael Burns
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.
Operational Amplifiers
2017-10-16 PHYS351001 L8 Michael Burns
Math Functions – Analog Computers
1970 Compumedic Analog Computer Model 6F13 Module
1960 EC-1 Module
Operational Amplifiers
2017-10-16 PHYS351001 L8 Michael Burns
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
Operational Amplifiers
2017-10-16 PHYS351001 L8 Michael Burns
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
Operational Amplifiers
2017-10-16 PHYS351001 L8 Michael Burns
Simulate other components
Humongous inductor – Simulate making a “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
Operational Amplifiers
2017-10-16 PHYS351001 L8 Michael Burns
Simulate other components
Humongous inductor – Simulate making a “Gyrator”
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)
Operational Amplifiers
2017-10-16 PHYS351001 L8 Michael Burns
Simulate other components
Humongous inductor – Simulate making a “Gyrator”
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.
Operational Amplifiers
2017-10-16 PHYS351001 L8 Michael Burns
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-| ~
Operational Amplifiers
2017-10-16 PHYS351001 L8 Michael Burns
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
Operational Amplifiers
2017-10-16 PHYS351001 L8 Michael Burns
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.
Operational Amplifiers
2017-10-16 PHYS351001 L8 Michael Burns
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+
Operational Amplifiers
2017-10-16 PHYS351001 L8 Michael Burns
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+
Operational Amplifiers
2017-10-16 PHYS351001 L8 Michael Burns
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)
Operational Amplifiers
2017-10-16 PHYS351001 L8 Michael Burns
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
Operational Amplifiers
2017-10-16 PHYS351001 L8 Michael Burns
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
Operational Amplifiers
2017-10-16 PHYS351001 L8 Michael Burns
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.
Operational Amplifiers
2017-10-16 PHYS351001 L8 Michael Burns
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
Operational Amplifiers
2017-10-16 PHYS351001 L8 Michael Burns
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.
Operational Amplifiers
2017-10-16 PHYS351001 L8 Michael Burns
Other Functions
Constant voltage source
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?
Operational Amplifiers
2017-10-16 PHYS351001 L8 Michael Burns
Other Functions
Constant voltage source
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