Agenda and Notes Today, during class! 9:30 a.m. Boeing

Space and Intelligence Systems (Matt and Matt)

4 extra credit assignments available at the bottom of http://hibp.ecse.rpi.edu/~connor/education/EILinks.html

Friday, Oct. 3 (EMPAC!), Open shop 2:00-5:00 p.m

Electronic InstrumentationExperiment 4 (continued)

Part A. Op Amp Basics Review

Part B. Adder and Differential Op Amp

Part C. Op Amp Limitations

What is an op amp? An inexpensive, versatile, integrated

circuit that is another basic building block to electronics (made of resistors and transistors)

Amplifier that has• Large open loop gain (intrinsic)• Differential input stage, inverting input (-)

and non-inverting input (+)• One output• Uses components in the feedback network

to control the relationship between the input and output

What does an Op-Amp do?

Performs “operations” on an input signal• Amplification

• Buffering

• Integration/Differentiation

• Addition/Subtraction

Open Loop/Closed Loop and Feedback

Open loop• Very high gain (intrinsic gain)• Poor stability• Open loop gain assumed to be infinite for ideal op amps

Closed loop• Uses feedback to add stability• Reduces gain of the amplifier• Output is applied back into the inverting (-) input• Most amplifiers are used in this configuration

Open loop gain

Feedback

ΣVin Vout-

+

Golden Rules of Op-Amp Analysis

Rule 1: VA = VB

• The output attempts to do whatever is necessary to make the voltage difference between the inputs zero.

• The op-amp “looks” at its input terminals and swings its output terminal around so that the external feedback network brings the input differential to zero.

Rule 2: IA = IB = 0• The inputs draw no current

• The inputs are connected to what is essentially an open circuit

1) Remove the op-amp from the circuit and draw two circuits (one for the + and one for the – input terminals of the op amp).

2) Write equations for the two circuits.

3) Simplify the equations using the rules for op amp analysis and solve for Vout/Vin

Steps in Analyzing Op-Amp Circuits

Why can the op-amp be removed from the circuit?

• There is no input current, so the connections at the inputs are open circuits.

• The output acts like a new source. We can replace it by a source with a voltage equal to Vout.

in

fin

in

fout R

RAV

R

RV

The Inverting Amplifier

g

f

ing

fout

R

RA

VR

RV

1

1

The Non-Inverting Amplifier

1in

out

V

VA

The Voltage Follower

High input impedanceLow output impedanceBuffer circuit

Ideal Differentiator

inf

in

f

in

f

in

out CRj

Cj

R

Z

Z

V

V

analysis

1

:

Amplitude changes by a factor of RfCin

dt

dVCRV in

infout

Time domain (like oscilloscope)

Frequency domain (like AC sweep)

Comparison of ideal and non-ideal

Both differentiate in sloped region.Both curves are idealized, real output is less well behaved.A real differentiator works at frequencies below c=1/RinCin

Ideal Integrator

finin

f

in

f

in

out

CRjR

Cj

Z

Z

V

V

analysis

1

1

:

Amplitude changes by a factor of 1/RinCf

)(1

DCinfin

out VdtVCR

V

Time domain (like oscilloscope)

Frequency domain (like AC sweep)

What happens to a capacitor at DC?

Miller (non-ideal) Integrator

If we add a resistor to the feedback path, we get a device that behaves better, but does not integrate at all frequencies.

Comparison of ideal and non-ideal

Both integrate in sloped region.Both curves are idealized, real output is less well behaved.A real integrator works at frequencies above c=1/RfCf

Comparison Differentiaion Integration original signal

v(t)=Asin(t) v(t)=Asin(t)

mathematically

dv(t)/dt = Acos(t) v(t)dt = -(A/cos(t)

mathematical phase shift

+90 (sine to cosine) -90 (sine to –cosine)

mathematical amplitude change

1/

H(j H(j jRC H(j jRC = j/RC electronic phase shift

-90 (-j) +90 (+j)

electronic amplitude change

RC RC

The op amp circuit will invert the signal and multiply

the mathematical amplitude by RC (differentiator) or 1/RC (integrator)

In Class Problem1. Which op amp below has a gain of

+5?

2. Op amp Analysis1. What are the golden rules for op amp

analysis?2. For the circuit to the right draw two

circuits (one for – input and one for + input)

3. Write the equation for each circuit

a) b) c)

In Class Problem1. Which op amp below has a gain of

+5? All of them! Topology may look different but the functionality is the same!

2. Op amp Analysis1. What are the golden rules for op amp

analysis?2. For the circuit to the right draw two

circuits (one for – input and one for + input)

3. Write the equation for each circuit V 1 V--

Z 1

V-- V out

Z fUse voltage divider

V+

Z 3

Z 2 Z 3V 2

Op Amps to know

Inverting Non-inverting Voltage Follower Differentiator Integrator Adder Differential (Subtracting)

Adders

12121

2

2

1

1

R

R

VV

VthenRRif

R

V

R

VRV

fout

fout

Output signal is the sum of the input signals (V1 and V2).

Weighted Adders Unlike differential amplifiers, adders are

also useful when R1 ≠ R2.

This is called a “Weighted Adder” A weighted adder allows you to combine

several different signals with a different gain on each input.

You can use weighted adders to build audio mixers and digital-to-analog converters.

Analysis of weighted adder

I2

I1

If

2

2

1

1

2

2

1

1

2

2

1

1

2

22

1

1121

0

R

V

R

VRV

R

V

R

V

R

V

VVR

VV

R

VV

R

VV

R

VVI

R

VVI

R

VVIIII

foutf

out

BAAA

f

outA

f

outAf

AAf

Differential (or Difference) Amplifier

in

f

in

fout R

RAVV

R

RV

)( 12

Analysis of Difference Amplifier(1)

:

:)1

in

fout

outinffoutinf

in

inf

f

fin

fBA

outinf

in

inf

fB

fin

inf

fin

outinf

B

fin

f

out

inBB

fin

fA

f

outB

in

B

R

R

VV

V

VRVRVRVRR

RV

RR

RV

RR

RVV

VRR

RV

RR

RV

RR

RR

RR

VRVR

V

RR

R

V

RV

VVforsolve

VRR

RV

R

VV

R

VVi

12

1212

1

11

2

1

:

11:)3

:

:)2

Analysis of Difference Amplifier(2)

Note that step 2(-) here is very much like step 2(-) for the inverting amplifier

and step 2(+) uses a voltage divider.

Op-Amp Limitations

Model of a Real Op-Amp Saturation Current Limitations Slew Rate

Internal Model of a Real Op-amp

• Zin is the input impedance (very large ≈ 2 MΩ)

• Zout is the output impedance (very small ≈ 75 Ω)

• Aol is the open-loop gain

Vout

V2

+V

-V

-+

V1

Vin = V1 - V2Zin

Zout

+

-

AolVin

Saturation Even with feedback,

• any time the output tries to go above V+ the op-amp will saturate positive.

• Any time the output tries to go below V- the op-amp will saturate negative.

Ideally, the saturation points for an op-amp are equal to the power voltages, in reality they are 1-2 volts less.

VVV out Ideal: -9V < Vout < +9V

Real: -8V < Vout < +8V

Additional Limitations Current Limits If the load on the op-amp is

very small, • Most of the current goes through the load• Less current goes through the feedback path• Op-amp cannot supply current fast enough• Circuit operation starts to degrade

Slew Rate• The op-amp has internal current limits and internal

capacitance. • There is a maximum rate that the internal capacitance

can charge, this results in a maximum rate of change of the output voltage.

• This is called the slew rate.

Analog Computers (circa. 1970)

Analog computers use op-amp circuits to do real-time mathematical operations (solve differential equations).

Using an Analog Computer

Users would hard wire adders, differentiators, etc. using the internal circuits in the computer to perform whatever task they wanted in real time.

Analog vs. Digital Computers In the 60’s and 70’s analog and digital computers competed. Analog

• Advantage: real time

• Disadvantage: hard wired Digital

• Advantage: more flexible, could program jobs

• Disadvantage: slower Digital wins

• they got faster

• they became multi-user

• they got even more flexible and could do more than just math

Now analog computers live in museums with old digital computers:

Mind Machine Web Museum: http://userwww.sfsu.edu/%7Ehl/mmm.htmlAnalog Computer Museum: http://dcoward.best.vwh.net/analog/index.html