1. ECE 221 Electric Circuit Analysis I Chapter 15 Operational Amplifiers
Op Amps
Herbert G. Mayer, PSU
Status 11/11/2015
Several Examples taken from Wikipedia and [1]

2. Syllabus Definition, History, Features Terminal Currents & Voltages
Ideal Op Amp
Noninverting Op Amp
Inverting Op Amp
Difference Op Amp
Summing Op Amp
References

3. Definition High Level
Operational Amplifiers, AKA Op Amps, amplify voltages:
They are nonlinear electronic devices that amplify the difference of their 2 input signals vp and vn , shown at output terminal vo
To function properly, sources +VCC and -VCC must satisfy: +VCC > -VCC, referred to as rails
Rails are frequently omitted from circuit drawings to reduce clutter; assumed to just be there
Though Op Amps are non-linear devices, like transistors, they are almost always used in their narrow, linear subrange
We can model them abstractly as a simple circuit element via a dependent voltage source

4. Op Amp Definition High Level
+Vcc
-Vcc vp vn + -
=> vo

5. Op Amp Definition
Operational Amplifiers, AKA Op Amps, are solid state, low-cost, integrated circuits performing electric transformations of two (or one of two) input signals: one inverting, the other noninverting
Transformations include
Amplification
Inversion
Summation
Difference
Differentiation – not discussed here
Integration
Comparison
A/D conversion
Characteristic is the high-gain amplification of 103 to 105 within a narrow band of input voltages vp - vn
Output is vo

6. Op Amp Definition
Amplifies the difference of its two input voltage signals vp and vn
The amplification factor A, known as gain, can be very high; the spectrum of voltages amplified is very narrow
With feedback loop, A can become quite low, often 1
Possible to hold one of the inputs down to 0 V, in which case solely the other input signal contributes to the amplified output signal vo

7. Op Amp Definition
Op Amps use external power supply, at positive +VCC and a at the second negative -VCC voltage pin
Not necessarily of same voltage, but usually absolute values are equal: |+VCC| = |-VCC|
Common power voltages are V to V, rarely over 20 V
Op Amps generally are used in their linear range of amplification; part of the curve with inclination A
Due to the high gain A, Op Amps operate only in a very narrow band of signal amplitudes, else saturate!
Without feedback loop (below) they saturate even more rapidly, i.e. without feedback the voltage difference of the inputs must be even more narrow for an Op Amp to operate linearly

8. Saturated Op Amp
From Wikipedia: sin() input signal saturates Op Amp, so vo becomes rectangular; only brief vertical lines show gain A at very low input voltage (difference)
Quick quiz: Is this Op Amp reading its sin() input signal vs at the inverting or noninverting input pin?

9. Op Amp History Built during ww2 with vacuum tubes
With V power supplies then
Solid state Op Amps since 1960s
Early solid state Op Amps: μA702
Bob Widlar 1964, retired at age 30 after invention!
9 transistors only in first solid state Op Amps!
μA741
Dave Fullagar 1968, Fairchild
Most popular Op Amp of all time
Packaged in 8-pin DIP
OP-07
Precision Monolithics, 1975
Special-purpose, high gain

10. Op Amp History, Early Contributors
Harold S. Black, US Western Electric, 1930s
Paul Voigt, UK, 1920s
Alan Blumlein, UK, 1930s
Hendrick Bode, US 1930s

11. Op Amp Nomenclature +Vin noninverting input, we’ll use: vp
-Vin inverting input, we’ll use: vn
+VS positive source, we’ll use: +VCC
-VS negative source, we’ll use: -VCC
+Vout output voltage, we’ll use: vo vo
+Vcc
-Vcc vp vn + -
Vout
+Vs
-Vs
+Vin
-Vin + -

12. More Likely You Just See:vo vp vn + -
Vout
+Vin
-Vin + -

13. Op Amp Circuit Symbol

14. Op Amp Circuit Symbol
Above circuit symbol shows enlarged shape of typical, contemporary Op Amp
Packaged in a DIP with 2 * 4 = 8 pins; DIP stands for dual in-line package
Two of the pins (offset null) are rarely used: they provide a method to scale function to overcome shifts caused by material aging
One pin is unused; AKA no connection
Leaves 5 pins for noninverting input +, inverting input -, output, and + - power supply

15. Actual Op Amp 741 Circuit Detail

16. Actual Op Amp 741 Circuit Detail
Circuit detail of typical μA741 with 20 transistors is way more complex than its ancestor of 9 transistors
Shown here purely for entertainment purposes; won’t analyze complex transistor actions in ECE 221
We’ll treat an Op Amp as a black-box, with defined functions
And that black box Op Amp will be idealized, e.g. assumed to be operating in linear region, in = ip = 0, and vp = vn
Those are the key equations for ideal Op Amp!
Yet ideal Op Amp operates largely like a real Op Amp within a narrow linear input range!

17. Op Amp Voltage Transfer Function

18. Terminal Current Names

19. Terminal Voltage Names

20. Ideal vs. Real Op Amps

21. Ideal Op Amp
Without feedback, the straight Op Amp generally saturates quickly, except for a very narrow band of input voltages vn and vp specifically: vp – vn
To reduce saturation, Op Amps generally feed some output signal from vo back to input signal vn
This negative feedback voltage subtracts from the actual input signal vn
Thus decreases gain and input voltage difference
And also decreases output voltage vo
Renders difference of vp and vn extremely small; we simplify by saying: vp = vn for ideal Op Amp!!!
Input resistance idealized to ∞ Ω, practically >> 1 MΩ
Idealized ip = in = 0 in our modeling of Op Amps

22. Ideal Op Amp Formulae vp = vn -- known as virtual Op Amp short!
ip = in = 0 A -- ∞ input Ω causes no current
A = ∞ -- infinite gain, real ~105 max
ip + in + io + iC+ + iC- = Kirchhoff C. Law
From ip = in = 0 it follows:
io = - ( iC+ + iC- )

23. Summary Ideal vs. Real Op Amp
Electric Parameters of Real Op Amp
Value
Large gain A
> 105
Small input currents ip, in
Close to 0 A
Small difference of input voltages: vp - vn
Close to 0 V
Steep incline of function vo = f( vp - vn )
High in linear range

24. Summary Ideal vs. Real Op Amp
Electric Parameters of ideal Op Amp
Value
Infinite gain A ∞
Zero Ampere input currents ip, in
0 A
Very small difference of input voltages: vp - vn
0 V
Extremely steep incline of function vo = f( vp - vn )

25. Type 1
Noninverting Op Amp

26. Type 1 Noninverting Op Amp, Ideal
The noninverting Op Amp has the positive input pin connected to the signal source vs via resistor Rp
The inverting pin is connected to the common node (ground) also via a resistor, named Rn
The input signal arriving at the noninverting input is often referred to as vs or vp
Output voltage vo is fed back to the inverting input pin via feedback resistor Rf generally decreasing gain A
Keeping in mind: ip = in = 0 for ideal Op Amp
Goal is to compute vo as function of input signal vs and the resistors; i.e. find the characteristic Op Amp function!

27. Type 1 Noninverting Op Amp, Generic

28. Type 1 Noninverting Op Amp, Generic
ia + if – in = – ia runs through Rn
ip = in = – in runs into - input
vn = vP = vs – no current in Rp
vs / vo = Rn / ( Rn + Rf ) – voltage divider
vo = vs * ( Rf + Rn ) / Rn
vo = vs * A
Characteristic function for output voltage vo of Noninverting Op Amp, with signal voltage vs

29. Type 1 Noninverting Op Amp
The characteristic function vo = vs * ( Rf + Rn ) / Rn states that output voltage vo is a scaled, direct replica of the input signal vS, scaled by gain A, with A = ( Rf + Rn ) / Rn
Note that the characteristic function vo is valid, since the Op Amp is (close to) ideal, operating in the linear region –i.e. NOT saturated!
Gain A can be controlled purely by resistors Rf and Rn for noninverting Op Amp

30. Type 1 Noninverting Op Amp
The amplification of a noninverting Op Amp with Rf and Rn is A = ( Rn + Rf ) / Rn
Provided Op Amp is ideal and is not saturated
That means, |vo| must be <= |+-VCC |
vs * ( Rf + Rn ) / Rn <= VCC
A = ( Rn + Rf ) / Rn <= |VCC / vs|

31. Type 1 Noninverting Op Amp, Sample 1
Design some noninverting Op Amp with gain A = 10
Both resistors at the input signal pins Rn and Rp are predefined to be 1 kΩ, and vs = 1 V
Compute vo and Rf
Check, whether under those conditions the Op Amp works in linear mode, assuming external power supplies of V

32. Type 1 Noninverting Op Amp, Sample 1

33. Type 1 Noninverting Op Amp, Sample 1
With vo / vs = A = ( Rf + Rn ) / Rn -- require gain A=10:
vo / vs = ( Rf + Rn ) / Rn = 10
10 * Rn = ( Rf + Rn )
Rf = 9 Rn
Rf = 9 kΩ
To see whether the Op Amp is saturated, compute vo and see, whether or not it falls within + - VCC:
vo = vs * ( Rf + Rn ) / Rn
vo = 1 * ( ) / 1 = 10 V -- is within range V

34. Type 1 Noninverting Op Amp, Sample 2
Assuming a signal voltage vs = 1.5 V, compute the lowest possible external power +-VCC that still enables the Op Amp to operate in linear mode
We take all other design parameters from Sample 1
Knowing that vo cannot be above |VCC|, we compute vo for vs = 1.5 V. This yields the lowest possible +VCC
Then we compute vo for vs = -1.5 V. This yields the most negative legal -VCC
Use formula vo = vs * ( Rf + Rn ) / Rn
With Rf, Rp, and Rn inherited from Sample 1

35. Type 1 Noninverting Op Amp, Sample 2

36. Type 1 Noninverting Op Amp, Sample 2
With vo = vs * ( Rf + Rn ) / Rn
Case 1 for largest vs = 1.5 V
vo+ = 1.5 * ( ) / 1 = 15 V
vo+ = 15 V
Case 2 for smallest vs = -1.5 V
vo- = -1.5 * ( ) / 1 = -15 V
Vo- = -15 V
Matching exactly and just barely our previous power situation + -VCC from Sample 1

37. Type 2
Inverting Op Amp

38. Type 2 Inverting Op Amp All Op Amps are Difference Op Amps!
Even if one of the inputs is connected to the common node, AKA ground
Inverting Op Amp has noninverting input pin grounded, which is at 0 V
Inverting input pin is connected to vs the actual input signal
Connection generally realized via resistor Rs AKA Rn in the literature
Output voltage vo is fed back, as is typical, to inverting input pin via resistor Rf

39. Type 2 Inverting Op Amp, Generic

40. Characteristic function for output voltage vo of Inverting Op Amp
Type 2 Inverting Op Amp
is + if - in = 0 A
vn = vP = 0 V -- even with Rp = 0 Ω at the +pin
is = ( vS - vn ) / RS = vS / RS
if = ( vo - vn ) / Rf = vo / Rf
with in = 0 it follows: is = -if
vo = - vS ( Rf / RS )
Characteristic function for output voltage vo of Inverting Op Amp

41. Type 2 Inverting Op Amp
The characteristic function vo = - vS ( Rf / RS ) states that the output voltage is a scaled inverted replica of the input signal vS
Note that the characteristic function vo is valid only if the Op Amp is close to ideal
The gain A can be controlled purely by the ratio of Rf to RS
Moreover, if Rf = RS then the Op Amp reaches a gain A = 1, i.e. it is reduced to a pure voltage inverter
We see it is easy to create a voltage inverter from a simplistic, inverting Op Amp

42. Type 2 Inverting Op Amp, Sample 1
In Sample 1 we use an inverting Op Amp with:
signal source of voltage va = 1 V
vn = vp , and vp being grounded = 0 V
Feedback resistor Rf = 100 kΩ
Resistor RS at signal source va is: RS = 25 kΩ
Source voltages +VCC and -VCC of V
Compute vo

43. Type 2 Inverting Op Amp, Sample 1

44. Type 2 Inverting Op Amp, Sample 1
i25 + i100 - in = 0 A
with in = 0 therefore:
i100 = -i25
i25 = (va - vn) / 25 k
i25 = (va – 0 ) / 25 k
i25 = 0.04 mA
i100 = (vo – vn) / 100k = -0.04
vo / 100k = -0.04
vo = -4 V within operating range

45. Type 2 Inverting Op Amp, Sample 1
Or less complicated than above, with characteristic function vo = - vS ( Rf / RS )
vo = -1 * ( 100 k / 25 k )
vo = -1 * 4
vo = -4 V

46. Type 2 Inverting Op Amp, Sample 2
In Sample 2 we use an inverting Op Amp with
signal source of voltage vs = 0.4 V
vn = vp, vp grounded = 0 V
Feedback resistor Rf = 80 kΩ
Resistor RS at signal source vs of: RS = 16 kΩ
Source voltages +VCC +10 V and -VCC of -15 V
Compute vo

47. Type 2 Inverting Op Amp, Sample 2

48. Type 2 Inverting Op Amp, Sample 2, Hard
i16 + i80 + in = 0 A
with in = therefore:
i = -i16
i = (vs - vn) / 16 k
i = (vs - 0) / 16 k
i = mA
i = (vo – vn) / 80k = mA
vo / 80k =
vo = -2 V within operating range

49. Type 2 Inverting Op Amp, Sample 2, Easy
Students use easier way of computing vo:
Use characteristic equation!
vo = -2 V

50. Type 3
Difference Op Amp

51. Type 3 Difference Op Amp, Terminology
Easy to get confused by terminology: Difference Amplifier vs. Differential Amplifier
When the circuit containing Op Amps is purely resistive, some literature uses the two terms interchangeably 
Literally, all Op Amps are Difference Amplifiers, as the output voltage vo is a function of the difference of the input voltages vp and vn
Some authors use the term Differential Amplifier as long as the gain is > 1; then when A = 1, switch to Difference Amplifier as Op Amp literally generates the difference vo = vp - vn with no gain!
We use Difference Amplifier to reduce confusion with Differentiating Amplifier, yet another type, not covered in ECE 221

52. Type 3 Difference Op Amp
For Difference Op Amp output voltage vo is proportional to the difference of voltages of input signals va and vb
Again we assume an ideal Op Amp . . .
And with feedback loop from vo to inverting input vn via feedback resistor Rf
Both input signals va and vb are scaled by resistors; one is Ra connected to inverting input, the other Rb
Noninverting input often has a separate resistor Rp to common node as well
Goal to express vo as a function of va and vb, specifically, as a function of the difference vb - va

53. Type 3 Difference Op Amp, Generic

54. Type 3 Difference Op Amp
(1) KCL, summing currents flowing to inverting pin:
( vn – va ) / Ra + ( vn – vo ) / Rf - in = 0
(2) Voltage Division:
vp = vn = vb * Rp / ( Rb + Rp )
Substitute vn (2) into (1), and with in = 0:
vb*Rp/(Ra*(Rb+Rp))-va/Ra + vb*Rp/(Rf*(Rb+Rp)) - vo/Rf = 0
vo = vb * (Rp*Rf / (Ra*(Rb+Rp) ) + Rp / (Rb+Rp)) - va* Rf / Ra
Characteristic function for output voltage vo of a general Difference Op Amp

55. Type 3 Difference Op Amp vo = Rf / Ra * ( vb - va )
Special case: Ra / Rf = Rb / Rp
Or specifically: Rb  Ra, and Rp  Rf then
vo = vb * (Rf*Rf + Ra*Rf) / (Ra*Ra+Ra*Rf) - va* Rf /Ra
vo = vb * Rf / Ra - va* Rf / Ra
vo = Rf / Ra * ( vb - va )
Characteristic function for output voltage vo of Difference Op Amp with Rb and Rp sized after Ra and Rf

56. Type 3 Difference Op Amp vo = ( vb - va ) Special case: Rf = Ra
vo = vb * Rf / Ra - va* Rf / Ra
vo = vb * Rf / Rf - va* Rf / Rf
vo = vb * 1 - va* 1
vo = ( vb - va )
Characteristic function for output voltage vo of Difference Op Amp with all R adjusted

57. Type 3 Difference Op Amp, Sample 1

58. Type 3 Difference Op Amp, Sample 1
i100 = -i20 as before, since in = 0
i20 = (va - vn) / 20k
i20 = ( ) / 20k = mA
i100 = (vo – vn) / 100k
i100 = vo/100k - 2/100k = mA
Vo/100k = 0.05 mA + 2/100k A = 0.07 mA
Vo = 7 V outside operating range

59. Type 4
Summing Op Amp

60. Type 4 Summing Op Amp, Generic
Summing Op Amp has multiple input signals all joining at the inverting or noninverting input pin
Shown here: 3 inputs va, vb, vc, at inverting input
Scaled by resistors Ra, Rb, Rc
Ideal Op Amp still requires currents in and ip = 0 A
Also, with noninverting input pin connected to common node (or ground), we know that vp = 0, and vn = vp = 0
Goal to compute vo

61. Type 4 Summing Op Amp, Generic

62. Type 4 Summing Op Amp, Generic
(vn-va) / Ra + (vn-vb) / Rb + (vn-vc) / Rc + (vn-vo) / Rf - in= 0
with vn = 0, in = 0 
vo / Rf = -va / Ra - vb / Rb - vc / Rc
vo = -Rf *( va / Ra + vb / Rb + vc / Rc )
Characteristic function for output voltage vo of a
Summing Op Amp with 3 inputs, scaled by resistors Ra, b, c

63. Type 4 Summing Op Amp, Generic
with Rs = Ra = Rb= Rc 
vo = -Rf * ( va + vb + vc ) / Rs
Characteristic function for output voltage vo of a
Summing Op Amp with 3 inputs, identical resistors RS
And with Rs = Rf 
vo = -( va + vb + vc )
Summing Op Amp with 3 inputs at inverting input, RS = Rf

64. Type 4 Summing Op Amp, Sample 1

65. Type 4 Summing Op Amp, Sample 1
(vn-va)/5k + (vn-vb)/25k + (vn-vo)/250k - in = 0
with va = 0.1 V, vb = 0.25 V
-0.1 / 5k – 0.25 / 25k = vo / 250k
vo / 250k = mA mA = mA
vo = -7.5 V

66. References
Nilsson, James W., and Susan A. Riedel, Electric Circuits, © Pearson Education Inc., ISBN 13:
Wiki page:
A. D. Blumlein, Improvements in and relating to Thermionic Valve Amplifiers, UK Patent 425,553, issued March 18, 1935
Hendrick Bode, Relations Between Attenuation and Phase In Feedback Amplifier Design, Bell System Technical Journal, Vol. 19, No. 3, July, 1940
Hendrick Bode: Amplifier, US Patent 2,123,178, issued July 12, 1938
Dave Fullagar: news/ /Voices-Dave-Fullagar-analog-IC-designer-and- entrepreneur