Transcript Op Amps

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]
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Syllabus
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Definition, History, Features
Terminal Currents & Voltages
Ideal Op Amp
Noninverting Op Amp
Inverting Op Amp
Difference Op Amp
Summing Op Amp
References
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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
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Op Amp Definition High Level
+Vcc
vp
vn
+
vo
-
=>
-Vcc
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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
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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
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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
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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 + - 5 V to + - 15 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
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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?
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Op Amp History
 Built during ww2 with vacuum tubes
 With 100-300 V power supplies then
 Solid state Op Amps since 1960s
 Early solid state Op Amps: μA702
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Bob Widlar 1964, retired at age 30 after invention!
9 transistors only in first solid state Op Amps!
 μA741
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Dave Fullagar 1968, Fairchild
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Most popular Op Amp of all time
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Packaged in 8-pin DIP
 OP-07
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Precision Monolithics, 1975
Special-purpose, high gain
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Op Amp History, Early Contributors
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Harold S. Black, US Western Electric, 1930s
Paul Voigt, UK, 1920s
Alan Blumlein, UK, 1930s
Hendrick Bode, US 1930s
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Op Amp Nomenclature
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+Vin noninverting input,
-Vin inverting input,
+VS positive source,
-VS negative source,
+Vout output voltage,
we’ll use: vp
we’ll use: vn
we’ll use: +VCC
we’ll use: -VCC
we’ll use: vo
+Vcc
+Vs
+Vin
vp
+
vo
Vout
-Vin
vn
-
-Vs
+
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-
-Vcc
More Likely You Just See:
+Vin
vp
+
vo
Vout
-Vin
+
vn
-
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-
Op Amp Circuit Symbol
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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
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Actual Op Amp 741 Circuit Detail
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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!
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Op Amp Voltage Transfer Function
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Terminal Current Names
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Terminal Voltage Names
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Ideal vs. Real Op Amps
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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
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Ideal Op Amp Formulae
vp = v n
ip = i n = 0 A
A=∞
-- known as virtual Op Amp short!
-- ∞ input Ω causes no current
-- infinite gain, real ~105 max
ip + in + io + iC+ + iC- = 0 -- Kirchhoff C. Law
From ip = in = 0 it follows:
io = - ( iC+ + iC- )
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Summary Ideal vs. Real Op Amp
Electric Parameters of Real Op Amp
Value
> 105
Large gain A
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 )
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High in linear range
Summary Ideal vs. Real Op Amp
Electric Parameters of ideal Op Amp
Value
∞
Infinite gain A
Zero Ampere input currents ip, in
0A
Very small difference of input voltages: vp - vn
0V
Extremely steep incline of function vo = f( vp - vn )
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∞
Type 1
Noninverting Op Amp
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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!
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Type 1 Noninverting Op Amp, Generic
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Type 1 Noninverting Op Amp, Generic
ia + if – in = 0
– ia runs through Rn
ip = in = 0
– 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
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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
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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|
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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 + -15 V
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Type 1 Noninverting Op Amp, Sample 1
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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 * ( 9 + 1 ) / 1 = 10 V -- is within range + - 15 V
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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
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Type 1 Noninverting Op Amp, Sample 2
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Type 1 Noninverting Op Amp, Sample 2
With vo = vs * ( Rf + Rn ) / Rn
Case 1 for largest vs = 1.5 V
vo+ = 1.5 * ( 9 + 1 ) / 1 = 15 V
vo+ = 15 V
Case 2 for smallest vs = -1.5 V
vo- = -1.5 * ( 9 + 1 ) / 1 = -15 V
Vo- = -15 V
Matching exactly and just barely our previous power
situation + -VCC from Sample 1
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Type 2
Inverting Op Amp
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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
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Type 2 Inverting Op Amp, Generic
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Type 2 Inverting Op Amp
is + if - in = 0 A
-- even with Rp = 0 Ω at the +pin
vn = vP
= 0V
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
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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
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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 + -10 V
Compute vo
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Type 2 Inverting Op Amp, Sample 1
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Type 2 Inverting Op Amp, Sample 1
i25 + i100 - in
= 0 A
with in
= 0
i100
= -i25
i25
= (va - vn) / 25 k
i25
= (va – 0 ) / 25 k
i25
= 0.04 mA
i100
= (vo – vn) / 100k
therefore:
= -0.04
vo / 100k = -0.04
vo
= -4 V
within operating range
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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
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Type 2 Inverting Op Amp, Sample 2
In Sample 2 we use an inverting Op Amp with
1.signal source of voltage vs = 0.4 V
2.vn = vp, vp grounded = 0 V
3.Feedback resistor Rf = 80 kΩ
4.Resistor RS at signal source vs of: RS = 16 kΩ
5.Source voltages +VCC +10 V and -VCC of -15 V
Compute vo
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Type 2 Inverting Op Amp, Sample 2
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Type 2 Inverting Op Amp, Sample 2, Hard
i16 + i80 + in = 0 A
with in
= 0
i80
= -i16
i16
= (vs - vn) / 16 k
i16
= (vs - 0)
i16
= 0.025 mA
i80
= (vo – vn) / 80k = -0.025 mA
vo / 80k
= -0.025
vo
therefore:
= -2 V
/ 16 k
within operating range
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Type 2 Inverting Op Amp, Sample 2, Easy
Students use easier way of computing vo:
Use characteristic equation!
vo = -2 V
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Type 3
Difference Op Amp
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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
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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
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Type 3 Difference Op Amp, Generic
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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
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Type 3 Difference Op Amp
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
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Type 3 Difference Op Amp
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
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Type 3 Difference Op Amp, Sample 1
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Type 3 Difference Op Amp, Sample 1
i100
= -i20
i20
= (va - vn)
i20
= ( 1 - 2 ) / 20k
i100
= (vo – vn) / 100k
i100
= vo/100k - 2/100k
= +0.05 mA
Vo/100k = 0.05 mA + 2/100k A
= 0.07 mA
Vo
= 7 V
as before, since in = 0
/ 20k
= -0.05 mA
outside operating range
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Type 4
Summing Op Amp
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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
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Type 4 Summing Op Amp, Generic
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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
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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 )
Characteristic function for output voltage vo of a
Summing Op Amp with 3 inputs at inverting input, RS = Rf
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Type 4 Summing Op Amp, Sample 1
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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
= -0.02 mA - 0.01 mA = -0.03 mA
vo = -7.5 V
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References
1. Nilsson, James W., and Susan A. Riedel, Electric Circuits, ©
2015 Pearson Education Inc., ISBN 13: 9780-13-376003-3
2. Wiki page: http://en.wikipedia.org/wiki/Operational_amplifier
3. A. D. Blumlein, Improvements in and relating to Thermionic
Valve Amplifiers, UK Patent 425,553, issued March 18, 1935
4. Hendrick Bode, Relations Between Attenuation and Phase In
Feedback Amplifier Design, Bell System Technical Journal, Vol.
19, No. 3, July, 1940
5. Hendrick Bode: Amplifier, US Patent 2,123,178, issued July 12,
1938
6. Dave Fullagar: http://www.edn.com/electronicsnews/4326905/Voices-Dave-Fullagar-analog-IC-designer-andentrepreneur
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