Electrical and Electronic Fundamental

R1 Rf R1 V - Rf Vo
V1 V1
Virtual I3
R2 I1 R2 short
V2 - Vo V2

+ I2

V+

Figure 8.13: Inverting summer and its virtual equivalent circuit

At the node V-, by applying KCL;

I1 + I2 = I3

V1- V- + V2 - V- = V- - Vo
R1 R2 Rf

and V- + V+ = 0

V1 - 0 + V2 - 0 = 0 - Vo
R1 R2 Rf

Thus,

Vo = - �RR1f V1 + Rf V2�
R2

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Example 8.7

Determine the output voltage of the circuit

V1=+0.2V 33kΩ 33kΩ
V1=-0.5V 22kΩ
V1=+0.8V 11kΩ -
+
Vo

Answer:

Vo = - �3333kk × 0.2 + 33k × (-0.5) + 33k × 0.8� = 1.6V
22k 11k

8.6 Multistage Amplifier

The performance obtainable from a single stage amplifier is might be
insufficient for many applications. Hence several stages may be combined
forming a multistage amplifier. When a number of stages are connected in
series or cascade, the overall gain is the product of the individual stage
gains. For example the output of the first stage is connected to form input
of second stage, whose output becomes input of third stage, and so on. In
order to analyze the circuits, we should determine the stage by stage of the
amplifiers output. Figure 8.14 shows the example of the multistage amplifies.
In this example, 3 stages of amplifier circuits involve. Those are combination
of one non inverting amplifier and two of inverting amplifier. The final output
can be found by determining the outputs of each stage from left to right.

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Rf Rf Rf Vo

R1 R2 R3
- - -
1 2 3
+ + +

V1

Figure 8.14: Multistage amplifier

Output of the amplifier 1,

Vo1 = �1 + RR1f � × V1

Output of the amplifier 2,

Vo2 = - �RR2f � × Vo1

Output of the amplifier 3,

Vo = - �RR3f � × Vo2
By substituting 1 and 2 into

Vo = - �RR3f � �- RR2f � �1 + RR1f � × V1= �RR3f � �RR2f � �1 + RR1f � × V1

The overall voltage gain is

Vo = Av = �RR3f � �RR2f � �1 + RR1f �
V1

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Example 8.8

i. Name the type of the amplifier of circuit 1,2 and 3.

ii. Calculate the output voltage of the amplifier, Vo1, Vo2 and Vo3 if V1 =
V2 = 1V

R1
100kΩ

R2 Vo1 R5 R7
2kΩ 20kΩ
10kΩ
V1 - R6 -
4kΩ 3
1
+ +

R4 Vo
200kΩ

R3

25kΩ

-
2
V2 + Vo2

Answer:
1. Op-amp 1: Inverting operational amplifier

Op-amp 2: Non-inverting operational amplifier
Op-amp 3: Inverting summing operational amplifier

2. For op-amp 1;
Vo1 = - �RR12� × V1 = - �11000� × 1 = -10V

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For op-amp 2;
Vo2 = �1 + RR43� × V2 = �1 + 22050kk� × 1 = 9V

For op-amp 3;

Vo = - �RR57 × Vo1 + R7 × Vo2� = - �220kk × (-10) + 20k × 9�
R6 4k

=-(-100 + 45) = 55V

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Summary
This chapter covers the operational amplifies such as
1. Op-Amp symbol, terminals and physical package
2. Ideal and practical op-amp characteristic such as input impedance,

output impedance, voltage gain and band width.
3. Virtual equivalent circuit for op amp that is the current at both of the

op-amp inputs is zero. I+ =I- = 0 the voltages at the two inputs are
equal. V+ = V-.
4. Open loop configuration does not required feedback circuit and
main apply as comparator
5. Close loop op-amp configuration has a negative feedback circuit
used to control the gain mainapplies to the inverting, non-inverting,
summer and substract amplifier.
6. Analyze and calculate the basic op-amp circuits such as
comparator, inverting, non-inverting,summer and substract amplifier.

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REFERENCES
Alexander, Charles K. and Sadiku, Matthew N. O., Fundamentals of Electric

Circuits 2019, 6th Edition, McGraw Hill.
Boylestad, Robert and Nashelsky, Louis, Electronic Devices and Circuit

Theory, 7th Edition, Prentice Hall.
Floyd, Thomas L., Electronic Devices 2012, 9th Edition, Prentice Hall.
Nilsson, James, W. and Riedel, Susan, A, Electric Circuits, 10th Edition,

Addison-Wesley Publishing Company.

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