op-amp

Voltage-series feedback amplifiers [Non-inverting Amplifier]:

The input voltage is applied to the non-inverting terminal and the inverting terminal
is grounded. The negative feedback is applied to the inverting terminal through the
resistor RF.

Note: Inverting terminal is grounded through a resistor R1.

The circuit diagram for Non-inverting Amplifier is as shown in figure.

Closed loop voltage gain:

The closed loop voltage gain AF can be obtained as follows.

We know that,

Open loop voltage gain is given by, = ……….. (1)
……….. (2)
……….. (3)

Closed loop voltage gain is given by, =


Gain of the feedback circuit is given by, =


From equation 1

=

= ( 1 − 2) ……….. (4)

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From figure

1 =

2 = = 1
+ 1

By substituting 1 2 in (4)

0 = ( − 1 )
1 +

0 = ( ( 1 + ) − 1 )
( 1 + )

0( 1 + ) + 1 = ( 1 + )

0( 1 + + 1) = ( 1 + )

= = ( 1+ ) (exact)
( 1+ + 1)

Since A is very large. Thus AR1≫ + 1

Thus 1 + + 1 ≈ R1

Hence

= = ( 1 + ) = ( 1 + )
( 1) ( 1)

= 1 + (ideal)
1

Dividing equation (3) both the numerator and denominator by + 1.

= = ( 1 + )⁄ + 1

+ 1⁄ + 1 + 1 ⁄ + 1

= =
1⁄
1 + + 1

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= 1 +

Where = 1
+ 1

The block diagram of the non-inverting amplifier is as shown in figure.

Input resistance with feedback:

Figure shows the op-amp equivalent circuit. Let Ri be the input resistance without
feedback and RF be the input resistance with feedback.

The input resistance with feedback is given by,

= …… (1)


We know that =


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Thus equation 1 becomes

= = …… (2)
⁄

However we know that = and =
1+

Thus (2) becomes = =
⁄

= = (1 + )
⁄1 +


= (1 + )

This means that input resistance of op-amp increases by a factor of (1 + )

times that without feedback.

Output resistance with feedback:

Output resistance is the resistance seen through the feedback amplifier from the
output terminals. The resistance can be obtained by using thevinin’s theorem. To

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find output resistance with feedback (ROF) reduce vin to zero, apply external

voltage vo then calculate resulting current io.

Thus = … (1)

We know that = 1 − 2

Since 1 = 0 and 2 =

= 0 −

= 0− 1
+ 1

= −

Since = ( 1 )

+ 1

Substituting in equation 3.

= +


Substituting io in equation 1

=
+


=
(1 + )

= (1
+ )

This means that output resistance decreases by a factor of 1 + . That is the

output resistance with feedback is less than output resistance with feedback.

Bandwidth with Feedback:

The gain Bandwidth product of a single break frequency op-amp is always constant.
Gain of the amplifier with feedback is less than gain without feedback. Therefore
bandwidth of amplifier with feedback fF must be larger than that without
feedback.
The frequency at which the gain A is 3dB less than from its value at 0 Hz is called
as break frequency.
The frequency at which the gain of the amplifier is equal to 1 is called as unity gain
bandwidth.

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Thus

=

Where,


A is called as open loop gain

Also =

Where,

ℎ ℎ
AF is called as closed loop gain.

Thus,

=

=

Since =
1+

=

⁄1 +

= (1 + )



= (1 + )

Where is the break frequency of the op-amp.
ℎ

=


=

Total output offset voltage with feedback:

When the temp & power supply are fixed, the output offset voltage is a function of
the gain of an op-amp.
Gain of the op amp with feedback is always less than the gain without feedback.
Therefore the total output offset voltage (VooT) must be smaller with feedback.

VooT = Total output offset volt without feedback

1 +

VooT = ±Vsat

1 +

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Where, ±Vsat= saturation voltage
A= open loop gain of the op-amp.
= gain of the feedback amplifier.

The output voltage of the op-amp without feedback can be either±Vsat, because A
is very high. VooT is same for inverting and non-inverting amplifier.
The negative feedback also reduces the

 Effect of noise
 Variations in supply voltages
 Changes in temperature on the output voltage

Voltage follower:
When the non-inverting amplifier is configured for unity gain is called as voltage
follower.
In other words voltage follower the output voltage follows the input voltage.
This circuit is similar to emitter follower. Even though voltage follower is
preferred since it is having high input impedance and output is exactly equal to
input.
To obtain voltage follower simply short the RF and open R1 as shown in figure

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From data AF=1

Thus, = = 1
1+

Hence, = 1 +

All the formulas changes into

AF=1

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Voltage-shunt feedback amplifiers [Inverting Amplifier]:

The input voltage drives the inverting terminal, and amplified as well as inverted
output signal also applied to the inverting input via feedback resistor RF.

Note: Non-inverting terminal is grounded and feedback circuit has RF and extra
resistor R1 is connected in series with the input signal source Vin.

The circuit diagram for Inverting Amplifier is as shown in figure.

Closed loop voltage gain:

The closed loop voltage gain AF can be obtained by writing Kirchhoff’s current
equation at the input node v2 as follows.

iin = iF + iB……….. (1)

Since Ri is very large and current iB is negligibly small. Thus iB=0

iin ≅ iF

Thus, − 2 = 2− …… (2)
We know that, 1
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1 2 =
−

1 − 2 =


Since 1 = 0

2 = −


Substituting the value of v2 in equation 2

− (− ) = (− ) −


1

+ = (− ) −


1

( + ) = ((− ) − ) 1


+ 1 + 1 = −


+ 1 + 1 = −

( + 1 + 1 ) = −

= = − (exact)….. (3)
+ 1+ 1

Since A is very large. Thus AR1≫ + 1

Thus 1 + + 1 ≈ R1

Hence

= = − (ideal) …… (4)

1

Note:

 The gain of the amplifier can be set to any value even to less than 1. (since in
non-inverting amplifier the least gain was 1).

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Dividing equation (3) both the numerator and denominator by + 1.

= = − + ⁄ + 1
1⁄ + 1 + 1 ⁄
+ 1

= = − ⁄ + 1
+ 1⁄ + 1
1

= − …. 5
1+

Where =
+ 1

= 1

+ 1

Note:
 The negative sign indicates phase inversion.
 The gain of the inverting amplifier is K times of the closed loop gain of non-

inverting amplifier. Where K< 1.

The block diagram of the inverting amplifier is as shown in figure.

In equation (5) 1 + ≫ 1 thus 1 + ≅

Then,

= − = −
1

Virtual ground: In figure the non-inverting terminal is grounded and the input

signal is applied to inverting terminal via resistor R1. Since the differential input is

ideally zero; thus the voltage at inverting terminal (v2) is equal to voltage at non-

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inverting terminal (v1). It means inverting terminal potential is approximately at
ground potential. Therefore inverting terminal is said to be virtual ground.

The ideal closed loop voltage can be obtained using the virtual ground concept is as
follows.

iin ≅ iF

− 2 = 2 −
1

However v1=v2=0

Therefore,

= −
1

Thus, = = −

1

Input resistance with feedback:

Easiest method of finding the input resistance is to millerize the feedback resistor
RF. Split RF in to its 2 Miller components as shown in fig.

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Millers theorem:

According to millers theorem the feedback resistor RF can be divided

into two components. Namely Rx and Ry.

From figure Rx = v2−0 and Ry = vo−0


WKT

= 1- 2

Since 1=0 therefore =- 2

Thus

= −



Now consider as

I= 2−

− −
I=

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I= (1+ )



I= (1+ )


= =
(1+ ) I

Similarly,

I= − 2


I= (1+ )



I= (1+ 1 )



(1 + 1 )=
I

(1 + )= I =

Thus, = 1 + ∥ ( )

In this circuit, the input resistance with feedback RiF is then

= 1 + ∥ ( ) (exact)
1+

Since A and Ri is very large.

∥ ( ) ≅ 0
1 +

Hence = 1 (ideal)

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Output resistance with feedback:

Output resistance is the resistance seen through the feedback amplifier from the
output terminals. The resistance can be obtained by using thevinin’s theorem.
To find output resistance with feedback (ROF) reduce vin to zero, apply external
voltage vo then calculate resulting current io.

Thus = … (1)

Applying KCL to the output node,

= + ….. (2)
Since ≫

Therefore ≅

The io can be determined by applying KVL to the output loop.
i.e. − − = 0
= −

= − …. (3)

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We know that = 1 − 2

Since 1 = 0 and 2 =

= 0 −

= 0− 1
+ 1

= −

Since = ( 1 )

+ 1

Substituting in equation 3.

= +


Substituting io in equation 1

=
+


=
(1 + )

= (1
+ )

This means that output resistance decreases by a factor of 1 + . That is the

output resistance with feedback is less than output resistance with feedback.

Bandwidth with Feedback:

The gain Bandwidth product of a single break frequency op-amp is always constant.

Gain of the amplifier with feedback is less than gain without feedback. Therefore

bandwidth of amplifier with feedback fF must be larger than that without

feedback.

= (1 + )… (1)

Where is the break frequency of the op-amp.
ℎ

=

= (2)
…

Substituting f0 in equation 1.


= (1 + )

⋅
=

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Where =
+ 1

= −

1+

Same closed loop gain, the closed loop bandwidth for the inverting amplifier is less
than the Non – inverting amplifier by a factor of K.

Total output offset voltage with feedback:

When the temp & power supply are fixed, the output offset voltage is a function of
the gain of an op-amp.
Gain of the op amp with feedback is always less than the gain without feedback.
Therefore the total output offset voltage (VooT) must be smaller with feedback.

VooT = Total output offset volt without feedback

1 +

VooT = ±Vsat

1 +

Where, ±Vsat= saturation voltage
A= open loop gain of the op-amp.

= gain of the feedback amplifier.

The output voltage of the op-amp without feedback can be either±Vsat, because A
is very high. VooT is same for inverting and non-inverting amplifier.

The negative feedback also reduces the
 Effect of noise
 Variations in supply voltages
 Changes in temperature on the output voltage

Current to voltage convertor:

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Consider the ideal voltage gain

= −
1

Therefore:

= − ( 1 )

= − ( 1 )

We know that =
1

= −

Thus the output voltage is proportional to the input current. And the above circuit
converts current into voltage.

Inverter:

It is a circuit which is used to produce a signal having same amplitude but opposite
polarity or having a phase shift of 180o. The inverting op-amp can be converted to
inverter by making RF =R1=R as shown in the below circuit.

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Voltage to Frequency converter:

Frequency to voltage converter is an electronic device which converts the
sinusoidal input frequency into a proportional current or output voltage. The basic
circuit includes operational amplifiers and RC circuits (Resistor Capacitor
networks). The operational amplifiers are used for signal processing. And the RC
networks are used to remove the frequency dependent ripples.
Applications:

1. Frequency to voltage converter in tachometers.
2. Frequency difference measurement.

Frequency to voltage converter:

A voltage-to-frequency converter (VFC) is an oscillator .Its frequency is linearly
proportional to the control voltage The voltage to frequency converter(VFC) is also
very useful for telemetry applications, since the VFC, which is cheap ,small, and
low-powered can be mounted.
Voltage-to-frequency converters are sometimes needed in some instrumentation
applications
Application:
Analog to Digital Conversion
Linear Voltage-Frequency Conversion

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Formula list for problems:

parameter Non-inverting amplifier Voltage Inverting amplifier.
follower
Voltage = ( 1+ ) (exact) − (exact)
gain ( 1+ + 1) = 1 + 1+ 1

= 1 + (ideal) = − (ideal)
1
1

= − 1 +
= − 1 +

Input = (1 + ) = = 1 + ∥ ( )(exact)
resistance 1+
Output
resistance = 1 (ideal)

Bandwidth = (1 = = (1
+ ) + )
Total
output = (1 + ) = = (1 + )
offset
voltage ⋅
(VooT) = =
±Vsat ±Vsat ±Vsat
VooT = 1 + VooT = VooT = 1 +

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