Operational Amplifiers – Complete Theory

Page 9 – Op-Amp Integrator

The Integrator is an operational amplifier circuit that produces an output proportional to the integral of the input signal.

                                               

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Circuit Components

• Input resistor (R)
• Feedback capacitor (C)
• Operational amplifier
• Input voltage Vin

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Virtual Ground Concept

Since the non-inverting terminal is grounded:

V− ≈ 0

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Input Current

The current through the input resistor is:

I = Vin / R

Because op-amp input current ≈ 0, this current flows through the capacitor.

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Capacitor Current Equation

Current through capacitor:

I = C ( dVout / dt )

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Derivation

Equating currents:

Vin / R = C ( dVout / dt )

Rearranging:

dVout / dt = Vin / RC

Integrating:

Vout = − (1 / RC) ∫ Vin dt

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Key Result

Output voltage is proportional to the integral of input voltage.

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Waveform Behavior

• Square input → Triangular output
• Constant input → Ramp output
• Sine input → Cosine output

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Applications

• Analog computers
• Signal processing
• Waveform generation
• Control systems

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GATE Important Points

• Integrator produces output proportional to ∫Vin dt
• Uses capacitor in feedback path
• Important for waveform conversion
• Common question: square → triangular waveform

 

Operational Amplifiers – Complete Theory

Page 8 – Differential Amplifier (Subtractor)

                                     

The Differential Amplifier is an operational amplifier circuit that amplifies the difference between two input signals.

It is also called a Subtractor Circuit.

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Purpose of Differential Amplifier

• Amplifies the difference between two voltages
• Rejects common noise signals
• Used in instrumentation and sensor circuits

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Circuit Components

• Four resistors (R1, R2, R3, R4)
• Two input voltages (V1 and V2)
• One operational amplifier

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Voltage at Non-Inverting Terminal

Using voltage divider:

V+ = (R4 / (R3 + R4)) × V2

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Voltage at Inverting Terminal

Using op-amp property:

V− ≈ V+

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Output Voltage Derivation

The general output equation is:

Vout = (R2 / R1) (V2 − V1)

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Special Case (Balanced Differential Amplifier)

If

R1 = R3 R2 = R4

Then

Vout = (R2/R1) (V2 − V1)

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Applications

• Instrumentation amplifiers
• Noise rejection circuits
• Sensor signal conditioning
• Data acquisition systems

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GATE Important Points

• Amplifies difference between two inputs
• Rejects common-mode signals
• Used in instrumentation amplifiers
• Important concept: Common Mode Rejection Ratio (CMRR)

 

Operational Amplifiers – Complete Theory

Page 7 – Summing Amplifier (Adder)

                               

                  

The Summing Amplifier is an op-amp circuit used to add multiple input voltages.

It is also called an Adder Circuit.

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Circuit Configuration

• Multiple input signals connected through resistors
• All inputs applied to the inverting terminal
• Non-inverting terminal grounded
• Feedback resistor connected from output to input

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Virtual Ground Concept

Since the non-inverting terminal is grounded:

V− ≈ 0

This node behaves like a virtual ground.

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Input Currents

Each input produces a current:

I1 = V1 / R1

I2 = V2 / R2

I3 = V3 / R3

Total current entering the node:

I = I1 + I2 + I3

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Output Voltage

The output voltage is:

Vout = −Rf ( V1/R1 + V2/R2 + V3/R3 )

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Special Case (Equal Resistors)

If

R1 = R2 = R3 = Rf

Then

Vout = − (V1 + V2 + V3)

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Applications

• Audio mixers
• Signal processing
• Digital to analog converters
• Analog computation

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GATE Important Points

• Summing amplifier adds multiple inputs
• Uses virtual ground concept
• Output is inverted
• Weighted sum possible

 

Operational Amplifiers – Complete Theory

Page 6 – Voltage Follower (Buffer Amplifier)

The Voltage Follower is a special case of the non-inverting amplifier.

In this circuit the output is directly connected to the inverting terminal.

                                        

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Circuit Configuration

• Input applied to non-inverting terminal (+)
• Output connected directly to inverting terminal (−)
• No external feedback resistor

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Virtual Short Concept

For an ideal op-amp:

V+ ≈ V−

Since

V+ = Vin

and output is connected to the inverting terminal:

V− = Vout

Therefore

Vout = Vin

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Voltage Gain

Av = Vout / Vin = 1

This is why it is called a Unity Gain Amplifier.

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Important Characteristics

• Very high input impedance
• Very low output impedance
• Gain = 1
• No phase inversion

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Applications

• Impedance matching
• Signal buffering
• Isolation between circuits
• Sensor interface circuits

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GATE Important Points

• Voltage follower = unity gain amplifier
• Gain = 1
• Used for impedance matching
• Output follows input voltage

 

Operational Amplifiers – Complete Theory

Page 5 – Non-Inverting Amplifier (Derivation)

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The Non-Inverting Amplifier is another important op-amp configuration.

In this circuit the input signal is applied to the non-inverting terminal (+) of the op-amp.

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Circuit Components

• Feedback resistor → Rf
• Ground resistor → R1
• Operational amplifier

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Virtual Short Concept

For an ideal op-amp:

V+ ≈ V−

Since input is applied to the non-inverting terminal:

V+ = Vin

Therefore

V− ≈ Vin

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Voltage at Inverting Terminal

The inverting terminal is connected to a voltage divider.

V− = Vout × ( R1 / (R1 + Rf) )

Since V− = Vin

Vin = Vout × ( R1 / (R1 + Rf) )

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Output Voltage

Rearranging the equation:

Vout = Vin ( 1 + Rf / R1 )

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Voltage Gain

Av = Vout / Vin = 1 + (Rf / R1)

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Important Properties

• No phase inversion
• High input impedance
• Stable amplifier configuration
• Gain controlled by feedback resistors

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GATE Important Points

• Gain = 1 + (Rf / R1)
• Input applied to non-inverting terminal
• Output in phase with input
• High input resistance

 

Operational Amplifiers – Complete Theory

Page 4 – Inverting Amplifier (Derivation)

                                       

The Inverting Amplifier is one of the most commonly used operational amplifier circuits.

In this configuration, the input signal is applied to the inverting terminal through a resistor.

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Circuit Components

• Input resistor → R_in
• Feedback resistor → R_f
• Operational amplifier

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Virtual Ground Concept

Because of the very high gain of the op-amp:

V₊ ≈ V_-

If the non-inverting terminal is grounded:

V_- ≈ 0

This point is called Virtual Ground.

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Current Through Input Resistor

I = V_in / R_in

Because input current of op-amp is approximately zero:

I_in ≈ 0

So the same current flows through the feedback resistor.

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Output Voltage

V_out = - I R_f

Substitute current value:

V_out = - (V_in / R_in) R_f

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Voltage Gain

A_v = V_out / V_in = - R_f / R_in

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Important Properties

• Output is 180° phase shifted
• Gain controlled by resistors
• Input resistance = R_in
• Stable amplifier configuration

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GATE Important Points

• Gain = −Rf / Rin
• Virtual ground concept
• Input current ≈ 0
• Output inverted signal

 

Operational Amplifiers – Complete Theory

                                          

Page 3 – Open Loop vs Closed Loop Operation

An operational amplifier can operate in two modes:

• Open Loop Operation
• Closed Loop Operation

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1. Open Loop Operation

In open loop operation, no feedback is used.

Vo = A (V+ − V−)

Where

• A = Open loop gain (very large)
• V+ = Non-inverting input
• V− = Inverting input

The open loop gain of an op-amp is extremely high:

A ≈ 10⁵ to 10⁶

Because of this large gain, even a very small input difference produces a large output voltage.

Applications

• Comparator circuits
• Zero crossing detectors
• Switching circuits

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2. Closed Loop Operation

Closed loop operation uses negative feedback.

Negative feedback stabilizes the gain and makes the amplifier linear.

Af = A / (1 + Aβ)

Where

• A = Open loop gain
• β = Feedback factor
• Af = Closed loop gain

Advantages

• Stable gain
• Improved bandwidth
• Reduced distortion
• Better linearity

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Comparison

Parameter	Open Loop	Closed Loop
Feedback	No feedback	Negative feedback used
Gain	Extremely high	Controlled gain
Stability	Poor	Good
Applications	Comparators	Amplifiers

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GATE Important Points

• Open loop gain ≈ 10⁵ – 10⁶
• Closed loop gain controlled using feedback
• Negative feedback improves stability
• Most op-amp circuits use closed loop configuration