Operational Amplifiers – Complete Theory

Page 2 – Op-Amp Equivalent Circuit

The practical op-amp can be represented using an equivalent circuit model. This model helps analyze op-amp circuits easily.

                                         

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Equivalent Circuit Components

An operational amplifier can be modeled using three main elements:

• Input resistance (Ri)
• Voltage controlled voltage source
• Output resistance (Ro)

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Open Loop Gain Model

The output voltage of the op-amp is:

Vo = A (V+ − V−)

Where

• A = Open loop gain
• V+ = Non-inverting input
• V− = Inverting input

The open loop gain of practical op-amps is extremely high:

A ≈ 10⁵ to 10⁶

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Input Resistance

Input resistance is very high so that almost no current enters the amplifier.

Ri ≈ ∞

This implies

I+ ≈ 0 I− ≈ 0

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Output Resistance

The output resistance of an ideal op-amp is approximately zero.

Ro ≈ 0

This allows the op-amp to drive loads efficiently.

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Ideal Op-Amp Model

• Infinite input resistance
• Zero output resistance
• Infinite open loop gain
• Infinite bandwidth

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GATE Important Points

• Input current ≈ 0
• Open loop gain extremely large
• Output voltage proportional to input difference
• Equivalent model used for circuit analysis

 

Operational Amplifiers – Complete Theory

Page 1 – Introduction to Operational Amplifier

                                             

An Operational Amplifier (Op-Amp) is a high gain differential amplifier designed to amplify the difference between two input voltages.

It is one of the most important components in Analog Electronics and widely used in:

• Signal amplification
• Filters
• Oscillators
• Analog computation
• Instrumentation systems

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Basic Structure

An Op-Amp has three main terminals:

• Non-inverting input (+)
• Inverting input (−)
• Output terminal

The output voltage depends on the difference between the input voltages.

Vo = A (V+ − V−)

Where

• A = Open loop gain
• V+ = Non-inverting input voltage
• V− = Inverting input voltage

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Ideal Op-Amp Characteristics

• Infinite open loop gain (A → ∞)
• Infinite input resistance
• Zero output resistance
• Infinite bandwidth
• Infinite CMRR
• Infinite slew rate

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Important Concept

Because gain is extremely large:

V+ ≈ V−

This is called the Virtual Short Concept.

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GATE Important Points

• Op-Amp is a differential amplifier
• Output depends on difference of input voltages
• Virtual short concept is used in circuit analysis
• Open loop gain is extremely large (10⁵ – 10⁶)

 

Analog Electronics – Page 43

Current Shunt Feedback Amplifier – Complete Derivation

                                   

In a Current Shunt Feedback Amplifier:

• The output current is sampled.
• The feedback signal is applied in parallel (shunt) with input.

This configuration is commonly used in Current Amplifiers.

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Closed Loop Gain Derivation

Let:

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

Feedback current:

If = βIo

Input current becomes:

Ii = Is − If

Output current:

Io = A Ii

Substitute:

Io = A(Is − βIo)

Rearranging:

Io(1 + Aβ) = AIs

Closed loop gain:

Af = Io / Is = A / (1 + Aβ)

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Input Resistance with Feedback

Since feedback is applied in parallel, input resistance decreases.

Rif = Ri / (1 + Aβ)

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Output Resistance with Feedback

Current sampling increases output resistance.

Rof = Ro (1 + Aβ)

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Key Characteristics

• Input resistance decreases
• Output resistance increases
• Gain stabilizes
• Bandwidth improves

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Important GATE Points

• Current Shunt Feedback → Current Amplifier
• Closed loop gain = A / (1 + Aβ)
• Input resistance decreases
• Output resistance increases

 

Analog Electronics – Page 42

Current Series Feedback Amplifier – Complete Derivation

                                      

                                             

Current series feedback is a feedback configuration where:

• Output current is sampled
• Feedback signal is applied in series with input

This configuration is commonly used in transconductance amplifiers.

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Closed Loop Gain Derivation

Let

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

Feedback current:

If = βIo

Input current to amplifier:

Ii = Is − If

Output current:

Io = A Ii

Substitute feedback:

Io = A(Is − βIo)

Expand equation:

Io = AIs − AβIo

Rearranging:

Io + AβIo = AIs

Io(1 + Aβ) = AIs

Closed loop gain:

Af = Io / Is = A / (1 + Aβ)

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Input Resistance with Feedback

Series mixing increases input resistance.

Rif = Ri (1 + Aβ)

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Output Resistance with Feedback

Current sampling increases output resistance.

Rof = Ro (1 + Aβ)

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Key Characteristics

• Input resistance increases
• Output resistance increases
• Gain stabilizes
• Bandwidth improves

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Important GATE Points

• Current series feedback → Transconductance amplifier
• Closed loop gain = A / (1 + Aβ)
• Input resistance increases
• Output resistance increases

 

Analog Electronics – Page 41

Voltage Shunt Feedback Amplifier – Complete Derivation

                    

Voltage shunt feedback is a feedback configuration in which:

• Output voltage is sampled
• Feedback signal is applied in parallel (shunt) with the input

This configuration is commonly used in transresistance amplifiers.

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Closed Loop Gain Derivation

Let

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

Feedback voltage:

Vf = βVo

Input signal current becomes:

Ii = Is − If

Output voltage:

Vo = A Ii

Substituting feedback:

Vo = A(Is − βVo)

Rearranging:

Vo + AβVo = AIs

Vo(1 + Aβ) = AIs

Closed loop gain:

Af = A / (1 + Aβ)

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Input Resistance with Feedback

Shunt mixing reduces input resistance.

Rif = Ri / (1 + Aβ)

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Output Resistance with Feedback

Voltage sampling reduces output resistance.

Rof = Ro / (1 + Aβ)

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Key Characteristics

• Input resistance decreases
• Output resistance decreases
• Gain becomes stable
• Bandwidth increases

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Important GATE Points

• Voltage shunt feedback → Transresistance amplifier
• Closed loop gain = A / (1 + Aβ)
• Input resistance decreases
• Output resistance decreases
•         
  
  
  

 

Analog Electronics – Page 40

Voltage Series Feedback Amplifier – Complete Derivation

                                              

Voltage series feedback is one of the most widely used feedback configurations in amplifiers.

In this method:

• Output voltage is sampled
• Feedback signal is applied in series with input

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Closed Loop Gain Derivation

Let

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

Feedback voltage:

Vf = βVo

Input to amplifier:

Vi = Vs − Vf

Substitute feedback:

Vi = Vs − βVo

Output voltage:

Vo = AVi

Substitute Vi:

Vo = A(Vs − βVo)

Expand equation:

Vo = AVs − AβVo

Rearranging:

Vo + AβVo = AVs

Vo(1 + Aβ) = AVs

Closed loop gain:

Af = Vo / Vs = A / (1 + Aβ)

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Input Resistance with Feedback

Voltage series feedback increases input resistance.

Rif = Ri (1 + Aβ)

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Output Resistance with Feedback

Voltage sampling reduces output resistance.

Rof = Ro / (1 + Aβ)

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Key Characteristics

• Input resistance increases
• Output resistance decreases
• Gain becomes stable
• Bandwidth increases

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Important GATE Points

• Voltage series feedback → Voltage amplifier
• Closed loop gain = A / (1 + Aβ)
• Input resistance increases
• Output resistance decreases

 

Analog Electronics – Page 39

Four Types of Feedback Amplifiers

Feedback amplifiers are classified based on:

• How the feedback signal is taken from output
• How the feedback signal is applied to input

                                                 

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Types of Feedback Amplifiers

Feedback Type	Output Sampling	Input Mixing	Amplifier Type
Voltage Series	Voltage	Series	Voltage Amplifier
Voltage Shunt	Voltage	Parallel	Transresistance Amplifier
Current Series	Current	Series	Transconductance Amplifier
Current Shunt	Current	Parallel	Current Amplifier

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1. Voltage Series Feedback

• Output voltage sampled
• Feedback applied in series with input
• Increases input impedance
• Decreases output impedance

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2. Voltage Shunt Feedback

• Output voltage sampled
• Feedback applied in parallel
• Reduces input impedance
• Reduces output impedance

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3. Current Series Feedback

• Output current sampled
• Feedback applied in series
• Increases input impedance
• Increases output impedance

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4. Current Shunt Feedback

• Output current sampled
• Feedback applied in parallel
• Reduces input impedance
• Increases output impedance

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Important GATE Points

• Feedback classification depends on input mixing and output sampling
• Series mixing increases input resistance
• Shunt mixing decreases input resistance
• Voltage sampling reduces output resistance
• Current sampling increases output resistance