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

 

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