Tuesday, 10 March 2026

Operational Amplifiers – Complete Theory

Page 15 – Active Low Pass Filter

An Active Low Pass Filter allows low-frequency signals to pass while attenuating high-frequency signals.

Basic Components

Operational amplifier
Resistor (R)
Capacitor (C)

Working Principle

At low frequencies → capacitor behaves like open circuit
Signal passes through amplifier
At high frequencies → capacitor offers low impedance
Signal gets attenuated

Cutoff Frequency

fc = 1 / (2πRC)

Where:

R → Resistance
C → Capacitance

Frequency Response

Passband → constant gain
Cutoff frequency → gain reduces to 0.707
Roll-off → −20 dB/decade

Advantages of Active Filters

No inductors required
High input impedance
Easy gain control

Applications

Audio signal processing
Noise filtering
Communication systems
Signal conditioning circuits

GATE Important Points

Cutoff frequency formula: fc = 1 / (2πRC)
Roll-off slope = −20 dB/decade
Uses op-amp + RC network
Important for frequency response questions

Operational Amplifiers – Complete Theory

Page 14 – Instrumentation Amplifier

An Instrumentation Amplifier is a precision amplifier designed to amplify small differential signals while rejecting common-mode noise.

.. .

Main Characteristics

Very high input impedance
High Common Mode Rejection Ratio (CMRR)
Accurate gain control
Low noise amplification

Basic Structure

The instrumentation amplifier uses three operational amplifiers.

Two op-amps act as input buffers
One op-amp acts as differential amplifier

Gain Equation

Vout = (1 + 2R / Rg) × (V2 − V1)

Where:

Rg → Gain controlling resistor
V1, V2 → Input voltages

Why Instrumentation Amplifier?

Amplifies very small signals
Rejects common-mode noise
Provides stable and accurate gain

Applications

Biomedical instruments (ECG, EEG)
Sensor signal amplification
Data acquisition systems
Industrial measurement systems

GATE Important Points

Uses three op-amps
Very high input impedance
High CMRR
Gain controlled by resistor Rg

Operational Amplifiers – Complete Theory

Page 13 – Precision Rectifier (Super Diode)

A Precision Rectifier is an op-amp circuit that rectifies signals without the voltage drop normally caused by diodes.

It is also called a Super Diode.

Problem with Normal Diodes

Ordinary diode rectifiers require a minimum voltage:

Vd ≈ 0.7 V (Silicon diode)

This causes errors when rectifying small signals.

How Precision Rectifier Works

Uses an operational amplifier with a diode
Op-amp compensates the diode voltage drop
Allows rectification of very small signals

Operation

Positive Input Cycle

Op-amp output drives the diode forward biased
Output follows input signal

Negative Input Cycle

Diode becomes reverse biased
Output becomes zero

Key Advantage

Rectifies signals even smaller than 0.7 V.

Applications

AC voltmeters
Signal detectors
Peak detection circuits
Audio signal processing

GATE Important Points

Also called Super Diode
Eliminates diode threshold voltage error
Used for small signal rectification
Improves measurement accuracy

Operational Amplifiers – Complete Theory

Page 12 – Schmitt Trigger

The Schmitt Trigger is a comparator circuit with positive feedback. It introduces hysteresis, which improves noise immunity.

Why Schmitt Trigger?

Removes noise from signals
Prevents multiple switching
Provides stable digital output

Positive Feedback

In this circuit, a portion of the output is fed back to the input.

Positive feedback creates two switching thresholds.

Upper Threshold Voltage (UTP)

UTP = (R1 / (R1 + R2)) × Vsat

This is the voltage at which the output switches from negative saturation to positive saturation.

Lower Threshold Voltage (LTP)

LTP = − (R1 / (R1 + R2)) × Vsat

This is the voltage at which the output switches from positive saturation to negative saturation.

Hysteresis Width

Hysteresis = UTP − LTP

This difference creates a dead band that removes noise.

Transfer Characteristic

When Vin > UTP → Output = +Vsat
When Vin < LTP → Output = −Vsat
Between UTP and LTP → Output remains unchanged

Applications

Wave shaping circuits
Noise filtering
Square wave generation
Switching circuits

GATE Important Points

Uses positive feedback
Introduces hysteresis
Two threshold voltages (UTP & LTP)
Improves noise immunity

Operational Amplifiers – Complete Theory

Page 11 – Op-Amp Comparator

A Comparator is an operational amplifier circuit that compares two voltages and produces an output indicating which one is larger.

Basic Principle

If Vin > Vref → Output goes to positive saturation
If Vin < Vref → Output goes to negative saturation

Output Levels

Vout = +Vsat when Vin > Vref

Vout = -Vsat when Vin < Vref

Types of Comparators

Non-inverting comparator
Inverting comparator
Zero-crossing detector

Zero Crossing Comparator

When reference voltage is zero:

Vref = 0

The circuit detects when the input signal crosses zero.

Applications

Analog to digital converters
Level detection circuits
Wave shaping circuits
Signal detection systems

GATE Important Points

Comparator operates in open-loop mode
Output saturates at ±Vsat
Very high gain of op-amp
Used in zero-crossing detection

Operational Amplifiers – Complete Theory

Page 10 – Op-Amp Differentiator

The Differentiator is an operational amplifier circuit that produces an output proportional to the rate of change of the input signal.

Circuit Components

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

Virtual Ground Concept

Since the non-inverting terminal is grounded:

V− ≈ 0

Capacitor Current

Current through capacitor:

I = C ( dVin / dt )

Because op-amp input current ≈ 0, this same current flows through the feedback resistor.

Voltage Across Resistor

Using Ohm’s law:

Vout = − I R

Substituting current:

Vout = − RC ( dVin / dt )

Key Result

Output voltage is proportional to the derivative of input voltage.

Waveform Behavior

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

Applications

Edge detection circuits
Waveform shaping
High-pass filter circuits
Signal processing

GATE Important Points

Differentiator produces output proportional to dVin/dt
Capacitor connected at input
Resistor in feedback path
Acts as a high-pass circuit

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.

Circuit Components

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

Virtual Ground Concept

Since the non-inverting terminal is grounded:

V− ≈ 0

Input Current

The current through the input resistor is:

I = Vin / R

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

Capacitor Current Equation

Current through capacitor:

I = C ( dVout / dt )

Derivation

Equating currents:

Vin / R = C ( dVout / dt )

Rearranging:

dVout / dt = Vin / RC

Integrating:

Vout = − (1 / RC) ∫ Vin dt

Key Result

Output voltage is proportional to the integral of input voltage.

Waveform Behavior

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

Applications

Analog computers
Signal processing
Waveform generation
Control systems

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.

Purpose of Differential Amplifier

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

Circuit Components

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

Voltage at Non-Inverting Terminal

Using voltage divider:

V+ = (R4 / (R3 + R4)) × V2

Voltage at Inverting Terminal

Using op-amp property:

V− ≈ V+

Output Voltage Derivation

The general output equation is:

Vout = (R2 / R1) (V2 − V1)

Special Case (Balanced Differential Amplifier)

R1 = R3 R2 = R4

Then

Vout = (R2/R1) (V2 − V1)

Applications

Instrumentation amplifiers
Noise rejection circuits
Sensor signal conditioning
Data acquisition systems

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.

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

Virtual Ground Concept

Since the non-inverting terminal is grounded:

V− ≈ 0

This node behaves like a virtual ground.

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

Output Voltage

The output voltage is:

Vout = −Rf ( V1/R1 + V2/R2 + V3/R3 )

Special Case (Equal Resistors)

R1 = R2 = R3 = Rf

Then

Vout = − (V1 + V2 + V3)

Applications

Audio mixers
Signal processing
Digital to analog converters
Analog computation

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.

Circuit Configuration

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

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

Voltage Gain

Av = Vout / Vin = 1

This is why it is called a Unity Gain Amplifier.

Important Characteristics

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

Applications

Impedance matching
Signal buffering
Isolation between circuits
Sensor interface circuits

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)

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.

Circuit Components

Feedback resistor → Rf
Ground resistor → R1
Operational amplifier

Virtual Short Concept

For an ideal op-amp:

V+ ≈ V−

Since input is applied to the non-inverting terminal:

V+ = Vin

Therefore

V− ≈ Vin

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) )

Output Voltage

Rearranging the equation:

Vout = Vin ( 1 + Rf / R1 )

Voltage Gain

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

Important Properties

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

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.

Circuit Components

Input resistor → R_in
Feedback resistor → R_f
Operational amplifier

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.

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.

Output Voltage

V_out = - I R_f

Substitute current value:

V_out = - (V_in / R_in) R_f

Voltage Gain

A_v = V_out / V_in = - R_f / R_in

Important Properties

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

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

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

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

Comparison

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

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.

Equivalent Circuit Components

An operational amplifier can be modeled using three main elements:

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

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

Input Resistance

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

Ri ≈ ∞

This implies

I+ ≈ 0 I− ≈ 0

Output Resistance

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

Ro ≈ 0

This allows the op-amp to drive loads efficiently.

Ideal Op-Amp Model

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

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

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

Ideal Op-Amp Characteristics

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

Important Concept

Because gain is extremely large:

V+ ≈ V−

This is called the Virtual Short Concept.

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.

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)

Input Resistance with Feedback

Since feedback is applied in parallel, input resistance decreases.

Rif = Ri / (1 + Aβ)

Output Resistance with Feedback

Current sampling increases output resistance.

Rof = Ro (1 + Aβ)

Key Characteristics

Input resistance decreases
Output resistance increases
Gain stabilizes
Bandwidth improves

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.

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)

Input Resistance with Feedback

Series mixing increases input resistance.

Rif = Ri (1 + Aβ)

Output Resistance with Feedback

Current sampling increases output resistance.

Rof = Ro (1 + Aβ)

Key Characteristics

Input resistance increases
Output resistance increases
Gain stabilizes
Bandwidth improves

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.

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)

Input Resistance with Feedback

Shunt mixing reduces input resistance.

Rif = Ri / (1 + Aβ)

Output Resistance with Feedback

Voltage sampling reduces output resistance.

Rof = Ro / (1 + Aβ)

Key Characteristics

Input resistance decreases
Output resistance decreases
Gain becomes stable
Bandwidth increases

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

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)

Input Resistance with Feedback

Voltage series feedback increases input resistance.

Rif = Ri (1 + Aβ)

Output Resistance with Feedback

Voltage sampling reduces output resistance.

Rof = Ro / (1 + Aβ)

Key Characteristics

Input resistance increases
Output resistance decreases
Gain becomes stable
Bandwidth increases

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

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

1. Voltage Series Feedback

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

2. Voltage Shunt Feedback

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

3. Current Series Feedback

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

4. Current Shunt Feedback

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

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

Saturday, 7 March 2026

Analog Electronics – Page 38

Bandwidth Improvement Using Negative Feedback

One of the major advantages of negative feedback in amplifiers is the increase in bandwidth.

Although feedback reduces gain, it significantly increases the frequency range over which the amplifier operates.

Gain with Negative Feedback

The closed-loop gain of a feedback amplifier is given by:

Af = A / (1 + Aβ)

Where

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

Bandwidth Relation

For an amplifier without feedback:

BW = fH − fL

When negative feedback is applied:

BWf = BW (1 + Aβ)

This means bandwidth increases by the same factor by which gain decreases.

Gain-Bandwidth Product

The gain-bandwidth product of an amplifier remains approximately constant.

A × BW = Constant

After feedback:

Af × BWf = A × BW

Example Problem

If an amplifier has:

Open loop gain A = 100
Bandwidth = 10 kHz
Feedback factor β = 0.04

Find new bandwidth.

1 + Aβ = 1 + (100 × 0.04)

1 + Aβ = 5

BWf = 10 kHz × 5 = 50 kHz

Important GATE Points

Negative feedback increases bandwidth
Gain reduces but stability improves
Gain × Bandwidth remains constant
Used in most practical amplifier circuits

Analog Electronics – Page 37

Feedback Amplifiers – Concept and Types

Feedback is the process of taking a portion of the output signal and returning it to the input of the amplifier.

Feedback is widely used in electronic circuits to control gain, improve stability, and reduce distortion.

Basic Feedback System

A feedback amplifier consists of three main parts:

Input Signal
Amplifier (Gain Block)
Feedback Network

Output → Feedback Network → Returned to Input

Types of Feedback

1. Positive Feedback

Feedback signal is in phase with input
Used in oscillators
Increases gain

Positive feedback can cause instability.

2. Negative Feedback

Feedback signal is opposite phase
Used in amplifiers
Improves stability
Reduces distortion

Advantages of Negative Feedback

Stabilizes amplifier gain
Reduces noise
Improves bandwidth
Reduces distortion

Types of Feedback Amplifiers

Voltage Series Feedback
Voltage Shunt Feedback
Current Series Feedback
Current Shunt Feedback

Important GATE Points

Negative feedback improves bandwidth
Gain becomes more stable
Distortion reduces significantly
Used in most practical amplifiers

Analog Electronics – Page 36

Bode Plot and Bandwidth of Amplifiers

A Bode plot is a graphical representation of amplifier gain versus frequency on a logarithmic scale.

It helps engineers understand how amplifier gain changes over different frequencies.

What is a Bode Plot?

Frequency is plotted on logarithmic scale
Gain is plotted in decibels (dB)
Shows amplifier frequency response clearly

Gain(dB) = 20 log₁₀(Av)

Important Frequency Points

Lower cutoff frequency → fL
Upper cutoff frequency → fH
Midband region → constant gain

Bandwidth

Bandwidth is the frequency range where the amplifier gain remains approximately constant.

Bandwidth (BW) = fH − fL

Slope of Bode Plot

At cutoff frequencies, gain begins to decrease.

Single pole system → −20 dB/decade
Two pole system → −40 dB/decade

Example Problem

If an amplifier has:

Lower cutoff frequency fL = 200 Hz
Upper cutoff frequency fH = 200 kHz

Find bandwidth.

BW = fH − fL

BW = 200000 − 200

BW ≈ 199.8 kHz

Important GATE Points

Bode plot uses logarithmic frequency scale
Gain expressed in decibels
Cutoff occurs at 0.707 of midband gain
Slope is −20 dB/decade for single pole systems

Analog Electronics – Page 35

High Frequency Analysis of Amplifiers

At high frequencies, the gain of an amplifier decreases due to internal capacitances of the transistor.

The main capacitances affecting high frequency response are:

Base–Emitter Capacitance (Cπ)
Collector–Base Capacitance (Cμ)

Transistor Internal Capacitances

These capacitances form RC networks which limit amplifier bandwidth.

Cπ → between base and emitter
Cμ → between collector and base

Miller Effect

The collector-base capacitance appears multiplied at the input due to voltage gain.

This phenomenon is called Miller Effect.

CM = Cμ (1 − Av)

Where:

CM = Miller capacitance
Cμ = Collector-base capacitance
Av = Voltage gain

Effect of Miller Capacitance

Increases input capacitance
Reduces bandwidth
Limits high frequency operation

Upper Cutoff Frequency

The upper cutoff frequency occurs when gain drops to 0.707 of midband gain.

fH = 1 / (2πRC)

Example Problem

If:

Cμ = 2 pF
Voltage gain Av = −100

Find Miller capacitance.

CM = Cμ (1 − Av)

CM = 2 pF (1 − (−100))

CM = 2 × 101 = 202 pF

Important GATE Points

Miller effect increases input capacitance
High voltage gain increases Miller capacitance
Bandwidth decreases due to Miller effect
Common base amplifier avoids Miller effect

Analog Electronics – Page 34

Low Frequency Analysis of Amplifiers

At low frequencies, the gain of an amplifier decreases due to the effect of capacitors present in the circuit.

The capacitors responsible for low frequency behavior are:

Input Coupling Capacitor
Emitter Bypass Capacitor
Output Coupling Capacitor

1️⃣ Input Coupling Capacitor

The input capacitor blocks DC and allows AC signals to pass to the amplifier.

At low frequencies:

Capacitive reactance increases
Signal attenuation occurs
Amplifier gain decreases

XC = 1 / (2πfC)

2️⃣ Emitter Bypass Capacitor

The emitter bypass capacitor provides AC ground for the emitter resistor.

At low frequencies:

Capacitor reactance becomes large
Emitter degeneration increases
Voltage gain reduces

3️⃣ Output Coupling Capacitor

This capacitor transfers AC output to the load while blocking DC.

At low frequencies:

Capacitive reactance increases
Output signal reduces

Lower Cutoff Frequency

The lower cutoff frequency occurs when gain drops to 0.707 of midband gain.

fL = 1 / (2πRC)

Example Problem

If coupling capacitor C = 10 µF and resistance R = 1 kΩ.

Find lower cutoff frequency.

fL = 1 / (2πRC)

fL = 1 / (2π × 1000 × 10×10⁻⁶)

fL ≈ 15.9 Hz

Important GATE Points

Low frequency roll-off caused by capacitors
Three capacitors control low-frequency response
Coupling capacitors block DC components
Bypass capacitor increases voltage gain

Analog Electronics – Page 33

Frequency Response of Amplifiers

The frequency response of an amplifier describes how the gain of the amplifier varies with signal frequency.

It shows the relationship between amplifier gain and input signal frequency.

Three Frequency Regions

Low Frequency Region
Mid Frequency Region
High Frequency Region

Low Frequency Region

Gain decreases at low frequencies
Due to coupling capacitors and bypass capacitors
Capacitive reactance becomes large

Mid Frequency Region

Gain remains constant
Amplifier operates normally
This region is called midband gain

High Frequency Region

Gain decreases at high frequencies
Due to internal transistor capacitances
Parasitic capacitances affect performance

Cutoff Frequencies

The frequencies at which the gain drops to 70.7% of maximum gain are called cutoff frequencies.

Lower cutoff frequency = fL
Upper cutoff frequency = fH

Gain at cutoff = 0.707 × Maximum Gain

Bandwidth

The bandwidth of an amplifier is the difference between upper and lower cutoff frequencies.

Bandwidth = fH − fL

Example Problem

If an amplifier has:

Lower cutoff frequency = 100 Hz
Upper cutoff frequency = 100 kHz

Find bandwidth.

BW = fH − fL

BW = 100000 − 100

BW = 99.9 kHz

Important GATE Points

Midband gain is constant
Cutoff occurs at 70.7% gain
Bandwidth determines amplifier performance
Higher bandwidth means better amplifier

Analog Electronics – Page 32

Multistage Amplifiers

A multistage amplifier consists of two or more amplifier stages connected in cascade to increase overall gain.

Each stage amplifies the signal and passes it to the next stage.

Why Multistage Amplifiers are Used

To obtain very high voltage gain
To improve power amplification
To drive heavy loads
To increase signal strength

Types of Coupling

RC Coupling
Transformer Coupling
Direct Coupling

Overall Voltage Gain

The total gain of a multistage amplifier is the product of gains of individual stages.

Av = Av1 × Av2 × Av3 × ... × Avn

Gain in Decibels

Gain is often expressed in decibels.

Gain(dB) = 20 log10(Av)

For multistage amplifier:

Total Gain(dB) = Gain1 + Gain2 + Gain3

Example Problem

A two-stage amplifier has gains:

First stage gain = 50
Second stage gain = 20

Find total voltage gain.

Av = Av1 × Av2

Av = 50 × 20

Av = 1000

Gain in dB

Gain(dB) = 20 log10(1000)

Gain = 60 dB

Key Points for GATE

Overall gain is multiplication of stage gains
In decibel scale gains are added
RC coupling is most common in voltage amplifiers
Transformer coupling used for power amplification

Analog Electronics – Page 31

Comparison of BJT Amplifier Configurations

There are three basic transistor amplifier configurations:

Common Emitter (CE)
Common Base (CB)
Common Collector (CC)

Each configuration has different gain and impedance characteristics.

1️⃣ Common Emitter Amplifier

High voltage gain
High current gain
Moderate input impedance
Moderate output impedance
Phase shift = 180°

Voltage gain: Av ≈ − (hfe RL) / hie

2️⃣ Common Base Amplifier

High voltage gain
Current gain ≈ 1
Low input impedance
High output impedance
No phase shift

Current gain: α ≈ Ic / Ie

3️⃣ Common Collector Amplifier (Emitter Follower)

Voltage gain ≈ 1
High current gain
Very high input impedance
Low output impedance
No phase shift

Voltage gain ≈ 1

Important Comparison Table

Parameter	CE	CB	CC
Voltage Gain	High	High	≈1
Current Gain	High (β)	≈1 (α)	High
Input Impedance	Medium	Low	High
Output Impedance	Medium	High	Low
Phase Shift	180°	0°	0°

GATE Important Points

CE is most widely used amplifier
CB used for high-frequency applications
CC used as buffer amplifier

Analog Electronics – Page 30

Common Emitter Amplifier Gain using h-Parameters

The hybrid h-parameter model is widely used for analyzing transistor amplifiers. It simplifies the transistor into a linear two-port network.

CE h-Parameter Model

The equations for h-parameter model are:

V1 = h11 I1 + h12 V2

I2 = h21 I1 + h22 V2

For Common Emitter configuration:

h11 = hie (Input resistance)
h21 = hfe (Current gain)
h12 = hre (Reverse voltage gain)
h22 = hoe (Output admittance)

Voltage Gain Derivation

Output voltage:

Vo = − Ic RL

Collector current:

Ic = hfe Ib

Input voltage:

Vi = Ib hie

Therefore voltage gain:

Av = Vo / Vi

Av = − (hfe RL) / hie

Current Gain

Current gain of CE amplifier:

Ai = Ic / Ib = hfe

Power Gain

Power gain is:

Ap = Av × Ai

Example Problem

Given:

hfe = 120
hie = 1.2 kΩ
RL = 3 kΩ

Find Voltage Gain

Av = − (hfe RL) / hie

Av = − (120 × 3000) / 1200

Av = −300

Key Observations

Voltage gain increases with higher hfe
Voltage gain increases with larger load resistance
Higher hie reduces voltage gain

Analog Electronics – Problems Page 29

h-Parameter Analysis & Gain Calculations

This section focuses on transistor amplifier analysis using hybrid h-parameters such as hie, hfe, hre and hoe. These parameters are widely used in small-signal analysis of BJT amplifiers.

Basic h-Parameter Equations

V1 = h11 I1 + h12 V2

I2 = h21 I1 + h22 V2

For common emitter configuration:

h11 = hie (input resistance)

h21 = hfe (current gain)

h12 = hre (reverse voltage gain)

h22 = hoe (output admittance)

Voltage Gain Formula

The approximate voltage gain of a CE amplifier using h-parameters is:

Av ≈ − (hfe RL) / hie

Problem 1

A transistor amplifier has hfe = 100, hie = 1 kΩ and load resistance RL = 4 kΩ. Calculate voltage gain.

Solution

Av = − (hfe × RL) / hie

Av = − (100 × 4000) / 1000

Av = −400

Problem 2

If hie = 2 kΩ and input voltage Vin = 10 mV, find base current.

Solution

Ib = Vin / hie

Ib = 10mV / 2kΩ

Ib = 5 μA

Problem 3

If hfe = 120 and base current Ib = 20 μA, find collector current.

Solution

Ic = hfe × Ib

Ic = 120 × 20 μA

Ic = 2.4 mA

Problem 4

A CE amplifier has hie = 1.5 kΩ, hfe = 80 and RL = 3 kΩ. Calculate voltage gain.

Solution

Av = − (hfe × RL) / hie

Av = − (80 × 3000) / 1500

Av = −160

Problem 5

If reverse voltage gain hre is very small, what is the effect?

Solution

Reverse feedback becomes negligible
Amplifier analysis becomes simpler
Voltage gain mainly depends on hfe and hie

Analog Electronics – Problems Page 28

Rapid Fire MCQs for GATE Revision

This section contains quick multiple choice questions from analog electronics useful for rapid revision before examinations.

MCQ 1

The thermal voltage at room temperature is approximately:

A) 10 mV B) 25 mV C) 50 mV D) 100 mV

Answer: B (25 mV)

MCQ 2

Which diode is used for voltage regulation?

A) Tunnel diode B) Zener diode C) LED D) Photodiode

Answer: B (Zener diode)

MCQ 3

In a BJT transistor:

A) IC = β IB B) IC = α IB C) IC = IE D) IC = IB

Answer: A

MCQ 4

The input impedance of an ideal op-amp is:

A) Zero B) Infinite C) Very small D) 1 kΩ

Answer: B (Infinite)

MCQ 5

The output impedance of an ideal op-amp is:

A) Infinite B) Zero C) 1 kΩ D) Very large

Answer: B (Zero)

MCQ 6

The Barkhausen criterion for oscillation is:

A) Aβ = 0 B) Aβ = 1 C) Aβ = 10 D) Aβ = 100

Answer: B

MCQ 7

The oscillator commonly used for audio frequency generation is:

A) Hartley oscillator B) Colpitts oscillator C) RC phase shift oscillator D) Crystal oscillator

Answer: C

MCQ 8

The cutoff frequency of an RC filter is:

A) 1/RC B) 1/(2πRC) C) RC D) 2πRC

Answer: B

MCQ 9

Which oscillator uses capacitive divider?

A) Hartley oscillator B) Colpitts oscillator C) Wien bridge oscillator D) RC oscillator

Answer: B

MCQ 10

The ripple factor of full wave rectifier is:

A) 0.121 B) 0.482 C) 1.21 D) 2

Answer: B

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Tuesday, 10 March 2026

Operational Amplifiers – Complete Theory

Page 15 – Active Low Pass Filter

Basic Components

Working Principle

Cutoff Frequency

Frequency Response

Advantages of Active Filters

Applications

GATE Important Points

Operational Amplifiers – Complete Theory

Page 14 – Instrumentation Amplifier

Main Characteristics

Basic Structure

Gain Equation

Why Instrumentation Amplifier?

Applications

GATE Important Points

Operational Amplifiers – Complete Theory

Page 13 – Precision Rectifier (Super Diode)

Problem with Normal Diodes

How Precision Rectifier Works

Operation

Key Advantage

Applications

GATE Important Points

Operational Amplifiers – Complete Theory

Page 12 – Schmitt Trigger

Why Schmitt Trigger?

Positive Feedback

Upper Threshold Voltage (UTP)

Lower Threshold Voltage (LTP)

Hysteresis Width

Transfer Characteristic

Applications

GATE Important Points

Operational Amplifiers – Complete Theory

Page 11 – Op-Amp Comparator

Basic Principle

Output Levels

Types of Comparators

Zero Crossing Comparator

Applications

GATE Important Points

Operational Amplifiers – Complete Theory

Page 10 – Op-Amp Differentiator

Circuit Components

Virtual Ground Concept

Capacitor Current

Voltage Across Resistor

Key Result

Waveform Behavior

Applications

GATE Important Points

Operational Amplifiers – Complete Theory

Page 9 – Op-Amp Integrator

Circuit Components

Virtual Ground Concept

Input Current

Capacitor Current Equation

Derivation

Key Result

Waveform Behavior

Applications

GATE Important Points

Operational Amplifiers – Complete Theory

Page 8 – Differential Amplifier (Subtractor)

Purpose of Differential Amplifier

Circuit Components

Voltage at Non-Inverting Terminal

Voltage at Inverting Terminal

Output Voltage Derivation

Special Case (Balanced Differential Amplifier)

Applications

GATE Important Points

Operational Amplifiers – Complete Theory

Page 7 – Summing Amplifier (Adder)

Circuit Configuration

Virtual Ground Concept