Operational amplifiers and applications notes — Unit 4

An Operational Amplifier (Op-Amp) is a highly complex, multi-stage, direct-coupled transistor amplifier packaged inside a single microscopic silicon Integrated Circuit (IC). The most famous standard is the ubiquitous IC 741.

⇒The Schematic Symbol

An Op-Amp is drawn as a triangle pointing left-to-right. It has two inputs and one output.

• Inverting Input (-): A signal applied here is amplified and flipped completely upside down (180° phase shift) at the output.
• Non-Inverting Input (+): A signal applied here is amplified but remains right-side up (0° phase shift) at the output.
• Power Supplies (+Vcc and -Vee): Op-Amps almost always require a dual-polarity power supply (e.g., +15V and -15V) so they can output both positive and negative voltage swings.

⇒The Core Function

Fundamentally, an Op-Amp is a Differential Amplifier. It calculates the difference between the voltage at the (+) pin and the (-) pin, and multiplies that tiny difference by a massive gain factor.

V_out = Aol * (V_in+ - V_in-)

Where Aol is the Open-Loop Gain (typically 100,000 to 1,000,000).

To mathematically analyze complex Op-Amp circuits easily, engineers assume the Op-Amp behaves perfectly according to the 'Ideal Model'. Real Op-Amps (like the 741) approach these ideals very closely.

⇒The Two Golden Rules of Analysis

Because of these ideal characteristics, circuit analysis relies entirely on two absolute rules:

RULE 1: No current ever flows INTO the input terminals. (I_in = 0)
RULE 2: The Op-Amp forces the voltage at the (-) pin to perfectly equal the voltage at the (+) pin via feedback. (V+ = V-)

This is the most critical concept in Op-Amp mathematics. It arises from the fact that the Open-Loop Gain (Aol) is practically infinite.

⇒The Derivation

We know: V_out = Aol * (V+ - V-)
Rearranging: (V+ - V-) = V_out / Aol

Since Aol is roughly Infinity, any finite number divided by infinity is ZERO.
(V+ - V-) = 0
Therefore: V+ = V-

⇒The Virtual Short

The Op-Amp internally adjusts its massive output voltage until the difference between the two input pins is exactly zero. The two pins act as if they are short-circuited together, even though no physical wire connects them. This is a 'Virtual Short'.

⇒Virtual Ground

If you physically connect the (+) pin to actual Ground (0V), then by the rule of the Virtual Short, the (-) pin will also magically act as if it is exactly at 0V. We call the (-) pin a 'Virtual Ground'. It has a voltage of 0V, but it cannot absorb any current.

An Op-Amp without any connections between its output and its inputs is running in 'Open-Loop'. Because its gain is 200,000, even a tiny 1mV input will try to drive the output to 200 Volts! It instantly hits the power supply limits (e.g., ±15V) and clips. This makes it useless as a linear amplifier.

⇒The Taming Mechanism

To make the Op-Amp usable and controllable, we must use Negative Feedback. We physically connect a resistor from the Output pin directly back to the Inverting (-) Input pin.

• Some of the output signal is fed back into the inverting pin.
• Because the inverting pin flips the signal, the returning feedback actually FIGHTS against the original input.
• This deliberate self-canceling reduces the massive 200,000 gain down to a precise, mathematically controllable number (like 10 or 50) dictated entirely by the external resistors.

⇒Benefits of Negative Feedback

• Allows exact control over the Voltage Gain (Closed-Loop Gain).
• Drastically increases the bandwidth.
• Reduces distortion and electrical noise.
• Stabilizes the amplifier against temperature changes and manufacturing defects.

The most fundamental Op-Amp circuit. It amplifies the input signal and flips it completely upside down (180° phase shift).

1. The (+) pin is connected directly to Ground (0V).
2. Input voltage (Vin) goes through resistor R1 into the (-) pin.
3. A feedback resistor (Rf) connects the Output back to the (-) pin.

⇒Mathematical Derivation of Gain

Because (+) is grounded at 0V, the (-) pin becomes a Virtual Ground (0V).

Current flowing from Vin to the (-) node (Virtual Ground):
I_in = (Vin - 0) / R1 = Vin / R1

Because the Op-Amp inputs draw ZERO current (Rule 1), ALL of I_in must flow up and through the feedback resistor (Rf) to the output.
I_f = I_in

Current flowing through Rf from the (-) node to Vout:
I_f = (0 - Vout) / Rf = -Vout / Rf

Equating the two:
Vin / R1 = -Vout / Rf

Rearranging for Voltage Gain (Av = Vout / Vin):
Av = - Rf / R1
Vout = - (Rf / R1) * Vin

The gain is perfectly dictated by the ratio of the two external resistors. The minus sign proves the 180° inversion.

This configuration amplifies the signal without flipping it (0° phase shift). It also provides an astronomically high input impedance.

1. Input voltage (Vin) is applied DIRECTLY to the (+) pin.
2. Resistor R1 connects the (-) pin to Ground.
3. Feedback resistor (Rf) connects the Output back to the (-) pin.

⇒Mathematical Derivation of Gain

By the Virtual Short rule, the voltage at the (-) pin perfectly matches the (+) pin. So the voltage at node (-) is exactly Vin.

Current flowing from (-) node down to Ground through R1:
I1 = (Vin - 0) / R1 = Vin / R1

Current flowing from Vout back through Rf into the (-) node:
I_f = (Vout - Vin) / Rf

Because the Op-Amp draws zero current, I_f must equal I1:
(Vout - Vin) / Rf = Vin / R1

Divide both sides by Vin:
(Vout/Vin - 1) / Rf = 1 / R1
(Av - 1) / Rf = 1 / R1
Av - 1 = Rf / R1

Voltage Gain (Av):
Av = 1 + (Rf / R1)
Vout = [ 1 + (Rf / R1) ] * Vin

Notice there is no minus sign. The gain is always positive and always greater than or equal to 1.

A special, extreme case of the Non-Inverting amplifier where we remove R1 entirely (making it infinite) and replace Rf with a solid wire (making it zero).

1. Vin is applied to the (+) pin.
2. A solid wire connects the Output directly back to the (-) pin.

⇒Mathematical Behavior

Using the Non-Inverting formula: Av = 1 + (Rf / R1)
If Rf = 0 and R1 = ∞ : Av = 1 + (0 / ∞) = 1
Vout = Vin

The output voltage is the EXACT same as the input voltage. It doesn't amplify the voltage at all. So why is this circuit so heavily used?

⇒The Ultimate Impedance Buffer

It acts as a bodyguard. It has practically infinite input impedance (so it draws zero current from delicate sensors) and near-zero output impedance (so it can blast massive current into a heavy motor or speaker). It transfers the voltage signal from a weak system to a heavy system without destroying the weak system.

An Op-Amp can be configured to mathematically ADD multiple separate voltage signals together. This is the foundation of analog audio mixing desks.

1. The (+) pin is grounded.
2. Three separate inputs (V1, V2, V3) connect to the (-) pin through three separate resistors (R1, R2, R3).
3. A single feedback resistor (Rf) connects Output to the (-) pin.

⇒Mathematical Derivation

The (-) pin acts as a Virtual Ground (0V). The currents from all three input branches meet at this node and combine.

I1 = V1 / R1
I2 = V2 / R2
I3 = V3 / R3

Total current entering the node: I_total = I1 + I2 + I3
Because Op-Amp draws no current, I_total is forced through Rf:
Vout = - I_total * Rf

Vout = - Rf * [ (V1/R1) + (V2/R2) + (V3/R3) ]

⇒The Direct Adder

If we make all resistors mathematically identical (R1 = R2 = R3 = Rf = R), the equation beautifully simplifies to:

Vout = - (V1 + V2 + V3)

It literally outputs the exact mathematical sum of the inputs (inverted).

Op-Amps can perform calculus. An integrator circuit outputs a voltage that is proportional to the mathematical integral (area under the curve) of the input voltage over time.

It is an Inverting Amplifier, but the Feedback Resistor (Rf) is replaced with a Capacitor (C).

⇒Mathematical Derivation

Node (-) is a Virtual Ground (0V).
Current through input resistor R:  I_in = Vin / R

Current through a capacitor is dictated by the rate of voltage change:
I_c = C * d(Vc)/dt
Since one side of C is at 0V and the other is at Vout:
I_c = C * d(0 - Vout)/dt = -C * (dVout / dt)

Equating currents (I_in = I_c):
Vin / R = -C * (dVout / dt)

Rearranging to solve for Vout:
dVout = -(1 / RC) * Vin * dt
Integrate both sides:
Vout = -(1 / RC) ∫ Vin dt

⇒Applications

If you input a constant DC voltage (a flat line), the integral of a constant is a sloping line (a Ramp). If you input a Square wave, the output is a Triangle wave. Used heavily in analog radar and waveform generators.

The exact opposite of the Integrator. It outputs a voltage proportional to the rate of change (derivative) of the input voltage.

The positions of the Resistor and Capacitor are swapped.
Input goes through Capacitor (C), Feedback is a Resistor (R).

⇒Mathematical Derivation

Node (-) is Virtual Ground (0V).
Current through input capacitor C:
I_in = C * d(Vin - 0)/dt = C * (dVin / dt)

Current through feedback resistor R:
I_R = (0 - Vout) / R = -Vout / R

Equating currents (I_in = I_R):
C * (dVin / dt) = -Vout / R

Rearranging to solve for Vout:
Vout = -RC * (dVin / dt)

⇒Applications

If you input a Triangle wave (constant slope), the derivative of a constant slope is a flat horizontal line. When the slope flips, the line flips. A Triangle wave input produces a Square wave output. Used to detect sharp edges or sudden spikes in sensor readings.

When transmitting delicate analog signals over long cables, electromagnetic noise from power lines (50Hz hum) infects the wire. This noise hits BOTH the inverting and non-inverting inputs of the Op-Amp at the exact same time.

⇒The Differential Magic

Because the Op-Amp amplifies the DIFFERENCE between the two inputs (V+ minus V-), if a +5mV noise spike hits BOTH pins simultaneously, the difference is (5mV - 5mV) = 0. The Op-Amp completely ignores the noise and does not amplify it. This ability is called Common Mode Rejection.

⇒Mathematical Definition

CMRR = Ad / Acm

Where:
Ad = Differential Gain (The massive gain applied to the intended signal difference)
Acm = Common-Mode Gain (The tiny, accidental gain applied to signals that are identical on both pins)

• In an Ideal Op-Amp, Acm = 0, therefore CMRR = Infinity.
• In a practical 741 Op-Amp, CMRR is typically around 90 dB (31,600 times better at amplifying difference than amplifying noise).

If you input a perfect, instantaneous square wave that jumps from 0V to 5V in zero seconds, you would expect the output to also jump instantly. However, the internal circuitry of an Op-Amp contains microscopic capacitors that take time to charge up.

⇒Definition

The Slew Rate is the maximum possible rate of change of the output voltage of an Op-Amp. It is the absolute speed limit of the amplifier.

Slew Rate (SR) = max ( dVout / dt )
Unit: Volts / microsecond (V/μs)

⇒The Consequence of Slew Rate

If you feed a high-frequency sine wave into an Op-Amp, the wave demands the output to swing up and down incredibly fast. If the wave demands a speed faster than the Slew Rate, the Op-Amp cannot keep up. The output wave gets heavily distorted, turning a beautiful curved sine wave into a jagged, sloped triangle wave.

• IC 741 Slew Rate: 0.5 V/μs. (It takes 20 microseconds just to swing 10 Volts. Highly sluggish, useless for high-frequency RF).
• Modern high-speed Op-Amps: > 5000 V/μs.

What happens if we completely remove the negative feedback resistor and run the Op-Amp in Open-Loop? It becomes a Comparator—a digital decision-making circuit.

⇒The Mechanism

Because the Open-Loop gain is immense (200,000), any microscopic difference between the inputs causes the output to instantly slam into the power supply rails (+Vsat or -Vsat, typically ±14V).

If Voltage at (+) is even 1mV HIGHER than (-):
Output = +Vsat (Logic HIGH)

If Voltage at (-) is even 1mV HIGHER than (+):
Output = -Vsat (Logic LOW)

⇒Applications: The Thermostat

Connect a reference voltage of 2V to the (-) pin. Connect a temperature sensor to the (+) pin. As the room warms up, the sensor voltage rises. The moment the sensor hits 2.001V, the (+) pin is higher than the (-) pin. The output violently swings to +15V, turning on the air conditioning relay instantly.

⇒End-of-Unit Verification

• Draw the symbol of an Op-Amp and define the functions of the Inverting and Non-Inverting inputs.
• List 5 characteristics of an Ideal Op-Amp and compare them with the practical values of the IC 741.
• State the two Golden Rules of ideal Op-Amp analysis.
• Mathematically prove the concept of Virtual Ground using infinite open-loop gain.
• Explain why Negative Feedback is absolutely mandatory for an Op-Amp to operate as a linear amplifier.
• Draw the circuit diagram and derive the voltage gain formula `Av = -Rf/R1` for an Inverting Amplifier.
• Draw the circuit diagram and derive the voltage gain formula `Av = 1 + (Rf/R1)` for a Non-Inverting Amplifier.
• Explain the circuit architecture and purpose of a Voltage Follower (Buffer).
• Derive the output voltage equation for a 3-input Summing Amplifier.
• Sketch an Integrator circuit and mathematically prove that `Vout = -(1/RC) ∫ Vin dt`.
• Define CMRR and Slew Rate, providing their formulas and typical units.