🧮 Op-Amp Applications: Mathematical Operations

We've learned the basics of op-amps. Now let's see how they can perform actual math operations on signals.

These circuits don't just amplify. They add, subtract, integrate, and differentiate.

And they do it continuously, in real-time, with analog voltages.

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➕ Summing Amplifier – Adding Voltages

The Problem

You have multiple input signals and need their weighted sum.

Examples:

• Audio mixing
• Multi-sensor averaging
• Control systems

The Circuit

The Math

Using virtual ground at the inverting input:

$$V_{out} = -R_f \left( \frac{V_1}{R_1} + \frac{V_2}{R_2} + \frac{V_3}{R_3} \right)$$

Simple Case: Equal Resistors

If $R_1 = R_2 = R_3 = R$ and $R_f = R$:

$$V_{out} = -(V_1 + V_2 + V_3)$$

Perfect adder! (with inversion)

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🎚️ Weighted Summing Example

Audio mixer with 3 channels:

• Voice: $V_1 = 1V$, $R_1 = 10k\Omega$
• Music: $V_2 = 0.5V$, $R_2 = 20k\Omega$ (quieter)
• Effects: $V_3 = 0.2V$, $R_3 = 50k\Omega$ (background)

With $R_f = 10k\Omega$:

$$V_{out} = -10k \left( \frac{1V}{10k} + \frac{0.5V}{20k} + \frac{0.2V}{50k} \right)$$ $$V_{out} = -(1V + 0.25V + 0.04V) = -1.29V$$

Each channel has its own "volume control" via resistor selection!

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Real-World Tip

The summing amplifier is the heart of analog audio mixers. Professional mixing consoles use hundreds of these circuits.

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➖ Difference Amplifier – Subtracting Voltages

The Problem

You need $V_{out} = V_2 - V_1$

Why?

• Sensor differential measurements
• Noise cancellation
• Bridge circuits

The Circuit

The Math

For a unity-gain difference amplifier where $R_1 = R_2 = R_f = R$:

$$V_{out} = V_2 - V_1$$

Perfect subtraction!

General Case (with gain)

If $R_f = R_1 \cdot A$ and $R_3 = R_2$:

$$V_{out} = A(V_2 - V_1)$$

You get amplified difference.

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🌡️ Temperature Differential Example

Two temperature sensors:

• Sensor A: $2.5V$ (Room temp)
• Sensor B: $2.7V$ (Near heat source)

Difference amplifier with gain = 10:

$$V_{out} = 10(2.7V - 2.5V) = 10 \times 0.2V = 2V$$

Now your ADC can easily measure the temperature difference with high resolution!

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Watch Out

The difference amplifier is sensitive to resistor matching. Use 1% tolerance resistors or better for accurate subtraction.

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📈 Integrator – The Accumulator

What It Does

The output is the time integral (accumulation) of the input.

If input is constant → output ramps linearly
If input is zero → output holds value
If input reverses → output ramps in opposite direction

The Circuit

The Math

$$V_{out}(t) = -\frac{1}{RC} \int_0^t V_{in}(\tau) \, d\tau$$

Simpler Understanding

The rate of change of output voltage:

$$\frac{dV_{out}}{dt} = -\frac{V_{in}}{RC}$$

• Positive input → output ramps down
• Negative input → output ramps up
• Larger input → faster ramp

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⚡ Integrator Example: Converting Current to Voltage

Current source: $I_{in} = 1mA$ (constant)

Integrator with $R = 10k\Omega$, $C = 1\mu F$:

Time constant: $RC = 10k \times 1\mu F = 10ms$

$$\frac{dV_{out}}{dt} = -\frac{V_R}{RC} = -\frac{I_{in} \times R}{RC} = -\frac{I_{in}}{C}$$ $$\frac{dV_{out}}{dt} = -\frac{1mA}{1\mu F} = -1000 \, V/s = -1V/ms$$

After 5ms: $V_{out} = -5V$

Perfect for measuring total charge!

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🎯 Real Applications of Integrators

Application	Why Integration?
Charge amplifier	Convert sensor current to voltage
Waveform generation	Convert square wave → triangle wave
Control systems	PI controllers need integration
Analog computers	Solving differential equations
Signal processing	Area under curve measurement

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Practical Issue

Real integrators drift due to:

• Op-amp input bias current
• DC offsets

Solution: Add a large resistor (MΩ range) in parallel with capacitor to provide DC feedback without affecting AC integration.

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📉 Differentiator – The Rate Detector

What It Does

Output is proportional to the rate of change of input.

Slow changes → small output
Fast changes → large output
Constant input → zero output

The Circuit

The Math

$$V_{out}(t) = -RC \frac{dV_{in}}{dt}$$

The output is the derivative (rate of change) of the input!

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🏃 Differentiator Example: Edge Detection

Input: Square wave ($0V \to 5V$ step in 1μs)

Differentiator with $R = 10k\Omega$, $C = 100pF$:

$$V_{out} = -RC \frac{dV_{in}}{dt} = -10k \times 100pF \times \frac{5V}{1\mu s}$$ $$V_{out} = -1\mu s \times 5 \times 10^6 V/s = -5V$$

At the rising edge, you get a sharp spike!

This is how edge detectors work.

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⚠️ Problem with Practical Differentiators

Differentiators amplify noise.

Why?
High-frequency noise has rapid changes → large derivatives → huge noise at output.

Solution

Add a small resistor in series with the capacitor to limit high-frequency gain.

(Add a small resistor (100Ω - 1kΩ) in series with the input capacitor to limit high-frequency gain and prevent instability)

This creates a band-pass differentiator that works on your signal frequencies but ignores high-frequency noise.

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🎯 Real Applications of Differentiators

Application	Why Differentiation?
Velocity from position	Rate of change = speed
Edge detection	Find transitions in signals
High-pass filtering	Block DC, pass AC changes
Frequency doubling	In FM demodulation
Acceleration sensing	Rate of velocity change

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Stability Warning

Differentiators are inherently unstable and noisy in pure form. Always add damping (series resistor) in practical circuits.

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📊 Comparison Table

Circuit	Function	Input-Output Relationship	Key Component
Summing Amplifier	Addition	$V_{out} = -(V_1 + V_2 + ...)$	Multiple input resistors
Difference Amplifier	Subtraction	$V_{out} = V_2 - V_1$	Balanced resistor network
Integrator	Accumulation	$V_{out} = -\frac{1}{RC}\int V_{in} dt$	Feedback capacitor
Differentiator	Rate of change	$V_{out} = -RC \frac{dV_{in}}{dt}$	Input capacitor

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🔬 Design Guidelines

For Summing Amplifiers

1. Use resistors in 1% tolerance range
2. Keep input impedances similar (unless weighting is desired)
3. Consider input bias current effects with high-value resistors

For Difference Amplifiers

1. Match resistor ratios precisely (this is critical!)
2. Use resistor networks for better matching
3. Consider instrumentation amplifier for better CMRR

For Integrators

1. Add parallel resistor for DC stability (typically 1-10 MΩ)
2. Add reset switch to discharge capacitor
3. Use low-leakage capacitors (polypropylene or polystyrene)

For Differentiators

1. Always add series input resistor for stability
2. Limit bandwidth to avoid noise amplification
3. Consider using RC high-pass filter instead for simple applications

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🧪 Lab Exercise Ideas

1. Build a 3-input audio mixer

   • Use potentiometers for volume control
   • Mix three audio sources
   • Observe how resistor values affect mixing

2. Temperature difference monitor

   • Use two temperature sensors
   • Build difference amplifier
   • Measure temperature gradient across a heatsink

3. Triangle wave generator

   • Square wave input to integrator
   • Observe triangle wave output
   • Change RC values to modify frequency

4. Edge detector

   • Practical differentiator circuit
   • Square wave input
   • Observe output spikes at edges

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✅ Key Takeaways

• Summing amplifiers add multiple signals with individual gain control
• Difference amplifiers subtract signals and reject common-mode noise
• Integrators accumulate signals over time (area under curve)
• Differentiators detect rate of change (slope)
• Each circuit has practical limitations that must be addressed
• These form the basis of analog computers and signal processing

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🎓 Looking Ahead

These mathematical operations are building blocks for:

• Active filters (our next topic!)
• Control systems (PID controllers)
• Analog signal processing
• Instrumentation amplifiers
• Waveform generators

Master these, and you're ready to design sophisticated analog systems! 🚀

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📚 Further Reading

• Experiment with different RC time constants
• Try cascading integrators and differentiators
• Research analog computers (they used these extensively!)
• Look into practical considerations for precision applications

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