Unit 5 - Notes

ECE221

Unit 5: Operational Amplifier Applications I

1. Linear Arithmetic Circuits

These circuits utilize the negative feedback configuration of the operational amplifier to perform mathematical operations such as addition, multiplication by a constant, and averaging.

1.1 Inverting Summing Amplifier

The summing amplifier produces an output voltage equal to the weighted algebraic sum of two or more input voltages.

Circuit Configuration:
- Multiple inputs ( $V_1, V_2, ... V_n$ ) connect to the inverting terminal via input resistors ( $R_1, R_2, ... R_n$ ).
- The non-inverting terminal is grounded.
- A feedback resistor ( $R_f$ ) connects the output to the inverting input.
Derivation:
- Virtual ground concept: The inverting terminal is at 0V.
- KCL at the summing node: $I_1 + I_2 + ... + I_n = I_f$ .
- $\frac{V_1}{R_1} + \frac{V_2}{R_2} + ... + \frac{V_n}{R_n} = -\frac{V_{out}}{R_f}$
Output Equation:
$V_{out} = -\left( \frac{R_f}{R_1}V_1 + \frac{R_f}{R_2}V_2 + ... + \frac{R_f}{R_n}V_n \right)$

1.2 Scaling Amplifier

This is a variation of the summing amplifier where inputs are weighted differently.

Function: Each input voltage is multiplied by a different gain factor (scale) before summing.
Condition: $R_1 \neq R_2 \neq R_n$ .
Example: In a Digital-to-Analog Converter (DAC), binary weighted resistors are used (e.g., $R, 2R, 4R, 8R$ ) to scale bits according to their significance.

1.3 Averaging Amplifier

This circuit calculates the mathematical average of the input voltages.

Condition: All input resistors are equal ( $R_1 = R_2 = ... = R_n = R$ ).
Gain Setting: The gain is set such that $\frac{R_f}{R} = \frac{1}{n}$ , where $n$ is the number of inputs.
Output Equation:
$V_{out} = -\frac{1}{n}(V_1 + V_2 + ... + V_n)$
(Note: The output is the inverted average).

2. Instrumentation Amplifier

The instrumentation amplifier (In-Amp) is a precision differential voltage amplifier. It is preferred over standard difference amplifiers for reading low-level signals from transducers (strain gauges, thermocouples).

2.1 Key Characteristics

High Input Impedance: Both inputs connect directly to non-inverting terminals of Op-Amps.
High CMRR (Common Mode Rejection Ratio): Rejects noise common to both lines.
Adjustable Gain: Gain can be set by a single external resistor ( $R_g$ ).

2.2 Circuit Architecture

The standard 3-Op-Amp topology consists of two stages:

Input Stage (Buffers): Two Op-Amps ( $A_1, A_2$ ) operating as non-inverting buffers with a shared gain resistor $R_g$ .
Output Stage (Difference Amp): One Op-Amp ( $A_3$ ) configured as a standard difference amplifier with unity gain (usually).

2.3 Output Equation

If the internal resistors of the input stage are $R_1$ and the resistors of the output difference stage are $R_2$ and $R_3$ :

V_{out} = (V_2 - V_1)\left( 1 + \frac{2R_1}{R_g} \right)

(Assuming the difference amplifier stage has unity gain).

3. Voltage to Current Converters (Transconductance Amplifiers)

These circuits convert an input voltage signal into a proportional output current, irrespective of the load resistance.

3.1 V-to-I Converter with Floating Load

The load ( $Z_L$ ) is placed in the feedback loop, meaning it is not directly connected to the ground.

Configuration: The input voltage $V_{in}$ is applied to the non-inverting terminal. The load connects between the inverting terminal and the output. A feedback resistor $R$ connects the inverting terminal to ground.
Operation: The voltage at the inverting terminal is forced to be $V_{in}$ (virtual short).
Current Equation:
$I_L = \frac{V_{in}}{R}$
Application: Driving low-voltage meters, LEDs. Limitation: The load is not grounded.

3.2 V-to-I Converter with Grounded Load (Howland Current Pump)

Allows the load to be grounded, which is necessary for many applications.

Configuration: Uses a combination of positive and negative feedback.
Operation: A balanced resistor network ensures that the voltage drop across the current sensing resistor compensates for the load voltage, maintaining constant current.
Equation: Assuming balanced resistors:
$I_L = \frac{V_{in}}{R}$
Stability: Requires precise resistor matching to maintain high output impedance.

4. Current to Voltage Converter (Transresistance Amplifier)

Converts an input current into a proportional output voltage.

Configuration:
- Input current source ( $I_{in}$ ) connects to the inverting input.
- Non-inverting input is grounded.
- Feedback resistor $R_f$ connects output to inverting input.
Operation: Since input impedance is infinite, all $I_{in}$ flows through $R_f$ . The inverting input is a virtual ground.
Output Equation:
$V_{out} = -I_{in} \times R_f$
Application: Photodiode amplifiers, current sensing in DACs.

5. Calculus Circuits

5.1 The Integrator

Produces an output voltage proportional to the time integral of the input voltage.

Circuit: Resistor $R$ at the input, Capacitor $C$ in the feedback loop.
Equation:
$V_{out} = -\frac{1}{RC} \int_{0}^{t} V_{in} dt + V_{initial}$
Frequency Response: It acts as a Low Pass Filter.
Practical Modifications: Ideally, DC gain is infinite (capacitor is open circuit at DC), leading to saturation due to offset voltages. A parallel resistor $R_f$ (shunt) is added across $C$ to limit low-frequency gain.

5.2 The Differentiator

Produces an output voltage proportional to the rate of change of the input voltage.

Circuit: Capacitor $C$ at the input, Resistor $R$ in the feedback loop.
Equation:
$V_{out} = -RC \frac{d V_{in}}{dt}$
Frequency Response: It acts as a High Pass Filter.
Stability Issues: Pure differentiators are unstable and amplify high-frequency noise.
Practical Modifications: A small resistor ( $R_1$ ) is placed in series with the input capacitor to limit high-frequency gain, turning it into an integrator at very high frequencies (Miller Integrator).

6. Introduction to Active Filters

Active filters use Op-Amps combined with resistors and capacitors (RC) to filter signals. Unlike passive filters (RLC), they provide gain, have high input impedance, low output impedance, and do not require bulky inductors.

6.1 First Order Low Pass Butterworth Filter

Allows low frequencies to pass and attenuates high frequencies.

Circuit: A non-inverting amplifier with an RC network at the non-inverting input (Resistor in series, Capacitor to ground).
Cut-off Frequency ( $f_c$ ): The frequency where gain drops by 3dB.
$f_c = \frac{1}{2\pi RC}$
Roll-off: -20 dB/decade.
Butterworth Characteristic: Maximally flat passband response (no ripple).

6.2 First Order High Pass Butterworth Filter

Allows high frequencies to pass and attenuates low frequencies.

Circuit: Similar to Low Pass, but the R and C positions at the input are swapped (Capacitor in series, Resistor to ground).
Cut-off Frequency ( $f_c$ ):
$f_c = \frac{1}{2\pi RC}$
Roll-off: +20 dB/decade.

6.3 Band Pass Filter (BPF)

Passes a specific band of frequencies.

Wide Band Pass: Constructed by cascading a Low Pass Filter and a High Pass Filter.
- Condition: $f_{c(HP)} < f_{c(LP)}$ .
Narrow Band Pass: Usually implemented using a "Multiple Feedback" topology. Characterized by Quality Factor ( $Q$ ).

6.4 Band Reject Filter (Notch Filter)

Attenuates a specific band of frequencies.

Construction: Created by summing the outputs of a Low Pass Filter and a High Pass Filter in parallel.
- Condition: $f_{c(LP)} < f_{c(HP)}$ .
Application: Removing 50Hz/60Hz power line hum.

6.5 All Pass Filter (Phase Shifter)

Passes all frequencies with unity gain but changes the phase of the signal.

Function: Used for phase equalization or delay.
Lag vs. Lead: Depending on the R-C arrangement in the feedback/input, the output phase lags or leads the input phase from $0^\circ$ to $180^\circ$ .
Equation: Magnitude $|V_{out}/V_{in}| = 1$ . Phase shift $\phi$ depends on frequency.

7. Waveform Generators

Op-Amps can be configured as oscillators (non-sinusoidal) using positive feedback (Schmidt Trigger action) and RC timing networks.

7.1 Square Wave Generator (Astable Multivibrator)

Principle: An Op-Amp comparator with hysteresis charges and discharges a capacitor.
Operation:
1. Output is at $+V_{sat}$ . Capacitor charges through $R$ toward $+V_{sat}$ .
2. When capacitor voltage ( $V_c$ ) exceeds the upper threshold voltage ( $V_{UT}$ ) set by feedback resistors, the Op-Amp switches to $-V_{sat}$ .
3. Capacitor discharges toward $-V_{sat}$ . When $V_c < V_{LT}$ (lower threshold), it switches back.
Frequency: Determined by the RC time constant and the feedback fraction $\beta$ .
$f = \frac{1}{2RC \ln(\frac{1+\beta}{1-\beta})}$

7.2 Triangular Wave Generator

Construction: Created by cascading a Square Wave Generator and an Integrator.
Operation:
1. The square wave feeds the integrator.
2. When square wave is $+V_{sat}$ , integrator output ramps down linearly ( $V_{out} = -kt$ ).
3. When square wave is $-V_{sat}$ , integrator output ramps up linearly.
Feedback: The output of the integrator is usually fed back to the comparator to trigger the switch, creating a self-oscillating loop.

7.3 Sawtooth Wave Generator

A sawtooth wave has a slow linear rise (ramp) and a very fast fall (flyback).

Circuit Modification: Based on the triangular generator but with asymmetric charging/discharging paths.
Mechanism: A diode and a small resistor are placed in parallel with the integrator resistor. This makes the time constant for one half-cycle significantly smaller than the other, resulting in a rapid drop after a slow rise.

7.4 Voltage Controlled Oscillator (VCO)

An oscillator where the output frequency is directly proportional to an input control voltage ( $V_{c}$ ).

Concept:
- Standard square/triangle generators use a constant resistor to charge the capacitor.
- In a VCO, the charging current is controlled by $V_{c}$ .
Op-Amp Implementation: Ideally uses a programmable current source controlled by $V_c$ to charge the timing capacitor.
Relationship:
$f_{out} = k \times V_{control}$
Application: FM modulation, Phase Locked Loops (PLL).