Unit 5 - Notes

ECE221 8 min read

Unit 5: Operational amplifier applications I

1. Summing, Scaling, and Averaging Amplifiers

Operational amplifiers can be configured to perform mathematical operations such as addition, scaling (multiplication by a constant), and averaging of multiple input signals. These are typically implemented using an inverting amplifier configuration.

Summing Amplifier (Adder)

Concept: Produces an output voltage that is the proportional sum of two or more input voltages.
Circuit: Multiple input voltages ( $V_1, V_2, V_3$ ) are connected to the inverting terminal via respective input resistors ( $R_1, R_2, R_3$ ). A feedback resistor ( $R_f$ ) connects the output to the inverting input.
Equation: $V_{out} = -R_f \left( \frac{V_1}{R_1} + \frac{V_2}{R_2} + \frac{V_3}{R_3} \right)$
If $R_1 = R_2 = R_3 = R$ , then $V_{out} = -\frac{R_f}{R} (V_1 + V_2 + V_3)$ . If $R_f = R$ , $V_{out} = -(V_1 + V_2 + V_3)$ .

Scaling Amplifier

Concept: A summing amplifier where each input is multiplied by a different weight or scale factor.
Equation: By choosing different values for $R_1, R_2, R_3$ , each input signal is scaled differently. $V_{out} = - \left( k_1V_1 + k_2V_2 + k_3V_3 \right)$ where $k_n = \frac{R_f}{R_n}$ .

Averaging Amplifier

Concept: Produces an output that is the mathematical average of the input voltages.
Equation: Let $R_1 = R_2 = R_3 = R$ . To find the average of $n$ inputs, set the feedback resistor $R_f = \frac{R}{n}$ .
Result: $V_{out} = - \frac{V_1 + V_2 + V_3}{3}$ (for 3 inputs).

2. Instrumentation Amplifier

An instrumentation amplifier is a type of differential amplifier that has been outfitted with input buffer amplifiers, eliminating the need for input impedance matching. It is used in measurement applications (like transducers, Wheatstone bridges, and biomedical sensors).

Key Characteristics: Very high input impedance, high Common Mode Rejection Ratio (CMRR), low DC offset, low drift, and low noise.
Standard 3-Op-Amp Configuration:
- Two non-inverting amplifiers act as input buffers, providing high input impedance.
- The outputs of these buffers are fed into a standard differential amplifier (the third op-amp).
Gain Equation:
$V_{out} = \left( 1 + \frac{2R_1}{R_G} \right) \left( \frac{R_3}{R_2} \right) (V_{in2} - V_{in1})$
Where $R_G$ is a variable gain-setting resistor, allowing the gain to be easily adjusted without affecting CMRR.

3. Voltage to Current (V-to-I) Converters

V-to-I converters produce a current proportional to the input voltage, acting as a voltage-controlled current source.

V-to-I Converter with Floating Load

Concept: The load is connected in the feedback loop, not referenced to ground.
Operation: The input voltage $V_{in}$ is applied to the non-inverting terminal. A resistor $R_1$ is connected from the inverting terminal to ground. The load $Z_L$ connects the output to the inverting terminal.
Equation: Since the inverting terminal is at $V_{in}$ (virtual ground principle), the current through $R_1$ is $I = \frac{V_{in}}{R_1}$ . This exact current flows through the floating load.
Result: $I_L = \frac{V_{in}}{R_1}$ (independent of load $Z_L$ ).

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

Concept: Used when one end of the load must be tied to ground.
Operation: Utilizes both positive and negative feedback to maintain a constant current through a grounded load.
Equation: By matching resistor ratios in the positive and negative feedback paths, the load current becomes strictly dependent on the input voltage. $I_L = \frac{V_{in}}{R}$ .

4. Current to Voltage (I-to-V) Converter

Also known as a Transimpedance Amplifier, it converts an input current to a proportional output voltage.

Applications: Photodiode signal processing, DAC (Digital to Analog Converter) output stages.
Circuit: The input current $I_{in}$ is fed into the inverting terminal. The non-inverting terminal is grounded. A feedback resistor $R_f$ connects the output to the inverting terminal.
Equation: Due to virtual ground, $I_{in}$ flows entirely through $R_f$ .
Output: $V_{out} = -I_{in} \times R_f$ .

5. The Integrator

Produces an output voltage that is proportional to the mathematical integral of the input voltage over time.

Ideal Circuit: Resistor $R$ at the inverting input, Capacitor $C$ in the feedback loop.
Equation: $V_{out} = -\frac{1}{RC} \int V_{in} dt$ .
Practical Integrator (Lossy Integrator): The ideal integrator has infinite gain at DC (since capacitor acts as an open circuit), causing the op-amp to saturate. To prevent this, a large resistor $R_f$ is placed in parallel with the feedback capacitor $C$ to limit low-frequency gain.

6. The Differentiator

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

Ideal Circuit: Capacitor $C$ at the inverting input, Resistor $R$ in the feedback loop.
Equation: $V_{out} = -RC \frac{dV_{in}}{dt}$ .
Practical Differentiator: The ideal differentiator is inherently unstable and highly susceptible to high-frequency noise. A practical differentiator adds a small resistor $R_1$ in series with the input capacitor, and a small capacitor $C_f$ in parallel with the feedback resistor to limit high-frequency gain and ensure stability.

7. Introduction to Active Filters

Filters transmit signals of certain frequency ranges while attenuating others.

Passive Filters: Use only R, L, and C components. Cannot provide gain; inductors are bulky at low frequencies.
Active Filters: Use op-amps along with R and C components.
Advantages of Active Filters:
- Provide gain (no signal attenuation in the passband).
- No inductors required (reduces size and cost).
- High input impedance and low output impedance (prevents loading effects).
- Easier to tune and cascade.

8. Specific Active Filter Types

First Order Low Pass Butterworth Filter

Function: Passes frequencies below a cutoff frequency ( $f_c$ ) and attenuates higher frequencies.
Circuit: RC network at the non-inverting input, op-amp configured as a non-inverting amplifier.
Cutoff Frequency: $f_c = \frac{1}{2\pi RC}$
Roll-off: -20 dB/decade. Maximum flat response in the passband (Butterworth characteristic).

First Order High Pass Butterworth Filter

Function: Passes frequencies above $f_c$ and attenuates lower frequencies.
Circuit: Swap the R and C positions from the low-pass filter at the non-inverting input.
Cutoff Frequency: $f_c = \frac{1}{2\pi RC}$ . Roll-off is +20 dB/decade below $f_c$ .

Band Pass Filter

Function: Passes a specific range of frequencies between two cutoff frequencies ( $f_L$ and $f_H$ ) and attenuates the rest.
Wide Band Pass: Created by cascading a Low Pass Filter and a High Pass Filter ( $f_H > f_L$ ).
Narrow Band Pass: Built using multiple feedback loops around a single op-amp to achieve a high Quality Factor ( $Q > 10$ ).

Band Reject Filter (Notch Filter)

Function: Attenuates a specific band of frequencies and passes all others.
Implementation: Parallel combination of a low-pass and high-pass filter fed into a summing amplifier. A specific narrow-band reject filter is called a "Notch Filter," often implemented using a Twin-T network to remove a specific interfering frequency (e.g., 50/60 Hz power line noise).

All Pass Filter

Function: Passes all frequencies equally well (gain is unity across all frequencies) but alters the phase of the signal.
Application: Used for phase correction or introducing a time delay.
Phase Shift: Varies from $0^\circ$ to $-180^\circ$ as frequency changes.

9. Waveform Generators

Square Wave Generator (Astable Multivibrator)

Concept: A free-running oscillator that switches between positive and negative saturation voltages.
Circuit: An op-amp with both positive feedback (forming a Schmitt trigger) and negative feedback (RC timing circuit).
Operation: The capacitor $C$ charges and discharges through $R$ between the upper and lower threshold voltages set by the positive feedback divider.
Frequency: $f = \frac{1}{2RC \ln\left(\frac{1+\beta}{1-\beta}\right)}$ , where $\beta = \frac{R_1}{R_1+R_2}$ is the feedback fraction.

Triangular Wave Generator

Concept: Generates a triangular wave by integrating a square wave.
Circuit: Cascades a square wave generator (comparator/Schmitt trigger) with an integrator.
Operation: When the comparator output is $+V_{sat}$ , the integrator produces a negative-going ramp. When the comparator switches to $-V_{sat}$ , the integrator produces a positive-going ramp. Amplitude and frequency are controlled by the RC time constants and comparator thresholds.

Sawtooth Wave Generator

Concept: A variant of the triangular wave generator where the rise time and fall time are unequal.
Implementation: Achieved by altering the charging and discharging paths of the integrator capacitor. Placing a diode in series with a resistor across the integrator's input resistor creates different time constants for the positive and negative cycles.

10. Voltage Controlled Oscillator (VCO)

Concept: An oscillator whose output frequency is directly proportional to an external control voltage ( $V_c$ ).
Operation: The control voltage determines the charging/discharging current of the timing capacitor. A higher control voltage increases the charging current, causing the capacitor to reach the threshold faster, thus increasing the frequency.
Applications: Frequency Modulation (FM), Phase-Locked Loops (PLL), tone generation, and synthesizers.
Common ICs: NE566, LM331, or built using op-amps configured as a tunable integrator and Schmitt trigger combination.

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