Unit 5 - Notes

ECE131

Unit 5: Fundamentals of Filters and Operational Amplifiers

1. Fundamentals of Filters

Electrical filters are circuits designed to reject all unwanted frequency components of an electrical signal while allowing desired frequencies to pass. They are characterized by their cutoff frequency ( $f_c$ ), the frequency at which the output power drops to half of the input power (or the voltage drops to $0.707$ or $1/\sqrt{2}$ of the maximum).

1.1 Low-Pass Filter (LPF)

A low-pass filter allows signals with a frequency lower than the cutoff frequency to pass through and attenuates signals with frequencies higher than the cutoff frequency.

Circuit Construction (Passive RC): A resistor ( $R$ ) is placed in series with the input, and a capacitor ( $C$ ) is placed in parallel with the output.
Operation: At low frequencies, the capacitor acts as an open circuit (high reactance), allowing voltage to appear across the output. At high frequencies, the capacitor acts as a short circuit (low reactance), shunting the signal to the ground.
Cutoff Frequency Formula:
$f_c = \frac{1}{2\pi RC}$
Application: Audio amplifiers (to remove high-frequency noise/hiss), DC power supplies (to smooth AC ripples).

1.2 High-Pass Filter (HPF)

A high-pass filter permits signals with a frequency higher than the cutoff frequency to pass and attenuates signals with frequencies lower than the cutoff frequency.

Circuit Construction (Passive RC): A capacitor ( $C$ ) is placed in series with the input, and a resistor ( $R$ ) is placed in parallel with the output.
Operation: At low frequencies (including DC), the capacitor blocks the current (high reactance). At high frequencies, the capacitor offers low reactance, allowing the signal to reach the output resistor.
Cutoff Frequency Formula:
$f_c = \frac{1}{2\pi RC}$
Application: Coupling circuits in amplifiers (to block DC offset), audio tweeters.

1.3 Band-Pass Filter (BPF)

A band-pass filter passes frequencies within a certain range (band) and rejects frequencies outside that range. It has two cutoff frequencies: a lower cutoff ( $f_L$ ) and an upper cutoff ( $f_H$ ).

Circuit Construction: Can be created by cascading a High-Pass filter and a Low-Pass filter. The HPF sets the lower limit ( $f_L$ ) and the LPF sets the upper limit ( $f_H$ ).
Condition: $f_L < f_H$ .
Bandwidth (BW): $BW = f_H - f_L$ .
Resonant Frequency ( $f_r$ ): The geometric mean of the cutoffs: $f_r = \sqrt{f_L \times f_H}$ .
Application: Wireless transmitters and receivers (tuning to a specific radio station), signal processing.

2. Operational Amplifier Abstraction

2.1 Device Overview

An Operational Amplifier (Op-Amp) is a high-gain, DC-coupled, multi-stage differential voltage amplifier. It generally has two inputs and one output.

Terminals:

Inverting Input ( $-$ ): Signals applied here appear phase-inverted ( $180^\circ$ shift) at the output.
Non-Inverting Input ( $+$ ): Signals applied here appear in phase with the output.
Output ( $V_{out}$ ): Sourcing or sinking current.
Power Supply: Positive ( $+V_{CC}$ ) and Negative ( $-V_{EE}$ ) rails.

2.2 Device Properties (Ideal vs. Practical)

Property	Ideal Op-Amp Characteristics	Practical Op-Amp (e.g., 741)	Significance
Open Loop Gain ( $A_{OL}$ )	Infinite ( $\infty$ )	Very High ( $~10^5$ to $10^6$ )	Allows precise control using feedback.
Input Impedance ( $Z_{in}$ )	Infinite ( $\infty$ )	High ( $2 M\Omega$ to $10^{12} \Omega$ )	Draws zero current from the source circuit.
Output Impedance ( $Z_{out}$ )	Zero ( $0 \Omega$ )	Low ( $~75 \Omega$ )	Can drive loads without voltage drop.
Bandwidth (BW)	Infinite ( $\infty$ )	Finite ( $~1 \text{MHz}$ )	Amplifies all frequencies equally (Ideal).
Offset Voltage	Zero ( $0 \text{V}$ )	Non-zero ( $~2 \text{mV}$ )	Output is zero when inputs are equal.
CMRR	Infinite ( $\infty$ )	High ( $~90 \text{dB}$ )	Ability to reject common noise signals.

3. Simple Op-Amp Circuits & Concepts

3.1 The Virtual Ground Concept

This is the most critical concept for analyzing Op-Amp circuits with negative feedback.

Premise: Since the open-loop gain ( $A_{OL}$ ) is effectively infinite, and $V_{out} = A_{OL}(V_+ - V_-)$ , the differential voltage ( $V_d = V_+ - V_-$ ) required to produce a finite output voltage is infinitesimally small.
The Rule: $V_+ \approx V_-$ .
Virtual Ground: If the non-inverting terminal ( $V_+$ ) is connected to the physical ground ( $0\text{V}$ ), the inverting terminal ( $V_-$ ) stays at approximately $0\text{V}$ . It is a "virtual" ground because it is at $0\text{V}$ potential but is not mechanically connected to the earth.

3.2 Inverting Amplifier

In this configuration, the input signal is applied to the inverting terminal through a resistor, and the non-inverting terminal is grounded.

Circuit: Input $V_{in}$ connects to $R_1$ , which connects to the inverting input ( $V_-$ ). A feedback resistor $R_f$ connects $V_{out}$ to $V_-$ . $V_+$ is grounded.
Analysis:
1. $V_+ = 0\text{V}$ , so $V_- = 0\text{V}$ (Virtual Ground).
2. Current entering the Op-Amp inputs is zero ( $I_{in} \approx 0$ ).
3. Apply KCL at node $V_-$ :
  $\frac{V_{in} - 0}{R_1} + \frac{V_{out} - 0}{R_f} = 0$
Output Formula:
$V_{out} = -\left( \frac{R_f}{R_1} \right) V_{in}$
Key Feature: The output is amplified and phase-shifted by $180^\circ$ (indicated by the negative sign).

3.3 Non-Inverting Amplifier

The input signal is applied directly to the non-inverting terminal.

Circuit: Input $V_{in}$ connects to $V_+$ . Resistor $R_1$ connects $V_-$ to ground. Feedback resistor $R_f$ connects $V_{out}$ to $V_-$ .
Analysis:
1. $V_+ = V_{in}$ . Therefore, $V_- = V_{in}$ (Virtual Short).
2. Apply Voltage Divider rule at $V_-$ :
  $V_- = V_{out} \times \frac{R_1}{R_1 + R_f}$
3. Equating $V_- = V_{in}$ :
  $V_{in} = V_{out} \times \frac{R_1}{R_1 + R_f}$
Output Formula:
$V_{out} = \left( 1 + \frac{R_f}{R_1} \right) V_{in}$
Key Feature: Gain is always $\geq 1$ . No phase shift. High input impedance.

4. Arithmetic Op-Amp Circuits

4.1 Op-Amp as an Adder (Summing Amplifier)

Used to perform the mathematical addition of two or more input voltages. Usually configured in the inverting mode.

Circuit: Multiple inputs ( $V_1, V_2, \dots$ ) are connected to the inverting terminal via resistors ( $R_1, R_2, \dots$ ). There is a single feedback resistor $R_f$ .
Analysis (KCL at Virtual Ground):
$I_1 + I_2 + I_f = 0$
$\frac{V_1}{R_1} + \frac{V_2}{R_2} + \frac{V_{out}}{R_f} = 0$
Output Formula:
$V_{out} = -\left( \frac{R_f}{R_1}V_1 + \frac{R_f}{R_2}V_2 + \dots \right)$
Special Case: If $R_1 = R_2 = R_f$ , then $V_{out} = -(V_1 + V_2)$ . (Inverting Summer).

4.2 Op-Amp as a Subtractor (Difference Amplifier)

Used to find the difference between two input voltages.

Circuit: Input $V_1$ connects to the inverting terminal via $R_1$ (with feedback $R_2$ ). Input $V_2$ connects to the non-inverting terminal via $R_3$ (with $R_4$ to ground).
Analysis (Superposition):
To simplify, assume all resistors are equal ().
1. Set $V_2=0$ : Acts as inverting amp. $V_{out1} = -V_1$ .
2. Set $V_1=0$ : Acts as non-inverting amp. Voltage at $V_+$ is $V_2/2$ . Gain is $(1+1)=2$ . $V_{out2} = 2(V_2/2) = V_2$ .
3. Combine: $V_{out} = V_{out1} + V_{out2}$ .
Output Formula (Equal Resistors):
$V_{out} = V_2 - V_1$
General Formula:
$V_{out} = \frac{R_2}{R_1}(V_2 - V_1) \quad (\text{Assuming balanced resistor ratios})$

5. Op-Amp RC Circuits

5.1 Op-Amp Integrator

A circuit that performs the mathematical operation of integration with respect to time.

Circuit: Derived from the inverting amplifier by replacing the feedback resistor ( $R_f$ ) with a capacitor ( $C_f$ ).
Operation:
Current through input resistor $R$ : $I = V_{in}/R$ .
Current through capacitor: $I = -C_f (dV_{out}/dt)$ .
Equating currents (Virtual Ground):
$\frac{V_{in}}{R} = -C_f \frac{dV_{out}}{dt}$
Output Formula:
$V_{out} = -\frac{1}{RC_f} \int_{0}^{t} V_{in} \, dt + V_{initial}$
Waveform Response:
- Input: Square Wave $\rightarrow$ Output: Triangular Wave.
- Input: Sine Wave $\rightarrow$ Output: Cosine Wave ( $-90^\circ$ shift).

5.2 Op-Amp Differentiator

A circuit that performs the mathematical operation of differentiation.

Circuit: Derived from the inverting amplifier by replacing the input resistor ( $R_1$ ) with a capacitor ( $C_{in}$ ) and using a resistor $R_f$ for feedback.
Operation:
Current through capacitor: $I = C_{in} (dV_{in}/dt)$ .
Current through feedback resistor: $I = -V_{out}/R_f$ .
Equating currents:
$C_{in} \frac{dV_{in}}{dt} = -\frac{V_{out}}{R_f}$
Output Formula:
$V_{out} = -R_f C_{in} \frac{dV_{in}}{dt}$
Waveform Response:
- Input: Triangular Wave $\rightarrow$ Output: Square Wave.
- Note: Differentiators are susceptible to high-frequency noise and often require a small capacitor in parallel with $R_f$ for stability (practical design).

6. Op-Amp as a Comparator

6.1 Basic Comparator Principle

A comparator operates in the open-loop configuration (no feedback). It compares two voltages applied at its input terminals and drives the output to either the positive or negative saturation voltage.

Inputs: Reference Voltage ( $V_{ref}$ ) and Input Signal ( $V_{in}$ ).
Operation:
- If $V_+ > V_-$ (Non-inverting input is higher): $V_{out} = +V_{sat}$ (Positive Saturation, close to $+V_{CC}$ ).
- If $V_+ < V_-$ (Inverting input is higher): $V_{out} = -V_{sat}$ (Negative Saturation, close to $-V_{EE}$ or Ground).

6.2 Application: Anti-Lock Braking System (ABS)

Op-amp comparators are critical in the control electronics of automotive ABS to prevent wheel lockup during heavy braking.

Working Principle:

Sensing: A wheel speed sensor (typically a magnetic inductive sensor) generates a voltage signal proportional to the rotational speed of the wheel.
Signal Conditioning: This AC signal is converted to DC voltage proportional to speed ( $V_{speed}$ ).
Reference Setting: The system calculates a Reference Voltage ( $V_{ref}$ ) representing the vehicle's actual travel speed (or the speed of other wheels).
Comparison (The Op-Amp Role):
- An Op-Amp Comparator continually compares $V_{speed}$ (Wheel Speed) against $V_{ref}$ (Threshold/Vehicle Speed).
- Normal Braking: $V_{speed} \approx V_{ref}$ . The comparator output remains stable.
- Lockup Detected: If the wheel locks, its speed drops drastically, meaning $V_{speed} < V_{ref}$ (beyond a specific tolerance).
Actuation:
- When $V_{speed}$ drops below the threshold, the Comparator output flips state (e.g., High to Low).
- This logic signal triggers a solenoid valve to release hydraulic pressure from that specific brake caliper.
Loop: As the brake releases, the wheel spins up again ( $V_{speed}$ rises). The comparator flips back, reapplying pressure. This cycle repeats roughly 15-20 times per second, maintaining maximum traction without locking the tires.