Section 2.4: Operational Amplifiers (Op-amps) & Filters

Section 2.4: Operational Amplifiers (Op-amps) & Filters

Operational Amplifiers, commonly referred to as op-amps, are the building blocks of analog electronic circuits. An op-amp is a high-gain, direct-coupled voltage amplifier with differential inputs and a single-ended output. The device is represented in schematics as a triangle with five basic terminals: the inverting input (labeled with a minus sign, -), the non-inverting input (labeled with a plus sign, +), the positive power supply ($V_{CC}$ or $V_{DD}$), the negative power supply ($V_{EE}$ or $V_{SS}$), and the output terminal. In practice, op-amps amplify the voltage difference between their two input terminals.

Ideal Op-Amp Characteristics

To analyze and design op-amp circuits, engineers refer to the model of an ideal op-amp. This model makes several simplifying assumptions:

• Infinite Input Impedance ($Z_{in} = \infty$): No current flows into or out of the inverting and non-inverting input terminals. This prevents the op-amp from drawing current from the signal source, avoiding signal distortion or attenuation due to loading.
• Zero Output Impedance ($Z_{out} = 0$): The op-amp can supply any amount of current to the load without experiencing an internal voltage drop.
• Infinite Open-Loop Gain ($A_{OL} = \infty$): The amplifier's gain without feedback is infinite. A microscopic difference between the inputs will drive the output to its maximum limits (saturation).
• Infinite Bandwidth ($BW = \infty$): The op-amp can amplify signals of any frequency, from Direct Current (DC) to ultra-high radio frequencies, without signal loss.
• Infinite Common-Mode Rejection Ratio (CMRR): The op-amp completely rejects signals that are common to both inputs, such as ambient electrical noise or electromagnetic interference, while only amplifying the difference.
• Zero Offset Voltage ($V_{OS} = 0$): When both inputs are at the exact same voltage, the output is exactly zero volts.

The Golden Rules and Negative Feedback

When an op-amp operates with negative feedback—where a portion of the output voltage is returned to the inverting input—its behavior is governed by two Golden Rules:

1. Rule 1 (Voltage Rule): The op-amp will adjust its output to keep the voltage difference between the two input terminals at zero ($V_+ = V_-$).
2. Rule 2 (Current Rule): No current flows into either input terminal ($I_+ = 0$; $I_- = 0$).

Negative feedback stabilizes the circuit. By sacrificing the massive, unstable open-loop gain, negative feedback allows the overall circuit gain (known as the closed-loop gain, $A_V$) to be precisely set by external passive components, such as resistors.

Inverting Amplifier Configuration

In the inverting amplifier configuration, the input signal is applied to the inverting input through an input resistor, $R_{in}$, while the non-inverting input is connected directly to ground ($V_+ = 0\text{ V}$). A feedback resistor, $R_f$, is connected between the output and the inverting input.

Applying the Golden Rules, since $V_+ = 0\text{ V}$, the voltage at the inverting input terminal must also be zero ($V_- = 0\text{ V}$). This node is called a virtual ground because it is held at zero volts by the feedback loop without being directly wired to ground.

The current entering the inverting node through $R_{in}$ is: $I_{in} = \\frac{V_{in} - V_-}{R_{in}} = \\frac{V_{in}}{R_{in}}$

Because no current can enter the op-amp input terminal (Rule 2), this entire current must flow through $R_f$ toward the output: $I_f = I_{in} = \\frac{V_{in}}{R_{in}}$

The voltage at the output terminal is: $V_{out} = V_- - (I_f \\cdot R_f) = 0 - \\left(\\frac{V_{in}}{R_{in}} \\cdot R_f\\right) = -V_{in} \\cdot \\left(\\frac{R_f}{R_{in}}\\right)$

Thus, the closed-loop voltage gain of the inverting amplifier is: $A_V = \\frac{V_{out}}{V_{in}} = -\\frac{R_f}{R_{in}}$

The negative sign indicates that the output signal is 180 degrees out of phase with the input signal, resulting in a phase inversion.

Non-Inverting Amplifier Configuration

In the non-inverting amplifier configuration, the input signal $V_{in}$ is applied directly to the non-inverting input terminal ($V_+ = V_{in}$). The inverting input is connected to a voltage divider formed by $R_f$ and $R_{in}$ connected to ground.

By the Golden Rules, the voltage at the inverting input is driven to match the non-inverting input ($V_- = V_+ = V_{in}$). The voltage at $V_-$ is determined by the voltage divider formula: $V_- = V_{out} \\cdot \\left(\\frac{R_{in}}{R_{in} + R_f}\\right)$

Equating $V_-$ to $V_{in}$ and solving for $V_{out}$: $V_{in} = V_{out} \\cdot \\left(\\frac{R_{in}}{R_{in} + R_f}\\right) \\implies V_{out} = V_{in} \\cdot \\left(1 + \\frac{R_f}{R_{in}}\\right)$

Thus, the closed-loop voltage gain of a non-inverting amplifier is: $A_V = \\frac{V_{out}}{V_{in}} = 1 + \\frac{R_f}{R_{in}}$

Unlike the inverting configuration, the gain is positive, meaning the output is in-phase with the input, and the gain is always greater than or equal to one.

Voltage Follower (Unity Gain Buffer)

A voltage follower, also called a unity gain buffer, is a special case of the non-inverting amplifier where the feedback resistor $R_f$ is replaced with a direct wire ($R_f = 0\ \Omega$) and the input resistor $R_{in}$ is removed ($R_{in} = \infty\ \Omega$). Substituting these values into the non-inverting gain equation yields: $A_V = 1 + \\frac{0}{\\infty} = 1$

Therefore, $V_{out} = V_{in}$. The voltage follower provides no voltage amplification, but it is highly useful because it inherits the ideal op-amp's characteristics: extremely high input impedance and extremely low output impedance. This makes it an excellent buffer to isolate high-impedance signal sources (like sensors or transducers) from low-impedance loads, preventing signal attenuation caused by loading effects.

Comparators and Schmitt Triggers

When an op-amp is used without any feedback (open-loop) or with positive feedback, it behaves as a comparator. A comparator compares the voltages at its two inputs and drives the output to one of its saturation limits.

• If $V_+ > V_-$, the output saturates to the positive supply voltage ($+V_{sat}$ or $+V_{CC}$).
• If $V_+ < V_-$, the output saturates to the negative supply voltage ($-V_{sat}$ or $-V_{EE}$/ground).

Standard comparators are highly susceptible to noise. If the input signal is close to the reference voltage, small noise fluctuations can cause the output to rapidly switch back and forth, a phenomenon known as chatter. To prevent this, engineers use positive feedback to introduce hysteresis. A comparator with hysteresis has two separate threshold voltages: an Upper Threshold Voltage (UTV) and a Lower Threshold Voltage (LTV). The threshold changes based on the current state of the output. This circuit is called a Schmitt trigger. The output will only switch from high to low when the input exceeds the UTV, and it will only switch back to high when the input falls below the LTV, effectively filtering out noise.

Passive and Active Filters

A filter is a circuit designed to pass signals of certain frequencies while attenuating others. The frequency at which the filter's output power drops to half of its maximum value (which corresponds to a voltage drop to $70.7\%$ or $-3\text{ dB}$) is called the cutoff frequency ($f_c$).

• Passive Filters: Constructed using only passive components: resistors ($R$), capacitors ($C$), and inductors ($L$). Passive filters do not require an external power supply but suffer from signal attenuation (insertion loss) and are highly sensitive to load impedance.
• Active Filters: Combine passive resistor-capacitor ($RC$) networks with active devices, such as op-amps. Active filters offer several key advantages:
  1. They can provide signal gain rather than insertion loss.
  2. Their high input impedance and low output impedance isolate the filter from loading effects.
  3. They eliminate the need for large, heavy, and expensive inductors, especially at low frequencies, by using capacitors and op-amps to simulate inductive behavior.

Filter Types

1. Low-Pass Filter (LPF): Passes low-frequency signals and attenuates high-frequency signals. In a basic passive RC low-pass filter, the resistor is in series with the signal path, and the capacitor is in parallel to ground. As frequency increases, the capacitor's capacitive reactance ($X_C = \frac{1}{2\pi fC}$) decreases, shunting high frequencies to ground.
2. High-Pass Filter (HPF): Passes high-frequency signals and blocks low-frequency signals. In a basic passive RC high-pass filter, the capacitor is in series with the signal path, and the resistor is in parallel to ground. At low frequencies, high capacitive reactance blocks the signal, while at high frequencies, low reactance allows the signal to pass.

The formula for calculating the cutoff frequency ($f_c$) of a single-pole RC filter (either low-pass or high-pass) is: $f_c = \\frac{1}{2\\pi R C}$

1. Band-Pass Filter (BPF): Passes a specific range of frequencies between a lower cutoff frequency ($f_{c1}$) and an upper cutoff frequency ($f_{c2}$). Frequencies outside this band are attenuated.
2. Band-Stop (Notch) Filter: Attenuates a narrow range of frequencies while passing all other frequencies. This is commonly used to filter out specific noise, such as the $60\text{ Hz}$ hum from AC power lines.

Filter Order and Roll-off Rates

The rate at which a filter attenuates signals in the stopband is called the roll-off rate. Roll-off is determined by the filter's order, which corresponds to the number of reactive elements (capacitors or inductors) in the circuit:

• First-order (single-pole) filter: Attenuates at a rate of $-20\text{ dB/decade}$ (or $-6\text{ dB/octave}$).
• Second-order (two-pole) filter: Attenuates at a rate of $-40\text{ dB/decade}$ (or $-12\text{ dB/octave}$).
• Third-order (three-pole) filter: Attenuates at a rate of $-60\text{ dB/decade}$ (or $-18\text{ dB/octave}$).

A steeper roll-off provides sharper separation between the passband and stopband.

graph TD
    A["Active vs Passive Filters"] --> B["Passive Filters (Resistors, Capacitors, Inductors)"]
    A --> C["Active Filters (Op-Amps, Resistors, Capacitors)"]
    B --> B1["No Power Supply Required"]
    B --> B2["Insertion Loss (No Gain)"]
    B --> B3["Requires Inductors at Low Frequencies"]
    C --> C1["Requires Power Supply (+Vcc/-Vee)"]
    C --> C2["Provides Signal Gain"]
    C --> C3["No Inductors Needed (RC Networks only)"]
    C --> C4["High Input/Low Output Impedance (Isolation)"]