8.4 Operational Amplifiers & Active Filter Design

8.4 Operational Amplifiers & Active Filter Design

1. Ideal vs. Practical Op-Amp Characteristics

An Operational Amplifier (Op-Amp) is a high-gain DC-coupled differential amplifier. Under negative feedback, op-amp circuit operation is simplified by the Golden Rules of Op-Amps.

The Golden Rules of Ideal Op-Amps (with Negative Feedback)

1. Rule 1 (Infinite Input Resistance): No current enters either input terminal ($I^+ = I^- = 0\text{ A}$).
2. Rule 2 (Virtual Short Circuit): Negative feedback forces the differential input voltage to zero ($V^+ - V^- = 0 \implies V^+ = V^-$).

Ideal vs. Practical (741 BJT) Parameter Comparison

Key Non-Ideal Parameters

• Common-Mode Rejection Ratio ($CMRR$): Ability to reject common noise signals present at both input terminals.

$$CMRR = 20 \log_{10} \left| \frac{A_d}{A_c} \right|$$

• Slew Rate ($SR$): Maximum rate of output voltage change per unit time:

$$SR = \left. \frac{dV_o}{dt} \right|_{max} \quad [\text{V/}\mu\text{s}]$$

• Full-Power Bandwidth ($f_{max}$): Maximum undistorted frequency for a peak output voltage amplitude $V_p$:

$$f_{max} = \frac{SR}{2\pi V_p}$$

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2. Linear Op-Amp Configurations

A. Inverting Amplifier

• Voltage Gain: $A_v = -\frac{R_f}{R_i}$
• Input Impedance: $Z_{in} = R_i$
• Virtual Ground: Non-inverting input is grounded ($V^+ = 0\text{ V} \implies V^- = 0\text{ V}$).

B. Non-Inverting Amplifier

• Voltage Gain: $A_v = 1 + \frac{R_f}{R_i}$
• Input Impedance: $Z_{in} \approx \infty$
• Phase: In-phase output ($0^\circ$ phase shift).

C. Voltage Follower (Unity-Gain Buffer)

• $R_f = 0\ \Omega, R_i = \infty \implies A_v = +1$.
• Ideal buffer: $Z_{in} = \infty, Z_{out} = 0$.

D. Summing Amplifier

• Output Voltage:

$$V_o = -R_f \left( \frac{V_1}{R_1} + \frac{V_2}{R_2} + \frac{V_3}{R_3} + \dots + \frac{V_n}{R_n} \right)$$

E. Difference (Differential) Amplifier

• When matched resistors are used ($R_4/R_3 = R_2/R_1$):

$$V_o = \frac{R_2}{R_1} (V_2 - V_1)$$

F. Three Op-Amp Instrumentation Amplifier

• Consists of two non-inverting input buffer stages driving a differential amplifier stage.
• Overall Voltage Gain:

$$A_v = \left( 1 + \frac{2 R_1}{R_g} \right) \left( \frac{R_3}{R_2} \right)$$

• Advantages: Single resistor ($R_g$) gain control, ultra-high input impedance on both inputs, superior CMRR.

G. Op-Amp Integrator & Differentiator

• Integrator: $v_o(t) = -\frac{1}{R C} \int_{0}^{t} v_i(\tau) d\tau + v_o(0)$. Practical circuits include parallel $R_f$ across $C$ to limit DC gain.
• Differentiator: $v_o(t) = -R C \frac{dv_i(t)}{dt}$. Practical circuits add series $R_s$ with $C$ to suppress high-frequency noise amplification.

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3. Non-Linear Op-Amp Circuits & Comparators

A. Schmitt Trigger (Inverting with Positive Feedback)

Provides positive feedback to introduce hysteresis, eliminating output noise chattering.

• Upper Threshold Point ($V_{UTP}$):

$$V_{UTP} = +V_{sat} \cdot \frac{R_1}{R_1 + R_2}$$

• Lower Threshold Point ($V_{LTP}$):

$$V_{LTP} = -V_{sat} \cdot \frac{R_1}{R_1 + R_2}$$

• Hysteresis Voltage ($V_H$):

$$V_H = V_{UTP} - V_{LTP} = 2 V_{sat} \left(\frac{R_1}{R_1 + R_2}\right)$$

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4. Active Filter Design & Approximation Responses

Active filters utilize op-amps, resistors, and capacitors (no inductors) to shape frequency response.

Filter Approximation Characteristics

1. Butterworth: Maximally flat passband response; smooth monotonic attenuation ($-20\text{ dB/dec}$ per order $n$).
2. Chebyshev: Steep initial roll-off rate; exhibits equal-ripple passband variations.
3. Bessel: Linear phase shift (constant group delay); gradual amplitude roll-off; preserves pulse shapes.

Second-Order Sallen-Key Low-Pass Filter (LPF)

• Cutoff Frequency ($f_c$):

$$f_c = \frac{1}{2\pi \sqrt{R_1 R_2 C_1 C_2}}$$

• Equal-Component Design ($R_1 = R_2 = R, C_1 = C_2 = C$):

$$f_c = \frac{1}{2\pi R C}$$

• Passband Voltage Gain ($A_v$) & Quality Factor ($Q$):

$$A_v = 1 + \frac{R_f}{R_i}, \quad Q = \frac{1}{3 - A_v}$$

• For Second-Order Butterworth Response ($Q = 0.707 = \frac{1}{\sqrt{2}}$):

$$3 - A_v = \sqrt{2} \approx 1.414 \implies A_v = 1.586 \implies \frac{R_f}{R_i} = 0.586$$

• Roll-off Rate: $-40\text{ dB/decade}$ ($-12\text{ dB/octave}$).