Comprehensive Guide to First-Order Active Filters

General Properties and Role of Active Filters

Filters are electronic circuits designed to perform signal processing operations by attenuating specific frequency components deemed as the useless part of a signal, while allowing other components, known as the useful part, to pass through. A filter is defined as a linear quadripole that only transmits signals within a frequency range referred to as the bandwidth of the filter. Active filters are distinguished from passive ones by their composition, which includes active elements such as Operational Amplifiers (AOP) alongside passive elements like resistors and capacitors (RC). Within the operation of the filter, the frequencies contained in the input signal, denoted as $V_e$, are not modified; the filter acts primarily on the amplitude of these components.

Transfer Functions and Filtering Characteristics

Filtering is fundamentally a linear operation aimed at selecting specific portions of a signal's spectrum. Like all linear systems, filters are characterized by their transfer function or transmittance, noted as $F$. This function is mathematically expressed as the ratio of the output voltage to the input voltage in the complex frequency domain: $F(j\times\tan(\theta))$ or more specifically $F(j\times\text{omega}) = \frac{V_s(j\times\text{omega})}{V_e(j\times\text{omega})}$.

The primary characteristics that define an active filter include its cutoff frequency, its bandwidth (essential for describing band-pass and band-stop filters), its maximum voltage amplification coefficient, and its maximum gain. To a first approximation, these characteristics depend solely on the passive components used in the circuit design. The gain of the filter is linked to its transmittance by the logarithmic relationship $G_{dB} = 20 \times \log(|F(j\times\text{omega})|)$, where the gain is expressed in decibels as a function of the frequency $f$ or the pulsation $\omega$.

Bandwidth and Cutoff Frequency Definitions

The bandwidth at $-3\,dB$ is defined as the interval of frequencies for which the gain $G$ is greater than or equal to the maximum gain minus three, expressed as $G \ge G_{max} - 3\,dB$. This condition corresponds to the threshold where the module of the transmittance is greater than or equal to the maximum transmittance divided by the square root of two, written as $|F| \ge \frac{F_{max}}{\sqrt{2}}$.

Methodology for Plotting Bode Diagrams

Tracing the Bode diagrams corresponding to a transfer function $F(j\times\text{omega})$ requires a systematic approach. First, one must express the gain in decibels as $F_{db} = 20 \times \log(|F(j\times\text{omega})|)$ and the phase shift as $\phi = \text{arg}(F(j\times\text{omega}))$ as functions of the frequency or pulsation. This calculation necessitates the use of mathematical formulas for complex numbers and logarithms.

Subsequently, particular limits must be calculated to determine the various asymptotes of the diagram. The final plotting is performed on a semi-logarithmic scale. This specific scale is advantageous because it compresses high-frequency data while simultaneously preserving the clear representation of lower values, allowing for a comprehensive view of the filter's behavior across a wide spectrum.

First-Order Active Low-Pass Filters

A first-order low-pass filter is designed to allow low-frequency signals to pass while attenuating high-frequency signals. The canonical form of the transfer function for this type of filter is given by $F(j\times\text{omega}) = \frac{k}{1 + j\times\frac{\omega}{\omega_0}}$ or alternatively in terms of the Laplace variable $p$ as $F(p) = \frac{k}{1 + \frac{p}{\omega_0}}$. In these expressions, $k$ is a real constant (which can be positive or negative), $\omega$ is the pulsation, and $\omega_0$ is the cutoff pulsation.

The process for drawing the Bode diagram involves two main steps. Step 1 focuses on calculating the gain in decibels and the phase shift across different frequencies. Step 2 involves determining the limits relative to the cutoff pulsation $\omega_0$. Specifically, as $\omega$ approaches zero, the gain approaches $20 \times \log(k)$. At the cutoff pulsation $\omega = \omega_0$, the gain is $G_{max} - 3\,dB$, and the phase shift is $-\frac{\pi}{4}$ (assuming $k > 0$). As $\omega$ tends toward infinity, the gain decreases with a slope of $-20\,dB/decade$, and the phase shift tends toward $-\frac{\pi}{2}$.

First-Order Active High-Pass Filters

High-pass filters allow high-frequency signals to pass while blocking or attenuating lower frequencies. Their canonical transfer function is expressed as $F(j\times\text{omega}) = \frac{k \times j\times\frac{\omega}{\omega_0}}{1 + j\times\frac{\omega}{\omega_0}}$ or in the Laplace domain as $F(p) = \frac{k \times \frac{p}{\omega_0}}{1 + \frac{p}{\omega_0}}$. Similar to the low-pass filter, $k$ remains a real constant and $\omega_0$ represents the cutoff pulsation.

The Bode diagram for a high-pass filter is also established in two steps. In Step 1, the gain in decibels and phase shift are calculated. In Step 2, limits are calculated relative to $\omega_0$. For very low pulsations where $\omega$ approaches zero, the gain trends toward negative infinity with a positive slope of $+20\,dB/decade$, and the phase shift reaches $\frac{\pi}{2}$. At the cutoff pulsation $\omega = \omega_0$, the gain is exactly $3\,dB$ below its maximum, and the phase shift is $\frac{\pi}{4}$. As the pulsation $\omega$ goes to infinity, the gain stabilizes at its maximum value $G_{max} = 20 \times \log(k)$, and the phase shift trends toward zero.