Diseño de Filtros Activos Parte 1

Name: Diseño de Filtros Activos Parte 1
Uploaded: 2025-06-23T12:01:22.000Z
Duration: 44 min 34 s

Active Filters Design Using Sallen-Key Architecture

Introduction to Active Filters

The video introduces the practical design of active filters using the Sallen-Key architecture, while also mentioning alternative designs.

It emphasizes that the examples will focus on second-order filters, with techniques for achieving higher orders by cascading multiple filters.

Polynomial Approaches in Filter Design

The discussion includes polynomial approaches such as Bode and Chebyshev, highlighting their functional advantages and disadvantages.

The transfer function of an active filter is expressed as a quotient of two polynomials (numerator and denominator).

Stability Conditions for Filters

Two basic conditions for filter stability are outlined:

All coefficients of the denominator must be real and positive to ensure roots are in the left half-plane.

The degree of the denominator polynomial must be greater than or equal to that of the numerator.

Frequency Response Characteristics

Higher order filters approach ideal responses but increase complexity and cost.

Four types of transfer functions are represented: low-pass, high-pass, band-pass, and notch filters.

Key Parameters Affecting Filter Quality

Parameters like maximum attenuation (αp), minimum attenuation in stopband, and transition bandwidth between passbands are crucial for defining filter quality.

The concept of selectivity is introduced, defined by cutoff frequencies in both passband and stopband.

Discrimination Factors in Filter Performance

Discrimination factors indicate how well a filter can amplify signals within its passband while attenuating those outside it.

A discrimination factor (K_T), ranging from 0 to 1, reflects performance based on maximum/minimum attenuations across bands.

Understanding Filter Symmetry and Selectivity

Importance of Filter Symmetry

The symmetry of a filter is crucial, particularly in band-pass filters. If f_0^2 equals the product of f_2 times f_3 or f_1 times f_4 , it indicates perfect symmetry.

Attenuation Characteristics

Minimum attenuation occurs in the stopband for band-pass filters, while maximum attenuation is found in the passband. Conversely, notch filters exhibit maximum attenuation in the passband and minimum in the stopband.

Relationship Between Frequencies

The filter's selectivity depends on the relationship between central frequency squared and other frequencies discussed earlier. Selectivity is defined as the difference between two frequencies rather than their ratio: (F3 - F2)/(F4 - F1) .

Bandwidth and Quality Factor

Bandwidth is determined by f_3 - f_2 , while the quality factor (Q) reflects both selectivity and symmetry, calculated as Q = f_0 / b . Normalization can be applied by setting f_0 = 1 .

First Order Filters

Circuit Design for First Order Filters

The first-order filter circuit utilizes an operational amplifier with input impedance Z and resistance R_UA forming an inverting amplifier configuration. Transfer function derived from this setup shows a first-degree polynomial in its denominator.

Gain Calculation

Gain within its passband can be easily calculated as B/R_A . As S approaches zero, transfer function simplifies to yield gain at low frequencies based on resistances involved. Frequency cutoff follows similar principles: 1/(R_B C) .

Active vs Passive Filters

Distinction Between Active and Passive Filters

An active filter requires an operational amplifier or transistor for gain, contrasting with passive filters that do not utilize such components. This distinction is essential when designing circuits involving amplifiers.

Transfer Function Analysis

For first-order active filters, transfer functions are expressed concerning angular frequency ( jomega ). The gain formula remains consistent with previous calculations: 1 + R_B/R_A , maintaining frequency cutoff at similar values as before.

Transitioning Between Filter Types

Converting Low-Pass to High-Pass Filters

To create a high-pass filter from a low-pass design, simply swap resistors (R ↔ C). This maintains circuit integrity while altering response characteristics; both numerator and denominator polynomials remain of order one after transformation.

Common Design Technique

A common technique involves designing a low-pass filter first then converting it into a high-pass version by interchanging components—this method streamlines design processes significantly for engineers seeking efficiency in circuit creation.

Second Order Filter Dynamics

Overview of Second Order Filters

The second-order filter's transfer function features distinct numerators depending on type but consistently includes a second-degree polynomial denominator characterized by coefficients labeled sub 2, sub 1, and sub 0 across various configurations.

Behavior Under Limit Conditions

In analyzing behavior at limits (e.g., S approaching zero or infinity), insights into gain characteristics emerge clearly—both low-pass and high-pass configurations reveal constant gains under specific conditions related to their respective coefficients within transfer functions.

Analysis of Filter Characteristics

Independent Terms in Filters

The expression includes an independent term b_0 , which was absent in the high-pass filter, indicating a characteristic difference in filter design.

Factorization and Frequency Relationships

By factoring out a_0 from the denominator and the leading coefficient from the numerator, it can be shown that C_2/A_s is inversely related to the square of the cutoff frequency for low-pass, high-pass, or band-pass filters.

Gain Characteristics Across Filter Types

In low-pass filters, after factoring out terms, we find that b_0 A_s represents gain within the passband. Similar relationships hold for high-pass and band-pass filters with respective coefficients.

Symmetry in Band-Eliminated Filters

The unique aspect of band-eliminated filters is having two gains at low frequencies ( B_0/A_S0 and B_U/A_S2 ), which enhances symmetry but does not necessitate equal gains across different frequency ranges.

Characteristic Equations for Filter Design

The equations derived based on cutoff frequency and quality factor are fundamental for designing second-order active filters: low-pass, high-pass, band-pass, and notch (band-eliminated). These serve as foundational principles in filter design.

Root Analysis of Denominator Polynomials

Root Locations on Cartesian Axes

Analyzing roots on Cartesian axes reveals that if Q = 0.5 , there exists a double pole on the negative real axis indicating stability due to its location in the left half-plane.

Effects of Varying Quality Factor (Q)

If Q < 0.5 , poles split into two; one approaches infinity while another nears zero, risking instability as one pole may shift into the right half-plane if Q becomes too small.

Complex Conjugate Poles with Higher Q Values

For values where Q > 0.5, poles become complex conjugates with identical real parts but differing imaginary components; increasing Q leads to purely imaginary poles along the vertical axis when approaching infinity.

Negative Quality Factor Implications

A negative value for Q violates initial conditions requiring positive real coefficients; this results in complete instability within filter performance parameters. Thus, maintaining appropriate values for quality factor is crucial for stability analysis.

Examples of Low-Pass Filters

Constant Numerator Behavior

Active low-pass filters exhibit constant numerators independent of frequency; their zeros approach infinity as indicated by transfer function limits at zero frequency yielding maximum gain within passband regions.

Transfer Function Characteristics

The transfer function shows consistent gain around cutoff frequencies (~100 kHz); beyond this point (6 dB/octave drop), gain decreases significantly illustrating typical behavior patterns observed in such filters under varying conditions of quality factor (Q).

Understanding Filter Design and Quality Factor (Q)

The Role of Quality Factor (Q) in Filters

The size of Q affects filter stability; a larger Q indicates proximity to instability, as poles are close to the y-axis.

A high-pass filter's transfer function shows gain at high frequencies, with a defined behavior for Q = 1/2, where phase transitions from 0 to 180 degrees.

Second-order high-pass filters have numerators that match the order of their denominators, with zeros located at the origin.

Effects of High Q on Filter Performance

Simulations indicate that a very large Q introduces peaks around the cutoff frequency, risking filter stability.

In low-pass filters, gain is not constant near the cutoff frequency; it decreases before stabilizing at higher frequencies.

Band-Pass Filter Characteristics

For band-pass filters, quality factor is defined as f₀ divided by bandwidth. A Q of 1/2 suggests symmetry in gain across low and high frequencies.

Zeros in band-pass filters exist both at the origin and infinity; increasing Q narrows bandwidth but can create undesirable peaks in transfer function.

Implications of High Quality Factor

A very narrow bandwidth due to high Q results in significant distortion in gain characteristics, leading to poor filter performance.

As central frequency f₀ exceeds 100 kHz, maintaining zeros at both ends becomes crucial for effective filtering.

Analysis of Notch Filters

Various graphs illustrate how different values of Q affect notch filter performance; lower values yield better selectivity while higher values lead to amplification rather than attenuation within certain bands.