Diseño de Filtros Activos: Ejemplos. Parte 3

Introduction to Practical Analog Filter Design

Overview of Filters

• The video presents four examples of practical analog filter designs: high-pass, low-pass, band-pass, and band-stop filters.

• The aim is to serve as a guide for viewers in their own design projects.

Low-Pass Filter Example

• The first example discussed is a second-order low-pass filter using Sallen-Key architecture. The operational amplifier is assumed to be ideal.

• Two resistors (R1 = 1KΩ and R2 = 5KΩ) and two capacitors (C1 = 100nF and C2 = 200nF) are used in the design.

Calculating Key Parameters

• Viewers are tasked with calculating the cutoff frequency, gain for a quality factor of 0.5, and proposing resistor values for the desired gain.

• The cutoff frequency calculated is approximately 503.29 Hz, while the gain required for stability yields a value of about 6.676.

Adjusting Resistor Values

Equal Resistor Scenario

• If R1 and R2 are both set to 5KΩ and C1 and C2 to 100nF, recalculating results in a new cutoff frequency of approximately 318.31 Hz.

• Maintaining the same gain leads to an unstable filter due to a negative quality factor outcome; thus adjustments are necessary for stability.

Modifying Gain for Stability

Quality Factor Adjustments

• To achieve stability while maintaining polynomial approximation characteristics, modifications in gain are suggested based on previous findings from Butterworth approximations.

• A new gain value calculated is approximately 1.586, leading to adjusted resistor ratios that may not yield commercially available values but remain theoretically sound within the exercise context.

Recommendations on Resistance Values

• It’s advised that resistances should ideally be kept within kilo-ohm ranges to minimize current consumption while avoiding excessive noise from very high resistance values (mega-ohms).

Compensating Offset Currents

Impedance Considerations

• The input impedance seen from the non-inverting terminal should be around twice the resistance value used in filtering applications (approximately 2R).

• For an ideal configuration with specified gains and impedances equalized at certain thresholds (e.g., around 10K), specific resistor values can be derived through calculations ensuring they align closely with commercial availability standards while compensating offset currents effectively within circuit designs.

Designing a Fourth-Order Bandpass Filter

Overview of Filter Design

• The design focuses on ensuring that the gain and quality factor do not deviate significantly from the second-order polynomial approximation of Bartor, which is essential for practical applications.

• The architecture involves cascading filters: a second-order Butterworth low-pass filter, a high-pass filter also using Butterworth approximation, and an amplifier to adjust gain.

Frequency Response Characteristics

• The Bode diagram indicates that the passband gain is 10 dB, with upper and lower cutoff frequencies at 1.5 kHz and 500 Hz respectively; the central frequency f_0 will be discussed later.

• For second-order filters, the asymptotic slopes in the Bode plot should be -40 dB per decade. The order of filters (low-pass before high-pass or vice versa) depends on whether input noise is higher in low or high frequencies.

Symmetry in Filter Design

• To achieve perfect symmetry in the bandpass filter, the central frequency f_0 must equal the square root of the product of cutoff frequencies, resulting in approximately 866.025 Hz.

• The quality factor Q , defined as the ratio between central frequency and bandwidth (the difference between cutoff frequencies), yields a value of 0.86.

Component Selection for Filters

• Both filters are designed using Butterworth approximation with capacitors valued at C = 10 text nF . All resistances are equal for filtering purposes.

• For high-pass filtering at a lower cutoff frequency of 500 Hz, resistance needs to be approximately 10.6 kΩ; for low-pass filtering at 1500 Hz, it requires about 31.8 kΩ.

Final Adjustments and Considerations

• Both filters must maintain identical quality factors; thus they share an amplification stage where gain equals approximately 1.586 due to their shared components.

• Two systems of equations are formed to ensure proper compensation for bias currents while achieving desired gains; calculated resistor values include R_B around 16.81 kΩ and R_A around 28.7 kΩ for high-pass filtering.

Commercial Component Availability

• Suggested commercial resistor values close to calculated ones include options like R_B being either 15 kΩ or 18 kΩ; adjustments may be necessary based on available components.

• For low-pass filtering near a target resistance of about 31.8 kΩ, suitable commercial alternatives could be around 33 kΩ; further adjustments might involve capacitor values to align with design specifications.

Practical Implementation Notes

• While offset current compensation has minimal impact due to component proximity, careful selection remains crucial for maintaining performance standards across designs.

• Using potentiometers can allow fine-tuning but is generally discouraged due to potential issues such as tolerance variations over time affecting circuit stability and performance consistency.

Amplifier Design and Gain Calculation

Understanding Potentiometer Influence

• The design of an amplifier may require adjustments to potentiometers to achieve the desired output values, depending on the criticality of the application.

Gain Calculation for Amplifiers

• To achieve a gain of 10 decibels, which corresponds to a linear value of approximately 3.1622, calculations must be made based on the product of gains from both low-pass and high-pass filters.

• The calculated gain for the chosen non-inverting amplifier is approximately 1.257, leading to a resistor ratio (R2/R1) of about 0.257. This necessitates selecting R1 as 10K ohms, resulting in R2 being around 2.57K ohms, which is not commercially available.

Design Considerations

• An inverting amplifier could have been used instead; however, this would depend on phase restrictions within the passband that are not discussed here but relate to Bode plot analysis.

• If there are no phase restrictions (e.g., needing zero or 180 degrees), using an inverting configuration remains feasible while compensating for offset currents through appropriate resistance adjustments.

Alternative Amplifier Configurations

• Other configurations include cascading two inverting amplifiers if maintaining filter phase characteristics is essential while also addressing offset current compensation needs.

Notch Filter Design

• The final design involves creating a notch filter by interchanging cutoff frequencies between low-pass and high-pass filters while ensuring a slope of -40 dB/decade for low-pass and +40 dB/decade for high-pass responses.

• A summing amplifier structure combines outputs from both filters; this approach avoids complex filtering methods while achieving practical results with standard components like resistors valued at approximately 31.8K ohms and others adjusted accordingly based on commercial availability constraints.

Quality Factor and Gain Adjustments

• Both filters need to maintain a quality factor (Q) of about 1.586; thus, equations are set up to ensure that output currents remain balanced across configurations despite potential deviations from ideal resistor values due to market limitations on component availability.

• Ultimately, achieving equal gains across both filter paths ensures that when summed together via the summing amplifier, they meet specified gain requirements effectively at around two times amplification necessary for proper signal processing at designated decibel levels (10 dB).

Amplifier Design and Configuration

Capacitor Value Adjustment

• The discussion begins with the importance of adjusting the capacitor value to achieve optimal performance, highlighting various ranges and degrees of freedom in finding a suitable solution for the problem at hand.

Summing Amplifier Structure

• The speaker describes using a subtractive summing amplifier configuration, where all inputs are connected to the inverting input of the operational amplifier through their respective resistors. This setup ensures proper signal processing.

Resistance Configuration

• It is noted that the resistances seen by both low-pass and high-pass filters (R1) are equal to two resistors placed in parallel. This design choice enhances clarity while ensuring accurate current compensation.

Gain Calculation

• The gain of the amplifier is defined as the ratio between V₀ and the sum of signals from both filters, calculated as R2 divided by R1. This follows typical differential amplifier principles, emphasizing offset current compensation.

Final Gain Values

• The speaker sets R1 at 10K ohms and RS2 at 20K ohms, considering commercial resistor values. A selection of 22K yields a total gain of approximately 3.489 or about 10.85 decibels, which is acceptable compared to an initial target of 10 decibels based on design requirements.