Butterworth Filter : Design of Low Pass and High Pass Filters

Name: Butterworth Filter : Design of Low Pass and High Pass Filters
Uploaded: 2017-10-16T06:43:20.000Z
Duration: 32 min 49 s

Butterworth Filter Design and Characteristics

Introduction to Butterworth Filters

The video introduces the topic of Butterworth filter approximation, emphasizing its significance in filter design.

It highlights that the Butterworth filter is known for its flat passband and a roll-off rate of 20n dB per decade, where 'n' represents the order of the filter.

Designing Second Order Low Pass Filters

The transfer function for a second-order low pass filter is introduced, with 'k' as gain and 'ω_n' as cutoff frequency.

The discussion transitions to deriving this transfer function based on quality factor Q, which influences the filter's performance.

Cascading First Order Filters

The method of cascading two first-order low pass filters to create a second-order low pass filter is explained.

A mathematical expression for the output-to-input ratio (Vout/Vin) is provided, showcasing how it can be rearranged into a standard form.

Cutoff Frequency Considerations

It’s noted that when designing cascaded filters, the cutoff frequency shifts from its original value due to component values.

For an nth order filter made by cascading first-order filters, the cutoff frequency will also shift according to specific calculations.

Importance of Q Factor in Butterworth Design

In Butterworth designs with equal resistor and capacitor values, it’s stated that achieving a Q value greater than 0.5 is essential for effective filtering.

A higher Q indicates more peaking around the cutoff frequency; specifically, at Q = 0.707, amplitude reaches 1/√2 times maximum value.

Active Components in Filter Design

To achieve desired characteristics in a Butterworth filter design, active components are necessary for positive feedback.

Unity Gain Sallen Key Low Pass Filter Design

Overview of the Sallen Key Filter

The unity gain Sallen Key low pass filter design allows for gain adjustment by connecting a feedback resistor between the inverting terminal and output.

The transfer function can be derived using nodal analysis, and assistance is offered for those struggling with this derivation.

Transfer Function and Quality Factor

The transfer function resembles that of a generalized second-order filter, where omega_c is defined as 1/sqrtR_1 R_2 C_1 C_2 and quality factor Q is expressed in terms of resistors and capacitors.

Simplifying with equal values for resistors ( R_1 = R_2 = R ) and capacitors ( C_1 = C_2 = C ), leads to a new expression for the transfer function dependent on gain k .

Limitations of Design

A critical limitation exists where the gain cannot exceed 3; exceeding this value causes Q to approach infinity, leading to system instability. Thus, k < 3 .

For Butterworth filter design, it’s established that Q = 0.707 , which translates into specific calculations for gain ratios involving resistors.

Designing Butterworth Filters

To achieve a Butterworth filter design with specified parameters, calculations yield that if R_3 = 10kOmega, then R_4 should be approximately 5.86kOmega. This demonstrates how adjusting gain influences filter characteristics.

By cascading multiple second-order filters using polynomial expressions, higher order filters can be designed effectively. This method utilizes polynomials derived from the desired frequency response characteristics.

Higher Order Filter Design

For designing third-order Butterworth low pass filters, one must cascade a second-order Sallen Key filter with a first-order RC low pass filter while adhering to specific polynomial forms like (S + 1)(S^2 + S + 1) .

Setting cutoff frequency at 1 kHz involves calculating component values based on standard formulas relating resistance and capacitance to frequency specifications. For instance, choosing capacitor values leads to determining appropriate resistor values ensuring desired cutoff frequencies are met.

Achieving Desired Quality Factor

To ensure the quality factor Q = 1, adjustments in resistor ratios are necessary; specifically setting conditions such that certain relationships hold true among resistances used in the circuit design ensures stability at targeted performance levels.

Designing High Pass Filters Using Similar Techniques

Transitioning from Low Pass to High Pass Filters

High Pass Filter Design and Butterworth Filters

Transfer Function of High Pass Filter

The transfer function for the high pass filter is expressed in a specific formula. When R1 = R2 = R and C1 = C2 = C, the denominator matches that of a low pass filter.

The transfer function V_out/V_in is defined as K cdot S^2 / (S^2 + s cdot omega_C cdot (3 - K) + omega_C^2) , illustrating how it can be derived from standard parameters.

Design Specifications for Butterworth Filter

For a Butterworth filter design, the quality factor Q should equal 0.707. This ensures optimal performance in frequency response.

The ratio of resistors R4/R3 must be 0.586 to maintain the desired characteristics in the high pass filter design, mirroring procedures used in low pass designs.

Cascading Filters for Higher Orders

To create a third-order filter, cascade a second-order with a first-order filter; similarly, two second-order filters can form a fourth-order Butterworth high pass filter.

The same polynomial equations utilized in low pass filter designs are applicable here, allowing for consistency across different types of filters.

Practical Exercise

An exercise is proposed to design a fourth order Butterworth high pass filter with a cutoff frequency ( f_c ) of 10 kHz. Participants are encouraged to share their design values in the comments section below.

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