Active Filter Fundamentals

• This lecture focuses on active filters and their relation to passive filters.

Objectives of Active Filters

• Review the fundamentals of passive filters
• Understand the limitations of passive filters
• Learn how to overcome passive filter limitations by using active devices

Filter Review - Voltage Divider

• As frequency changes, components change their reactance
• This in turn alters the voltage divider ratio and output voltage
• Output voltage is always less than the input in a voltage divider
• The filter "gain" of a passive filter is always below 1
• Passive components cannot provide amplification

Passive Filter Review - Butterworth Filter

• The 1st order Butterworth Filter is a common type of filter
• Its frequency response is governed by the reactance of the capacitor
• Reactance is frequency-dependent
• The critical frequency formula is fc = 1 / (2πRC)
• Where f is frequency in Hertz and C is capacitance in Farads
• The formula for reactance is Xc = 1 / (2πfC)

Filter Roll-Off Rate

• The filter order can be determined by the number of reactive components.
• Each filter order has a 20dB/decade increase in roll-off rate.
• A comparison of the 1st, 2nd, and 3rd order Butterworth LPF's shows this increase.
• The mirror image would be true for HPF's.
• 1st Order: -20dB/dec
• 2nd Order: -40dB/dec
• 3rd Order: -60dB/dec

Passive Filter Review - Higher Order Filters

• A 1st order filter with a -20dB/decade roll-off is sometimes insufficient
• A higher order filter can be realized by cascading multiple lower order filters
• Cascading three 1st order LPFs creates a 3rd order LPF with a steeper roll-off of -60dB/Decade

Limitations of Realizing Higher Order Filters

• In theory, cascading filters involves connecting the input of the next stage to the output of the previous stage
• However, in practice, this is not so simple with passive filters
• For discussion, assume the LPF resistor is 100Ω and the reactive capacitance (Xc) is 100Ω at a particular frequency
• Analyzing one LPF can determine the expected behavior
• Since LPF #1, #2, and #3 are identical, ideally only one needs to be analyzed
• Based on the voltage divider formula: Vout = 0.5 Vin
• The filter should output half the input voltage
• In an ideal world, each filter functions independently, but this is not the case.
• The input impedance of LPF #2 & LPF #3 affects the overall behavior of LPF #1
• This drastically changes its behavior
• Decoupling each LPF is needed so the input impedance doesn't load down the preceding stages

Recall - Buffer Amplifier

• A buffer amplifier has a gain of 1
• Despite the gain of 1, it has extremely high bandwidth
• Buffer amplifier decouples impedances between different circuit stages
• Ideally, input terminals have infinitely high impedance and draw zero current
• Ideally, the output has zero impedance, and the output can supply unlimited current to the load.

Active Filters

• Active filters uses the high impedance input/low impedance output of a closed-loop opamp amplifier to function independently
• This creates an "Active" filter since there is an amplifying device
• Active filters enable gains higher than 1

Active Butterworth Filter

• An active 1st order Butterworth filter is realized by connecting a passive Butterworth filter to a non-inverting amplifier
• The net magnitude response for an active filter is the product of the passive filter and amplifier in DB
• This is equivalent to the sum of the dB parts

Active Butterworth Response

• Active filters can provide gains above 1 (above 0dB)
• The cutoff gain is offset above the 0dB line

Active 2nd Sallen-Key Filter

• Active filters can provide gains above 1 , and the cutoff gain is offset above the 0dB line.

Sallen-Key Filter Frequency Response

• Combined positive and negative feedback makes the filter's frequency response analysis complex
• The Damping Factor (DF) measures the Sallen-Key filter's immunity to gain variations near its cutoff frequency (Fc)
• Different values of DF provide different frequency response characteristics
• DF = 2 * R(RF / R1)

1st Order Sallen-Key LPF Type Comparison

• Butterworth: DF = 1.414, quite flat passband/stopband, medium roll-off, linear phase
• Chebyshev: DF = 0.767, very wavy passband, high roll-off, non-linear phase
• Bessel: DF = 1.732, rounded passband/stopband, low roll-off, optimally linear phase
• The Chebyshev filter is best for high roll-off rates, but with uneven passband and non-linear phase response
• The Bessel filter has poor roll-off or magnitude response flatness, but has excellent phase linearity
• The Butterworth filter is a good compromise, but needs cascading stages to achieve the desired roll-off.