Introduction
Introduction to Active Filters
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Filters process signals by passing selected frequencies while rejecting others
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Active filters use op-amps with passive RC circuits
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Filters categorized by output voltage variation with input frequency
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Four basic categories:
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Low-pass
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High-pass
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Band-pass
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Band-stop
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Advantages over passive filters:
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Gain provided by active elements
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High input impedance
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Low output impedance
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Filter Responses
Low-Pass Filter Response
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Passes frequencies from DC to critical frequency \(f_c\), attenuates others
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Passband: Frequencies with \(< -3 \, \text{dB}\) attenuation
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Critical frequency: \(f_c = \dfrac{1}{2 \pi R C}\) (single-pole \(RC\) filter)
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Roll-off: \(-20 \, \text{dB/decade}\) per pole
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Bandwidth: \(BW = f_c\) (ideal)
High-Pass Filter Response
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Passes frequencies above \(f_c\), attenuates below
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Critical frequency: \(f_c = \dfrac{1}{2 \pi R C}\) (single-pole \(RC\) filter)
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Roll-off: \(-20 \, \text{dB/decade}\) per pole
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Practical limitation: Op-amp’s internal \(RC\) circuits limit high-frequency response
Band-Pass Filter Response
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Passes frequencies between lower (\(f_{c1}\)) and upper (\(f_{c2}\)) critical frequencies
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Bandwidth: \(BW = f_{c2} - f_{c1}\)
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Center frequency: \(f_0 = \sqrt{f_{c1} f_{c2}}\)
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Quality factor: \(Q = \dfrac{f_0}{BW} = \dfrac{1}{DF}\) (DF: Damping factor)
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\(Q > 10\): Narrow-band; \(Q < 10\): Wide-band
Band-Stop Filter Response
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Rejects frequencies between \(f_{c1}\) and \(f_{c2}\), passes others
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Bandwidth: \(BW = f_{c2} - f_{c1}\)
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Opposite of band-pass filter
Filter Characteristics
Filter Response Characteristics
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Butterworth: Flat passband, \(-20 \, \text{dB/decade/pole}\), nonlinear phase
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Chebyshev: Steeper roll-off (\(> -20 \, \text{dB/decade/pole}\)), passband ripples
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Bessel: Linear phase, minimal overshoot, suitable for pulse waveforms
Damping Factor
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Determines response characteristic
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Defined as: \(DF = 2 - \dfrac{R_1}{R_2}\)
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Example: For second-order Butterworth, \(DF = 1.414\), so \(\dfrac{R_1}{R_2} = 0.586\)
Critical Frequency and Filter Order
Critical Frequency (\(f_c\)) for first-order RC filter: \[f_c = \frac{1}{2\pi RC}\]
Applies to both low-pass and high-pass single-pole configurations
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Roll-Off Rate:
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Butterworth filter: \(-20\, \text{dB/decade/pole}\)
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1st-order (1 pole): \(-20\, \text{dB/decade}\)
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2nd-order (2 poles): \(-40\, \text{dB/decade}\)
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3rd-order (3 poles): \(-60\, \text{dB/decade}\), etc.
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Filter Implementations
Single-Pole Low-Pass Filter
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Uses one \(RC\) circuit, roll-off \(-20 \, \text{dB/decade}\)
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Critical frequency: \(f_c = \dfrac{1}{2 \pi R C}\)
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Gain: \(A_{cl(NI)} = \dfrac{R_1}{R_2} + 1\)
Sallen-Key Low-Pass Filter
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Common second-order configuration (VCVS)
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Two-pole filter, roll-off \(-40 \, \text{dB/decade}\)
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Critical frequency: \(f_c = \dfrac{1}{2 \pi \sqrt{R_A R_B C_A C_B}}\)
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If \(R_A = R_B = R\), \(C_A = C_B = C\): \(f_c = \dfrac{1}{2 \pi R C}\)
Cascaded Low-Pass Filters
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Third-order: Cascade 2-pole Sallen-Key + 1-pole filter (\(-60 \, \text{dB/decade}\))
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Fourth-order: Cascade two 2-pole Sallen-Key filters (\(-80 \, \text{dB/decade}\))
Single-Pole High-Pass Filter
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Roll-off: \(-20 \, \text{dB/decade}\)
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Critical frequency: \(f_c = \dfrac{1}{2 \pi R C}\)
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Limitation: Op-amp’s internal \(RC\) limits high-frequency response
Sallen-Key High-Pass Filter
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Two-pole filter, roll-off \(-40 \, \text{dB/decade}\)
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\(R\) and \(C\) positions swapped compared to low-pass
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Response optimized by feedback resistors \(R_1\), \(R_2\)
Band-Pass and Band-Stop Filters
Cascaded Band-Pass Filter
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Cascade high-pass and low-pass Sallen-Key filters
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\(f_{c1} = \dfrac{1}{2 \pi \sqrt{R_{A1} R_{B1} C_{A1} C_{B1}}}\)
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\(f_{c2} = \dfrac{1}{2 \pi \sqrt{R_{A2} R_{B2} C_{A2} C_{B2}}}\)
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Center frequency: \(f_0 = \sqrt{f_{c1} f_{c2}}\)
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Suitable for wide bandwidth applications
Multiple-Feedback Band-Pass Filter
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Center frequency: \(f_0 = \dfrac{1}{2 \pi C \sqrt{\dfrac{R_1 + R_3}{R_1 R_2 R_3}}}\)
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Gain: \(A_0 = \dfrac{R_2}{2 R_1}\)
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Resistors:
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\(R_1 = \dfrac{Q}{2 \pi f_0 C A_0}\)
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\(R_2 = \dfrac{Q}{\pi f_0 C}\)
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\(R_3 = \dfrac{Q}{2 \pi f_0 C (2 Q^2 - A_0)}\)
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State-Variable Filter
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Uses summing amplifier and two integrators
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Center frequency: \(f_0 \approx \dfrac{1}{2 \pi R C}\)
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\(Q = \dfrac{1}{3} \left( \dfrac{R_5}{R_6} + 1 \right)\)
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Provides low-pass, high-pass, band-pass outputs
Biquad Filter
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Integrator, inverting amplifier, integrator
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Provides band-pass and low-pass outputs
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High \(Q\), bandwidth independent of \(f_0\)
Multiple-Feedback Band-Stop Filter
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Similar to band-pass, with modified resistor configuration
State-Variable Band-Stop Filter
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Created by summing LP and HP outputs
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Common application: 60 Hz noise rejection
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Center frequency set by integrator RC circuits
Measurements
Filter Response Measurements
Discrete Point Measurement
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Set constant input amplitude
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Sweep frequency in steps
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Record output at each frequency
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Plot response curve
Swept Frequency Measurement
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More efficient than discrete method
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Uses swept frequency generator
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Display options:
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Spectrum analyzer
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Oscilloscope in X-Y mode
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Provides continuous response curve
Summary
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Covered low-pass, high-pass, band-pass, band-stop filters
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Discussed Butterworth, Chebyshev, Bessel characteristics
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Analyzed single-pole, Sallen-Key, multiple-feedback, state-variable, biquad filters
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Explained damping factor, critical frequency, roll-off rates
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Described discrete point and swept frequency measurements