Active Low Pass Filter Design with Op Amps

This video explains five common filter approximations used in analog filter design: Butterworth (maximally flat passband, no stopband ripple, 20n dB/decade roll-off, used in audio processing), Chebyshev (rippled passband, monotonic stopband, faster roll-off than Butterworth), Inverse Chebyshev (flat passband, rippled stopband, same roll-off as Chebyshev), Elliptic (rippled passband and stopband, fastest roll-off but poorest step response), and Bessel (flat passband, monotonic stopband, slowest roll-off but linear phase response and best step response for non-sinusoidal signals).

Second-order filters are classified by their damping coefficient (ζ): Butterworth filters (ζ=0.707) provide the flattest passband response with no peak and asymptotic time response; Chebyshev filters (ζ<0.707) offer steeper roll-off but exhibit a peak at cutoff frequency and oscillatory behavior; Bessel filters (ζ=0.866) provide linear phase response with constant group delay but have a less distinct passband-stopband separation. The choice of filter depends on the specific application requirements.

Three main filter types differ in their frequency response characteristics: (1) Butterworth filters provide maximally flat passband response with no ripple but have slower roll-off; (2) Chebyshev filters offer steepest roll-off but introduce passband ripple (typically 0.5 dB or 3 dB); (3) Bessel filters maintain linear phase response with minimal overshoot but have the slowest roll-off. Each type has trade-offs between passband flatness, stopband attenuation, and transient response.

Four major filter types exist with distinct characteristics: Butterworth offers maximally flat response near the corner frequency with poles arranged evenly; Chebyshev provides sharper roll-off with ripples in either passband (Type 1) or stopband (Type 2); Bessel delivers maximally linear phase response ideal for time-domain applications; Elliptic provides the sharpest roll-off but introduces ripples on both sides. Selection depends on whether flatness, sharpness, or linear phase is prioritized in the application.

Filters are classified by their amplitude response shapes, each offering different trade-offs between performance characteristics. Butterworth filters provide maximally flat response with no ripple in passband or stopband but have gradual transitions. Chebyshev Type I introduces controlled ripple in the passband for steeper roll-off, while Chebyshev Type II (inverse) places ripple in the stopband. Elliptic filters achieve the steepest transitions but exhibit ripple in both bands. These response types allow engineers to select filters optimized for specific application requirements, balancing transition sharpness against ripple tolerance.

This section explores active filter configurations using operational amplifiers. Active filters offer advantages over passive designs including gains greater than 1, elimination of inductor requirements, and immunity to loading effects due to high input impedance and low output impedance. Common active filter topologies include: low-pass with RC input and feedback resistor; high-pass with RC series input; broadband bandpass created by cascading low-pass and high-pass filters with an inverting amplifier; bandpass using passive high-pass with active low-pass; and band-reject using parallel high-pass and low-pass filters with an inverting amplifier. Transfer functions for active filters include: low-pass H(s) = -K/(ωc+s), high-pass H(s) = -Ks/(ωc+s), cascaded bandpass H(s) = -K(ωc₂s)/(s² + (ωc₁+ωc₂)s + ωc₁ωc₂), and band-reject H(s) = K(s² + 2ωc₁s + ωc₁ωc₂)/[(s+ωc₁)(s+ωc₂)]. Cutoff frequencies depend on component values: low-pass 1/(R₂C), high-pass 1/(R₁C), bandpass edges at 1/(R₁C₁) and 1/(R₂C₂). Time-domain applications include low-pass as integrators and high-pass as differentiators.

The video covers three fundamental active filter types: Low-pass filters allow low frequencies to pass while attenuating high frequencies using a feedback RC network. High-pass filters block low frequencies and pass high frequencies with a capacitor at the input. Band-pass filters combine both principles, allowing only a specific frequency band to pass while attenuating frequencies above and below. Each configuration uses similar component values (10kΩ resistors and 100nF capacitors) but arranges them differently to achieve different frequency responses. The band-pass filter achieves second-order behavior with a center frequency of approximately 159Hz using the formula f = 1/(2πRC).

Active filters are classified into four main types based on their frequency response characteristics: (1) Low Pass Filter (LPF) - passes low frequencies and blocks high frequencies; (2) High Pass Filter (HPF) - passes high frequencies and blocks low frequencies; (3) Band Pass Filter (BPF) - passes a specific range of frequencies between two cutoff points; (4) Band Reject Filter (BRF), also called Notch Filter - blocks a specific range of frequencies while passing all others.

An active high pass filter is constructed similarly to the low pass filter but with the resistor and capacitor positions interchanged, allowing only high frequency components to pass while rejecting low frequencies. The voltage gain is determined by the feedback resistor (Rf) and input resistor (R1) according to the formula: Gain = 1 + Rf/R1. An active band pass filter allows only a specific band of frequencies between a lower cutoff frequency (fl) and an upper cutoff frequency (fh). It is constructed by cascading an active high pass filter (first stage) with an active low pass filter (second stage). An active band stop filter rejects a specific band of frequencies while allowing both low and high frequencies to pass. It consists of two stages: an active low pass filter and an active high pass filter, whose outputs are combined using a summing amplifier. When the low pass cutoff frequency is set lower than the high pass cutoff frequency, the resulting filter rejects frequencies between these two cutoff points.

A band-reject filter can be constructed by combining a low-pass filter and a high-pass filter, with both outputs connected to an operational amplifier summer. The low-pass filter passes frequencies up to ω₁ while blocking higher frequencies, and the high-pass filter passes frequencies above ω₂ while blocking lower frequencies. When these two signals are summed together at the operational amplifier input, the resulting circuit blocks frequencies between ω₁ and ω₂ while passing all other frequencies. This implementation requires careful selection of component values to achieve the desired cutoff frequencies.

The gain-bandwidth product (GBP) is a constant value for internally compensated op amps, where GBP = gain × bandwidth, allowing prediction of bandwidth for any closed-loop gain configuration; however, slew rate limiting—the maximum rate at which an op amp can change its output voltage—can become the dominant factor in determining frequency response, particularly for larger signals, as it depends on the signal amplitude and frequency through the relationship Slew Rate = 2π × V_peak × f, potentially causing waveform distortion when exceeded.

Operational amplifiers have two key limitations: the gain-bandwidth product and slew rate. The gain-bandwidth product states that maximum frequency equals the rated frequency divided by the desired gain (e.g., 741 rated at 600 kHz for gain of 1, so 60 kHz for gain of 10). Slew rate is the maximum rate of output voltage change, measured in volts per microsecond. When input signals change faster than slew rate allows, output cannot reach full amplitude, causing gain reduction at high frequencies. Both factors must be considered when selecting amplifiers for specific applications.

The LM741 operational amplifier suffers from several non-ideal characteristics including limited output voltage swing (only 0.8V from power supply rails), significant input offset voltage (1mV), slow slew rate (0.5V/μs), and limited gain-bandwidth product (1MHz), making it unsuitable for most modern circuits despite being a classic teaching example; modern alternatives like the MCP600x series (rail-to-rail I/O, 1.8-6V operation, 400μV offset), TLC081 (400μV offset), NE5532, and LM4562 offer vastly superior performance for practical applications.

Slew rate is the maximum rate of change of the output voltage, measured in volts per microsecond (V/μs), representing how quickly an op-amp can transition between voltage states. The original TL072 has a typical slew rate of 20 V/μs, while the generic counterfeit measured only 0.4 V/μs—50 times slower. This dramatically affects high-frequency applications, as the output signal becomes deformed rather than accurately reproducing the input. The maximum frequency at which an op-amp can accurately reproduce a sine wave without slew rate distortion is calculated using: fmax = (Slew Rate) / (2π × Vpeak). For a 20V peak-to-peak sine wave (10V peak), the original TL072 can handle approximately 318 kHz, while the generic op-amp can only handle about 6.37 kHz. Bandwidth refers to the frequency range over which an op-amp maintains its specified gain. When negative feedback is applied for a closed-loop gain of 100, bandwidth extends from approximately 10 Hz (open-loop) to about 30 kHz. The -3dB point indicates the bandwidth limit. For the generic op-amp, the -3dB point occurs at approximately 12 kHz, while the original TL072 maintains its gain until about 60 kHz. This demonstrates that counterfeit op-amps may have substantially reduced bandwidth, making them unsuitable for applications requiring high-frequency response. Generic op-amps are unpredictable and may fail to meet specifications, potentially causing circuit malfunction.

The Gain Bandwidth Product (GBP) is the constant product of an op-amp's gain and bandwidth, meaning increasing gain reduces bandwidth proportionally; for example, an op-amp with 1 MHz GBP configured at gain 1 has 1 MHz bandwidth, but at gain 10 it only has 100 kHz bandwidth. Slew rate, the maximum rate of output voltage change (typically measured in volts per microsecond), further limits bandwidth by causing distortion when output amplitude is too high for the frequency being amplified. Together, these parameters define the maximum usable frequency range for an op-amp circuit, and data sheets often include charts showing maximum output voltage versus frequency to help designers select appropriate components for their applications.