Digital Control Systems Waseem Gulsher BS (Evening) 11 Dec, 17 Signal Conditioning Lecture - 12 Digital Control Systems Waseem Gulsher

Active Filters Tuned op-amp are generally referred to active filters. There are four basic types of active filters, each with its own circuit configuration and frequency response curve. Low Pass Filter High Pass Filter Band Pass Filter Band Stop Filter

Low Pass Filter The low pass filter is designed to pass all frequencies from DC up to an upper cutoff frequency, f2.

High Pass Filter The high pass filter is designed to pass all frequencies that are above a lower cutoff frequency, f1. All frequencies normally applied to this circuit that are above f1 are passed by the circuit.

High Pass Filter

Band Pass Filter The band pass filter is designed to pass only the frequencies that fall between the values of f1 and f2.

Band Stop Filter The band pass filter (or Notch filter) is used to eliminate all signals within the stop band (between f1 and f2) while passing all frequencies outside this band. The function of the notch filter is the opposite of that of the band pass filter.

Band Stop Filter

General Terminology Pole A pole is nothing more than an RC circuit. Thus, a one-pole filter is one that contains a single RC circuit. A two-pole filter contains two RC circuits and so on.

Butterworth Filter The most commonly used type of active filter is the Butterworth filter. It has a 20dB/decade roll-off for each pole contained in the circuit.

Butterworth Filter The Butterworth filter is one that has relatively constant gain across the pass band of the circuit. The term flat response is used to describe this constant-gain characteristics. Flat-response mean the value of Av(dB) is relatively constant across the pass band of the circuit.

Butterworth Filter Since Butterworth filters have the best flat-response characteristics, they are often referred to as maximally flat, or flat-flat filters.

Chebyshev Filter Another type of active filter is the Chebyshev fliter. It has a higher roll-off rate than the Butterworth filter containing the same number of poles. A Chabyshev filter has a roll-off rate of 40dB/decade per pole. However, the frequency response curve of the Chebyshev filter is not flat across the pass band.

Butterworth & Chebyshev Response Comparison

Butterworth & Chebyshev Response Comparison Even though the Chabyshev filter has a roll-off rate that is closer to the ideal tuned circuit than the Butterworth filter, it is not used as often.

Butterworth & Chebyshev Response Comparison There are two reasons for this: The inconsistent gain of the Chebyshev filter is undesirable. By using a two-pole Butterworth filter, the roll-off rate of the Chebyshev filter can be achieved while still having a flat response.

The Single-Pole Low-Pass Filter The single-pole low-pass Butterworth filter is constructed in one of the two ways, as either High-gain circuit or A voltage follower (has a voltage gain of a 0dB, or unity)

The Single-Pole Low-Pass Filter

The Single-Pole Low-Pass Filter The circuit is simply a non-inverting amplifier with an added input shunt capacitor. Since the reactance of the capacitor decreases as frequency increases, the high- frequency response is limited. The value of f2 for this type of circuit is

Example Determine the bandwidth of the single- pole low-pass filter in figure below. Also determine the value of ACL for the circuit.

Solution Using the values of R1 and C1, the upper cutoff frequency is found as

Solution Since the amplifier is capable of working as a DC amplifier (0Hz), there is no value of f1 for the circuit. Thus, the bandwidth is equal to f2 or 48.23 Hz. Because the circuit is a non-inverting amplifier, its value of ACL is found using equation

Solution In this case, Rf =4.7kΩ and Ri =9.1kΩ. Therefore, the value of ACL for the circuit is found as

The Single-Pole Low-Pass Filter Although, the value of ACL in previous example was relatively low, it can be made much higher by changing the Rf : Ri ratio.

The Two-Pole Low-Pass Filter The two-pole low-pass filter has roll-off rate of 40 dB/decade.

The Two-Pole Low-Pass Filter Two commonly used two-pole low-pass filter configurations are shown. Each circuit has two RC circuits, R1 – C1 and R2 – C2. As the operating frequency increases beyond f2 , each RC circuit will be dropping ACL by 20 dB, giving a total roll-off rate of 40dB/decade when operated above f2.

The Two-Pole Low-Pass Filter The cutoff frequency is then found as For the two-pole low-pass filter to have a flat response curve, its value of ACL can be no greater than 1.586 (4 dB). This means that we cannot have a high- gain two-pole low-pass filter.

The Two-Pole Low-Pass Filter The circuit shown in previous figure can be designed for any value of ACL between 1 and upper limit of 1.586 (4 dB). This circuit is normally designed according to the following guidelines: R2 = R1 C2 = 2C1 Rf ≤ 0.586 Ri

Example Verify that the relationships between the component values (listed on previous slide) apply to the circuit shown below. Also determine the bandwidth of the filter and draw its response curve.

Solution First, let’s verify the relationships listed earlier. R1 and R2 are equal in value, while C2 is twice the value of C1. Also note that the value of Rf is approximately equal to 0.586Ri.

Solution The cutoff frequency (and thus the bandwidth) of the filter is found as

The Three-Pole Low-Pass Filter The roll-off rates of cascaded stages add to form the total roll-off rate for the amplifier. It would follow that a three-pole filter, which has a roll-off rate of 60 dB/decade, could be formed by cascading a single-pole filter with a two-pole filter. This circuit would have the combined roll- off rate of 20dB+40dB=60dB/decade.

The Three-Pole Low-Pass Filter

The Three-Pole Low-Pass Filter We again have the requirements for the values of ACL. The two-pole circuit must have a closed-loop gain of approximately 4dB while the single-pole filter must have a closed loop gain of 2dB. As long as both of these requirements are fulfilled, the filter will have a Butterworth response curve.

The Three-Pole Low-Pass Filter Also note that the two filters would be tuned to the same value of f2. All circuit calculations for the amplifier are the as those performed earlier. Active filters can have more than three poles.

The Three-Pole Low-Pass Filter When we have an active filter more than three poles, following points to be remembered: The filter will have a 20dB/decade roll-off for each pole. For example, a five- pole filter would have a roll-off rate of 5x20=100dB/decade. All stages will be tuned to the same cutoff frequency.

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