Analog Filter Design Reference

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Chapter 1. Introduction to Analog Filter

Chapter 2. Low-Pass Filters

Introduction to Low-Pass Filter

First Order and Odd Order Low-Pass Filter

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For odd order filters, we always need 1st order filter circuit in our cascaded filter design topology (see filter construction section). There are two common types of first order active filter circuit, i.e. the inverting and no-inverting one. Inverting type is shown in figure 1, and the non-inverting type is shown in figure 2.

(to be continued)

Quality Factor and Filter Design Parameters

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Filter Design Parameters
In designing filters, the specification requirement is usually the pass-band, transient-band, and the sop-band characteristics of the filter (See figure 1).

Figure 1. Filter Design Parameters

The pass-band is normally defined as the frequency range that the signal is not attenuated more than 3 dB. Because the pass-band and the stop-band is not clearly demarcated, there would be a transition-band where the attenuation increase before reaching the specified stop-band attenuation level.

Quality Factor
Quality factor, or known popularly as Q-factor, is another convenient way to specify a filter performance. Rather than specifying the n for the order of a certain filter type, it's more convenient to specify the Q-factor because we can directly express the actual performance of the filter we need.

For a pass band filter with mid frequency fm, quality factor Q is defined as the ratio of fm to the band width.

Q=fm/(fc2-fc1)

The band width is the pass area between cut-off frequency fc1 and fc2, where the signal is passed with no more than -3dB attenuation.

For low-pass and high-pass filters, Q represents the pole quality and is defined as:

Q=sqr(bi)/ai

High Qs can be graphically presented as the distance between the 0-dB line and the peak point of the filter’s gain response. An example is given in Figure 2, which shows a tenth-order Tschebyscheff low-pass filter and its five partial filters with their individual Qs.

Figure 2. Graphical Presentation of Quality Factor Q on a Tenth-Order Tschebyscheff Low-Pass Filter with 3-dB Passband Ripple

The gain response of the fifth filter stage peaks at 31 dB, which is the logarithmic value of Q5:
Q5[dB] = 20·logQ5

Solving for the numerical value of Q5 yields:
Q5=power(10,31/20)=35.48

which is within 1% of the theoretical value of Q = 35.85. The graphical approximation is good for Q > 3. For lower Qs, the graphical values differ from the theoretical value significantly. However, only higher Qs are of concern, since the higher the Q is, the more a filter inclines to instability.

References

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Johnson, David E., Johnson, J.R., and Moore, H.P., A Handbook of Active Filters, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1980.
D. Johnson and J.Hilburn, Rapid Practical Designs of Active Filters, John Wiley & Sons, 1975.
Kugelstadt, Thomas, Active Filter Design Techniques, part of Op Amps for Everyone, Design Reference, Texas Instruments, 2002.

Introduction to Low-Pass Filter

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Low-pass filter pass the the low frequency element of the signal, and block the high frequency element of the signal. The ideal frequency response should looks like a rectangular window with sharp corner (figure 1, the dashed line), but it's practically impossible. Many approach has been made to approach this ideal filter, such as Chebyshev, Butterworth, or Elliptic filters. Typical realistic response, Butterworth response, is shown by the solid line (Figure 1).

Figure 1. Ideal and Practical Filter Response.

About

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In this blog, I present many methods to design many types of analog electronic filters. Many types of filters: Chebyshev, Butterworth, Inverse Chebyshev, and Elliptic Filters will be covered. I assume that the reader has been familiar with analog electronics, especially with operational amplifier circuits. You can find many formulas, and step-by-step design guides here. I hope you enjoy it.

Filter Construction

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There are many ways to construct third or higher order analog filter, and one of the most popular method is by cascading first order and second order filter stages. For example, to construct fourth order filter, we can cascade two second order stages; and to construct fifth order filter, we can cascade two second order stages and a first order stage. Figure 1 show the general filter construction.

Figure 1. General Filter Construction

The infinite input and zero output approximation of active filter designed with op-amp make the cascading produce non-interacting stages, therefore the transfer function of each stage remain unchanged, giving the total cascaded output response equivalent to the multiplication product of their individual transfer functions.

The first order and the second order stages is easy to design, and using design reference presented here will be easy to construct many types of high filters (high-pass, low-pass, band-pass) with many approach (Butterworth, Chebyshev, Elliptic).

Analog Filter Elements

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Analog filter element can be passive or active. Passive filter uses inductor and capacitor, while active filter uses operational amplifier or any kind of amplification circuit. Passive filter is effective and efficient for high frequency because the size of inductor and capacitor will be small, but it's undesirable for low frequency because large capacitor and inductor consume large component and space, therefore it will be expensive. This blog will dedicate the design reference for active type, and focused on filter design with operational amplifier.

Using operational amplifier, the need for inductor to construct high order filter can be eliminated. The operational amplifier component and its symbol are shown in the figures below.

Figure 1. Operational Amplifier Integrated Circuit Chip

Figure 2. Operational Amplifier Symbol

In designing an active filter, we have to chose appropriate components to make our design meet the . Here are some consideration in choosing the op-amp for the active filter:

The open loop gain of the op-amp should be at least 50 times the filter gain [Melen, R., and H. Garland, Understanding IC Operational Amplifiers, Howard W. Sams and Co., New York, 1971].
The input impedance/resistance of the op-amp should be at least 100 times the largest resistor used in the circuit, assuming that we use 1% tolerance resistors.
Use op-amp with appropriate frequency response and slew rate. This information can be obtained from the data sheet of the op-amp from the manufacturer.

For resistors, you can use 5% tolerance for fourth or lower order filter, and 1% for higher order. In most design, mylar capacitor type is acceptable. Polystyrene and Teflon capacitor are better, but more expensive, use it for high performance filter. A cheap ceramic capacitor can be used for low-grade application.

Introduction to Analog Filter

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Frequency Response and Transfer Function
Analog filtering is done by analog electronic circuit, either active or passive circuits. Many realistic filter response can be seen in figures below:

Figure 1. Many approach realize realistic frequency response

Ideally, the response curve need to be a square window, so the frequency beyond the pass band will be completely discarded, but practically that's impossible. Many approaches has been searched to design the filter, to approximate as close as possible to the ideal filter response. The most popular of these approaches are Butterworth, Chebyshev, and Elliptic filter response.

About this Blog

2008-06-21T06:18:00.000-07:00

Hi all readers, here I present many methods in designing analog filters. You might find that this blog looks like a book, but that's what I want. Within this blog, you can find design reference for many types of analog filters. I assume that you have been familiar with analog electronics, especially with operational amplifier (op-amp) circuits.