25-27 Filter (Textbook Scan)

7/24/2019 25-27 Filter (Textbook Scan)

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Prof. Tzong-Lin Wu

EMC Laboratory

Department of Electrical Engineering

National Taiwan University

2011/2/21 MW & RF Design / Prof. T. -L. Wu

Filter


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Accept desired signal and reject signal outside the operating band

Low-pass, high-pass and bandpass filters.

Insertion loss method will be introduced

The insertion loss method is based on network synthesis techniques, and can be used

to design filters having a specific type of frequency response. The technique beginswith the design of a low-pass filter prototype that is normalized in terms of impedanceand cutoff frequency.

Impedance and frequency scaling and transformations are then used to convert thenormalized design to the one having the desired frequency response, cutoff frequency,and impedance level.

Additional transformations, such as Richard's transformation, impedance/admittanceinverters, and the Kuroda identities, can be used to facilitate filter implementation interms of practical components such as transmission lines sections, stubs, and resonantelements.

2MW & RF Design / Prof. T. -L. Wu


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5.1 Filter design by the insertion loss method

For minimum loss: binomial response,For sharpest cutoff: Chebyshev response

Linear phase needs sacrifices attenuation rate

Power loss ratio:


Thus, for a filter to be physically realizable its power loss ratio must be of the form.


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Maximally flat or binominal or Butterworth

4MW & RF Design

/ Prof. T. -L. Wu

This characteristic is also called the binomial or Butterworth response, and is optimum inthe sense that it provides the flattest possible passband response for a given filtercomplexity, or order.


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2011/2/21 MW & RF Design / Prof. T. -L. Wu 5

Maximally flat or binominal or Butterworth


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Chebyshev response or equal ripple


Chebyshev polynomial is used to specify the insertion loss of an N-order low-pass filter


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The above filters specify the amplitude response, but in some applications(such as multiplexing filters in frequency-division multiplexed communications system).It is important to have a linear phase response in the passband to avoid signal distortion.

A linear phase characteristic can be achieved with the following phase response:

which shows that the group delay for a linear phase filter is a maximally flat function.

Linear Phase


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2011/2/21 8MW & RF Design / Prof. T. -L. Wu

We will next discuss the design of low-pass filter prototypes which are normalized interms of impedance and frequency;

This type of normalization simplifies the design of filters for arbitrary frequency,impedance, and type (low-pass, high-pass, bandpass, or bandstop).

The low-pass prototypes are then scaled to the desired frequency and impedance,and the lumped-element components replaced with distributed circuit elements forimplementation at microwave frequencies.


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Maximally flat low-pass filter prototype

Consider the two-element low-pass filter prototype circuit shown in Figure 5.3; we willderive the normalized element values, L and C, for a maximally flat response.

The desired power loss ratio will be, for N = 2,


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Comparing to the desired response



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In principle, this procedure can be extended to find the element values for filters withan arbitrary number of elements, N, but clearly this is not practical for large N.



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Equal ripple low-pass filter prototype


Chebyshev polynomials have the property that

Desired response


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Chebyshev polynomial of order 2 is given as



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Note that (5.14) gives a value for R that is not unity, so there will be an impedancemismatch if the load actually has a unity (normalized) impedance;

this can be corrected with a quarter-wave transformer, or by using an additionalfilter element to make N odd. For odd N, it can be shown that R = 1.


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Filters having a maximally flat time delay, or a linear phase response, can be designed

in the same way, but things are somewhat more complicated because the phase of thevoltage transfer function is not as simply expressed as is its amplitude. Design valueshave been derived for such filters, however, again for the ladder circuits of Figure 5.4,and are given in Table 5.3 for a normalized source impedance and cutoff frequency

Linear Phase Low-Pass Filter Prototype


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5.2 Filter scaling and transformation

Impedance scaling (1 -> R0)


In the prototype design, the source and load resistances are unity.


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Frequency Scaling for Low Pass Filter



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Both impedance and frequency scaling


Frequency Scaling for Low Pass Filter


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Low-pass to high-pass transformation


Both impedance and frequency scaling


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Bandpass and bandstop transformation

31MW & RF Design/ Prof. T. -L. Wu

Fractional bandwidth


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The new filter elements are determined



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Band stop transformation


Then series inductors of the low-pass prototype are converted to parallel LC circuitshaving element values given by

The shunt capacitor of the low-pass prototype is converted to series LC circuits having

element values given by


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2011/2/21 37MW & RF Design / Prof T L Wu

25-27 Filter (Textbook Scan)

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