Davis W.A.Radio frequency circuit design.2001

90 FILTER DESIGN AND APPROXIMATION

ω D 0. The gain function for this type of filter is given by

where H0 1. The first 2n 1 derivatives of the denominator of this function are all zero at ω D 0, implying that it is maximally flat. The poles of this function all have a magnitude of 1 and are separated from one another on the unit circle by /n radians. Furthermore there are no poles on the jω-axis. At the edge of the pass band, the filter attenuates the power by 12 or 3 dB. A recursion formula for the filter elements that would produce this response can be found in a variety of references, one of which is [1]:

A set of filter values that is worth remembering is the three-pole Butterworth filter, where g1, g2, g3 D 1, 2, 1. The low-pass prototype starting with a shunt C has C1 D 1 F, L2 D 2 H, and C3 D 1 F.

Often minimum requirements are placed on the shape of the pass band. In this instance the minimum number of poles needed to produce a desired specifica-

tion is

log[ 10˛min/10 1 10˛max/10 1 ]

In this expression the maximum attenuation in the pass band 0 ω ωp is ˛max. The minimum attenuation in the stop band, ωs ω < 1, is ˛min.

5.4.2The Chebyshev Filter

The slope of the pass-band skirts for a given number of poles of a filter can be improved by allowing small ripples in the pass band. In antenna theory the DolphChebyshev amplitude weighting of the array elements provides the minimum beam width for a specified side lobe level. Similarly in filter design the Chebyshev function provides the maximum possible bandwidth for a given pass-band ripple or the minimum possible pass-band ripple for a given bandwidth. The Chebyshev (equal ripple) low-pass filter transducer gain function is

where ωc is the low-pass cutoff frequency. The value ε is a number <1 and is a measure of the pass-band ripple. The Chebyshev function, Tn x , oscillates

between C1 and 1 when its argument is less than 1. The poles of this transfer function lie on an ellipse with no jω axis poles.

For x > 1, Tn x rapidly becomes large. The Chebyshev function can be

Since Tn x < 1 in the pass band, the pass-band transfer function is

1 jH jω j2 1 1 C ε2

Outside the pass band, Tn x increases approximately exponentially.

The Chebyshev functions can be found in terms of a polynomial of its argument from a recursion formula:

The formula begins by setting T0 x D 1 and T1 x D x. Furthermore for n odd

If the maximum pass-band frequency is ωc and the minimum stop-band frequency beyond which the attenuation is always greater than ˛min is ωs, then the number of poles required in the function is n [1]:

1

arccosh 10˛max/10 1 1/2

arccosh ωs/ωc

92 FILTER DESIGN AND APPROXIMATION

Just as in the Butterworth approximation, there is a set of recursion formulas for the Chebyshev filter. Finding expressions for the g values for the filter requires first expanding the Chebyshev functions by its own set of recursion formulas. The low-pass prototype filter structure (for a given number n of reactive elements) is then equated to the nth-order filter function so that a correlation is made between the circuit and the function. Fortunately the hard work has been done in network synthesis books (e.g., [1,2]). The final recursion formulas are given below:

One important difference between the Butterworth and Chebyshev approximations is the value for gnC1. The unequal impedance levels for the even-order Chebyshev termination impedances is often avoided by simply restricting the choices of n for the Chebyshev function to odd values. The circuit element values for these two filter functions were found by using network synthesis techniques after determining the poles of the transfer function. Other filter functions are available and are briefly mentioned here.

5.4.3The Inverse Chebyshev Filter

In this filter, equal ripples are found in the stop band while the pass band is smooth. The inverse Chebyshev function is useful when it is necessary to control the attenuation over the entire stop band. The transfer function and the minimum number of poles needed to guarantee a maximum attenuation in the pass band

94 FILTER DESIGN AND APPROXIMATION

5.5THE ELLIPTIC FILTER

The low-pass filter can be characterized as having a pass band from ω D 0 to ω D ωp with an attenuation no greater than H0 plus a small ripple. In addition it is characterized as having a stop band from ωs to 1 with an insertion loss no less than some high value, ˛min (Fig. 5.5). In the Chebyshev filter the pass-band ripple is fixed to a certain maximum, but small, value while the attenuation in the stop band increases monotonically with ω. The inverse Chebyshev filter produces an equal ripple in the stop band and a monotonically decreasing insertion loss for ω going from ωp toward ω D 0. The elliptic function filter equal ripple response in both the pass band and in the stop band. This design provides a way of not throwing away excess stop-band attenuation at high frequencies by allowing redistribution of the attenuation over the whole stop band. As a consequence the rate of cutoff may be increased by putting some of the transmission zeros near the pass band. The cost for having equal ripple response in both the pass band and in the stop band is a slightly more complicated circuit topology for the elliptic filter (Fig. 5.5).

There is no simple recursion formula for the design of elliptic function filters. Typically tables of values are derived numerically [4,5] and are used for the lowpass prototype filter. These tabulated values have been incorporated in a program called ELLIPTIC. In this program the desired maximum attenuation level in the pass band, minimum attenuation in the stop band, the frequencies where the pass band ends and the stop band begins, and finally the number of poles, n, are balanced against each other to provide an elliptic filter design. If so desired, the program will produce a SPICE net list that can be used to analyze the design. In the PSPICE version of SPICE, the voltage is plotted using V(21) or VDB(21) to display the insertion loss on a linear or log scale, respectively.

5.6MATCHING BETWEEN UNEQUAL RESISTANCES

For a low-pass filter, perfect match cannot in principle be achieved when

functions, the constant, H0, is 1, since a passive filter cannot produce gain greater than 1. When the input and output resistance levels are equal, then H0 is 1. The ratio of the load to generator resistances introduces a constraint on H0. For Butterworth filters this constraint is

while for Chebyshev filters this constraint is given as follows [1]:

One might wonder if the generator and load consisted of complex impedances, what technique might be used for matching. Without getting too involved with that issue, it is known that such matching is not always possible. The impedances must be “compatible” for matching to occur. One thing a designer can do, though, is try to incorporate the reactive part of the load into the filter as much as possible.

5.6.1The Darlington Procedure

A doubly terminated filter can be designed for any physically realizable transfer function. A variety of different circuit realizations may be possible, but only one will be described. However, this particular realization method is widely used and provides practical filter design. Approximation theory determines the transfer function jH jω j2 that comes closest to the ideal filter characteristics. In a lossless, low-pass circuit with possibly unequal termination resistances (Fig. 5.6), the reflected power j jω j2 can be found:

FIGURE 5.6 Butterworth low-pass filter with unequal resistance terminations. When RG D 20 and RL D 80 , L1 D 127.75 H, C2 D 0.01804 F, and L3 D 43.400 H.

Once the reflection coefficient is determined, the Darlington synthesis procedure is used to obtain the circuit elements. The input impedance to the filter at any frequency is given in terms of the reflection coefficient:

The Cauer extraction technique for a ladder network can now be used. The polynomials in the numerator and denominator are arranged in descending powers of s. It will always be the case for a lossless transfer function that the highest power of the numerator and denominator will differ by at least 1. If the numerator is the higher-order polynomial, then an impedance pole at s D 1 (i.e., a series inductor) can be extracted from the impedance function. This is done by synthetic division. The fractional remainder is now inverted, and synthetic division again carried out to extract an admittance pole at s D 1 (i.e., a shunt capacitor). The process continues until only the load resistance or conductance remains.

As an example, consider a three-pole Butterworth filter with a 3 dB cutoff frequency at 1 rad/s. The input resistance RG D 20 and the output resistance is RL D 80 . The Butterworth transfer function is therefore

Now replace ω with js, factor the denominator into the six roots of 1, and recombine into two cubic factors where one factor contains the left half-plane roots and the other the right half-plane roots. This is the standard Butterworth polynomial:

In this case the denominator is readily factored analytically, but the roots of the numerator when H0 D6 1 must be found numerically. The program POLY can provide the complex roots of a polynomial with complex coefficients. In this example all values are calculated using double precision arithmetic, though for clarity only three or four significant figures are shown.

98 FILTER DESIGN AND APPROXIMATION

The reflection coefficient containing only left half-plane poles and zeros is

The input impedance is found from Eq. (5.51):

Extraction of the impedance pole at s D 1 is done by synthetic division:

6.387s

0.313s2 C 0.577s C 0.400 2s3 C 3.687s2 C 3.423s C 1.599

2s3 C 3.687s2 C 2.555s

0s3 C 0s2 C 0.868s C 1.599

The remainder is inverted, and an admittance pole at s D 1 is extracted:

0.361s

0.868s C 1.599 0.313s2 C 0.577s C 0.400 0.313s2 C 0.577s

0s2 C 0s C 0.868s C 0.400

By inversion again and performing synthetic division, once more another impedance pole at s D 1 is removed:

2.170s

0.400 0.868s C 1.599

The final remainder, 1.599/0.400 D 4.000 represents the normalized load resistance, which is the expected value. Hence L10 D 6.387 H, C02 D 0.361 F, and L30 D 2.170 H. The impedance level of the circuit is now adjusted from 1 to RG D 20 by multiplying all the inductances and dividing all capacitances by 20 . Thus L10 becomes L1 D 127.75 H, C02 becomes C2 D 0.01804 F, and L30 becomes L3 D 43.400 H. The final circuit is shown in Fig. 5.6. Verification of this circuit is shown by a SPICE analysis found in Fig. 5.7. Near zero frequency the insertion loss is 0.8 or 1.938 dB and at 1 rad/s (0.159 Hz); the loss has increased by 3 dB.

Easier analytical methods are available for the Chebyshev filter, and these are in fact used in the Chebyshev impedance transforming circuit described in