The insertion-loss frequency response of a lowpass filter provides a frequency response of a highpass filter, when in (7.36) and (7.37) v/vc is replaced with vc /v. The frequency response of a bandpass filter is obtained by replacing v/vc with the term (v /v 0 − v0 /v )/D, and that of a bandstop filter by replacing v /vc with the term D/(v /v 0 − v 0 /v ).

7.2.2 Design of Microwave Filters

At microwave frequencies we have two major problems in realizing the synthesized filters. First, good lumped elements do not exist; instead we must use distributed elements. The frequency behavior of the distributed elements is more complicated than that of the lumped ideal ones, which makes the filter synthesis difficult. However, design of a narrow-band filter is easy, because over a narrow bandwidth many distributed elements may be modeled by ideal inductors and capacitors. Second, the filter elements should be physically close to each other (in wavelength scale), which is often impossible in practice. Therefore, transmission line sections must be used to separate the filter elements.

Richards’ transformation is used to transform the lumped elements into sections of transmission lines. Richards’ transformation [9]

maps the v plane to the V plane. By replacing v with V, we can write the reactance of an inductor and the susceptance of a capacitor, respectively, as

This means that an inductor can be replaced with a short-circuited stub having an electrical length of bl and a characteristic impedance of L . Accordingly, a capacitor can be replaced with an open-circuited stub having an electrical length of bl and a characteristic impedance of 1/C. The cutoff frequency of the transformed filter is the same as that of the prototype (vc = 1), if V = 1 = tan b l , or l = lg /8. At frequencies where v differs a lot from vc , the response of the transformed filter differs considerably from the response of the prototype. The response is periodic, repeating every 4vc .

With the aid of the Kuroda identities [5, 10, 11] we can separate the transmission line stubs physically from each other, transform the series stubs

Figure 7.15 Design of a lowpass filter: (a) a lowpass filter prototype; (b) lumped inductors and capacitors converted to transmission line stubs (l = l g /8 at 3 GHz) using Richards’ transformation; (c) series stubs transformed to parallel stubs using a Kuroda identity; (d) microstrip layout of the filter; and (e) calculated frequency response.

the series stubs cannot be realized with microstrip lines, we transform them into parallel stubs using a Kuroda identity. In order to be able to do that, we first add a transmission line section with a length of lg /8 and a characteristic impedance of Z 0 = 1 into both ends of the filter. These sections do not affect the filter response because the generator and load are matched to these

164 Radio Engineering for Wireless Communication and Sensor Applications

lines. The second Kuroda identity of Figure 7.14, with Z 1 = 1.5963, Z 2 = 1, n 2 = 1 + Z 2 /Z 1 = 1.6264, n 2Z 1 = 2.596, and 1/(n 2Z 2 ) = 0.615,

is then used for both ends of the filter. The result is shown in Figure 7.15(c). Finally, the impedances are scaled by multiplying by 50V, and the line lengths are calculated to be lg /8 at 3 GHz. The stub characteristic impedances are 50n 2Z 2 = 81.3V, 50/C2 = 45.6V, and 81.3V, and the transmission line sections between the stubs have a characteristic impedance of 50n 2Z 1 = 129.8V. In order to find the exact line lengths and widths we should know the characteristics of the substrate and use (3.85) through (3.93). However, a schematic layout is presented in Figure 7.15(d), and its calculated frequency response is presented in Figure 7.15(e). When this response is compared with that of a filter made of lumped elements, we observe that in the passband the responses are nearly equal, but above the cutoff frequency the insertion loss of the microstrip filter increases more quickly and the response has another passband at 12 GHz.

As in the preceding example, in most filters we want to use only parallel (or series) elements. In the microwave region we often prefer parallel stubs. In addition to the Kuroda identities, this transformation can also be done using impedance or admittance inverters. The impedance inverters are also called the K -inverters and the admittance inverters the J -inverters. For example, an ideal quarter-wave transformer is both a K - and a J -inverter. It transforms a load impedance Z L into an impedance Z in = K 2/Z L or a load admittance Y L into an admittance Yin = J 2 /Y L . Here K is the characteristic impedance and J the characteristic admittance of the transformer. A quarterwave transformer, however, behaves approximately as an ideal inverter only over a narrow bandwidth, and therefore this method is suitable only for narrow-band filters. Figure 7.16 shows how a bandpass filter is transformed into a filter circuit consisting only of either series or parallel resonators. The characteristic impedance values of the impedance and admittance inverters are presented in Table 7.4.

R aG , R aL , L ak , and Cak can be chosen freely, as long as the resonance frequency of the resonant circuits is the same as the filter center frequency or v0 = 1/√L ak Cak and the impedance values of the inverters are calculated according to Table 7.4. We may also choose a value 1 for R aG and R aL , as well as for the inverters, and then calculate values for L ak and Cak starting from one end of the filter. The LC parallel or series resonant circuits can be realized, for example, with openand short-circuited transmission line stubs with a length of lg /4 or its multiple (see also Section 7.1.4). The characteristic impedance of a short-circuited stub with a length of l g /4 must

Figure 7.16 A bandpass filter, and the same filter transformed using impedance and admittance inverters.

Table 7.4

Characteristics of the Impedance and Admittance Inverters (k = 1 . . . N − 1)

be Z 0 = (p/4) √L ak /Cak , which may be difficult to realize because of its low value.

Example 7.3

Design a bandpass filter with the following specifications: maximally flat response with N = 3, frequency range from 950 to 1,050 MHz, filter impedance of 50V.