Question 8.5: Design a low-pass filter for fabrication using microstrip li...

From Table 8.4 the normalized low-pass prototype element values are

  g_1 = 3.3487 = L1,
  g_2 = 0.7117 = C2,
  g_3 = 3.3487 = L3,
  g_4 = 1.0000 = RL,

with the lumped-element circuit shown in Figure 8.36a.

We now use Richards’ transformations to convert series inductors to series stubs, and shunt capacitors to shunt stubs, as shown in Figure 8.36b. According

to (8.78), the characteristic impedance of a series stub (inductor) is L, and the characteristic impedance of a shunt stub (capacitor) is 1 /C. For commensurate line synthesis, all stubs are λ /8 long at  ω = ω_{c}.

(It is usually most convenient to work with normalized quantities until the last step in the design.)
The series stubs of Figure 8.36b would be very difficult to implement in mi-
crostrip line form, so we will use one of the Kuroda identities to convert these
to shunt stubs. First we add unit elements at either end of the filter, as shown
in Figure 8.36c. These redundant elements do not affect filter performance since
they are matched to the source and load (Z_{0} = 1). Then we can apply Kuroda
identity (b) from Table 8.7 to both ends of the filter. In both cases we have that

The result is shown in Figure 8.36d.
Finally, we impedance and frequency scale the circuit, which simply involves
multiplying the normalized characteristic impedances by 50 Ω and choosing the
line and stub lengths to be λ/8 at 4 GHz. The final circuit is shown in Figure 8.36e, with a microstrip layout in Figure 8.36f.

The calculated amplitude response of this filter is plotted in Figure 8.37, along with the response of the lumped-element version. Note that the pass-band characteristics are very similar up to 4 GHz, but the distributed-element filter has a sharper cutoff. Also notice that the distributed-element filter has a re-sponse that repeats every 16 GHz, as a result of the periodic nature of Richards’ transformation.