Synthesis of Distributed Circuits

The starting procedure for the synthesis of distributed circuits is the same as for the conventional synthesis of lumped parameter circuits. If a one-port network is to be synthesized, then a desired drivingpoint immittance H(s) must be defined first, where

s ¼ sþ jv (19:1)

is the complex frequency, s is the damping coefficient of the operating signal, and v is the operating angular frequency. If a two-port network is to be synthesized, then a desired transmittance T(s) must be defined first. According to conventional principles of network synthesis [1], for the one-port network, H(s) is

represented by

H(s) ¼ P(s) Q(s)

¼ ans n þ an1sn1 þ þ a1sþ a0

bmsm þ bm1sm1 þ þ b1sþ b0 (19:2)

where an and bm are constants determined by the network parameters Q(s) is a driving function P(s) is the response function

for a two-port network

T(s) ¼ P(s) Q(s)

¼ ans n þ an1sn1 þ þ a1sþ a0

bmsm þ bm1sm1 þ þ b1sþ b0 (19:3)

Both H(s) and T(s) should be examined for realizability [1] before proceeding. If the summation of even-order terms of P(s) is M1(s) and the summation of odd-order terms

of P(s) is N1(s), then

P(s) ¼ M1(s)þ N1(s) (19:4)

Similarly,

Q(s) ¼ M2(s)þ N2(s) (19:5)

For a one-port network the driving-point impedance is synthesized by [1]

Z(s) ¼ N1(s) M2(s)

(19:6)

or

Z(s) ¼ M1(s) N2(s)

(19:7)

For a two-port network [1], if P(s) is even, the transadmittance is

y21 ¼ P(s)=N2(s)1þ M2(s) N2(s)= ½ (19:8)

the open-circuit transfer admittance is

y21 ¼ P(s)N2(s) (19:9)

and the open-circuit output admittance is

y22 ¼ M2(s)N2(s) (19:10)

If P(s) is odd,

y21 ¼ P(s)=N2(s)1þ M2(s) N2(s)= ½ (19:11)

y21 ¼ P(s)M2(s) (19:12)

and

y22 ¼ N2(s)M2(s) (19:13)

In both cases,

y11 ¼ y21(s)n (19:14)

where n is the current-ratio transfer function from port 1 to port 2. From these y-or z-parameters, the required values for the network components, i.e., L, C, and R, can be determined [1]. In high-frequency circuits the L, C, and R may be synthesized using distributed circuit components.

The synthesis of distributed components in microstrip line and circuits is the emphasis of this chapter.

If the required capacitive impedance is jXC V, the capacitance is