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90 Ladder Network Analysis
port (node a in Figure 4.13), at which time I, could be added to I., and the
recursion could again proceed normally. Equation (4.7) applies to the bridged
subnetwork. Kirchhoff's law for node b in Figure 4.13 yields
Also, Figure 4.13 shows that
Substitution of these two equations into (4.7) yields
I,= 'K - lb' (4.38)
(4.39)
'KZ,-Vb(A-I)
Ib (4.40)
B+Z,
Only A and B need to be determined. But (4.7) shows that A=V. when Vb= I
and I b = 0; also, B= V. when Ib= I and Vb = O. Two analyses of the ladder
subnetwork using these output terminal conditions will provide A and B. Then
(4.40) yields I b , and (4.38) yields bridging current I,.
This procedure begins when the recursive ladder method encounters a
component-type code that indicates a node-b condition, as in Figure 4.13. Vb
and I K are saved and replaced by I and 0, respectively. The complex linear
update is allowed to find V. = A, and this is saved. Then the complex linear
update is restarted at node b in Figure 4.13, with Vb=O and I b= I; it is
allowed to find V,=B. Then I b is calculated according to (4.40); that and the
saved Vb value are used to restart the complex linear update from node b for
the third and last time. The subnetwork voltages and currents will then be
correct. Upon arrival at node a in Figure 4.13, I. is increased by I, according
to (4.38) and the ladder recursion method continued toward the ladder input
terminals.
Example 4.3. Suppose that the embedded subnetwork is the bridged-T
shown in Figure 4.14, with recursion variables I K=2 and Vb=3. To find A
according to its definition from (4.7), set 'b=O, V b = I, and find V•. Since
there is no current through the 40-ohm branch, node-d voltage to ground must
be I, and the branch current must be 1/30. Therefore, A=V.=1+50/30
=8/3. Setting Vb=O and Ib = I, node-d voltage to ground must be 40 and
10
, , ,
'a + Ie
"
,~o d ~o
·vv
•
"
"
b 'K '" 2
+ +
V, :- 30" Vb = 3
Figure 4.14.
A resistive bridged-T embedded network.