Page 189 -

The solution of this difference equation can be directly obtained by the
techniques discussed in Chapter 2 for obtaining solutions of homogeneous
difference equations. The physically meaningful solution is given by:

 2 

λ= + 1 Z 1 − Z 1 + ZZ  (6.93)

1
Z  2 4 2 1 
2  
and the voltage phasor at node n is then given by:
˜
˜
V = V λ n (6.94)
n s
We consider the model where Z = jωL and Z = 1/(jωC), respectively, for an
1
2
inductor and a capacitor. The expression for λ then takes the following form:
/
 υ   υ  12
4
2
2

λ = 1 −  − j  υ −  (6.95)
 2   4 
where the normalized frequency is defined by υ = ω ω =/ ω LC . We plot in
0
Figure 6.3 the magnitude and the phase of the root λ as function of the nor-
malized frequency.
As can be directly observed from an examination of Figure 6.3, the magni-
tude of λ is equal to 1 (i.e., the magnitude of is also 1) for υ < υV ˜ = 2, while
n cutoff
it drops precipitously after that, with the dropoff in the potential much
steeper with increasing node number. Physically, this represents extremely
short penetration through the ladder for signals with frequencies larger than
˜
the cutoff frequency. Furthermore, note that for υ < υ cutoff = 2, the phase of V n
increases linearly with the index n; and because it is negative, it corresponds
to a delay in the signal as it propagates down the ladder, which corresponds
to a finite velocity of propagation for the signal.
Before we leave this ladder circuit, it is worth addressing a practical con-
cern. While it is impossible to realize an infinite-dimensional ladder, the
above conclusions do not change by much if we replace the infinite ladder by
a finite ladder and we terminate it after awhile by a resistor with resistance
equal to LC/.

In-Class Exercise
Pb. 6.56 Repeat the analysis given above for the LC ladder circuit, if instead
we were to:
a. Interchange the positions of the inductors and the capacitors in the
ladder circuit. Based on this result and the above LC result, can
you design a bandpass filter with a flat response?

© 2001 by CRC Press LLC