DOE-2 Engineers Manual Version 2.1A - DOE-2.com

In terms of Laplace transfer functions,
and
V(s) = K (s) = R1 = 1,
i 1 (S) 1
\i(s) = K (s) = ---7- 1 --;:-
2 1+ s '
i 2 (s )
1
= 2 + S •
The z-transfer function equivalent to this Laplace transfer function is
K(z)
-11)] [ 1 -211
- e + - 2 e
1
-211 -1
- e z
(I1.63)
No amount of algebraic manipulation can force Eq. (I 1.62) to be equivalent to
Eq. (I1.63); however, the coefficients [aO, aI, and b1 as defined in Eq.
(I1.52)J are very close. For 11 = 0.1 in Eq. (11.62), the direct combination
of z-transfer functions, aO = 0.04614, a1 = 0.04463, and b1 = -0.81846. For
Eq. (11.63), the combination of Laplace transfer functions with subsequent
conversion to z-transfer functions, ao = 0.04683, al = 0.04381, and bl =
0.81873.
In these two examples, the z-transfer functions of the individual COl11-ponents
were simple enou gh so that they coul d be algebra i ca lly man i pu 1 a ted. In
problems involving real rooms, this is no longer true. For this reason, a
time-step method is employed. In this method, a unit-pulse input is still used
to find the transfer function of the network; the unit pulse is appl ied at the
location in the network corresponding to the input. Individual outputs from
each transfer function in the network are calculated at each time step. Some
of these outputs may act as inputs to other transfer functions. The output for
the process of interest is saved at each time step. This sequence of outputs
represents a z-transfer function for that process in that network. The transfer
function is in the form of Eq. (I1.52) with bO = 1 and all other bi = O.
There are various methods, one of which was described in Sec. 2.2.2.1, that a
z-transfer function in this form can be converted into a ratio of two polynomials.
As an example of this technique, the z-transfer function for the
circuit in Fi g. 11.8b will be obtained. The two transfer functions that make
11.53