BEHAVIOR OF SYSTEMS BETWEEN SAMPLING INSTANTS 217 
tions in regard to ripple component directly from the transform expres- 
sions. On the other hand, the partial-fraction-expansion method shows 
the major contributory terms producing output ripple somewhat more 
explicitly than does the modified z-transform method described in this 
section. 
8.5 Hidden Oscillations in Sampled-data Systems 
It has been mentioned previously that it is possible for sampled-data 
systems to have hidden oscillations! which are not detected by the 
inversion of the ordinary z transform of the output pulse sequence. This 
Output e(é) 

0 T 2T 3T 4T 5T oT 
Time 
Fic. 8.14. Output of system containing “hidden oscillations.” 
comes about from the fact that the continuous output may contain oscil- 
latory components whose frequency coincides exactly with the sampling 
frequency or integral multiples thereof. In this circumstance, the 
sampler takes values of the continuous hidden oscillation at the same 
relative phase each time, and the output of the sampler will be zero or a 
constant. For instance, if the sampler takes values of an output as shown 
in Fig. 8.14, no evidence of the presence of an increasing oscillation will 
be contained in the output c(nT’), even though the system is oscillating. 
As mentioned previously, this circumstance is hardly to be expected in 
practical situations and can readily be avoided in theoretical studies by 
making note of the continuous-system transfer function poles and deter- 
mining if their imaginary parts are integral multiples of 21/T. 
The condition for the presence of hidden oscillations is that the system 
have a continuous element in the control loop whose transfer function 
contains poles whose imaginary component is an integral multiple of the 
sampling frequency expressed in radians per second. This is seen by 
considering a continuous loop transfer function G(s)H(s) which contains, 
