General mathematical expression found for energy storage shows that for linear, passive networks there is a minimum possible energy storage corresponding to a prescribed impedance. The electromagnetic energy storage is determined at different excitation frequencies through analysis of the networks terminal and reactance characteristics

Smith, W. E.

English

NASA Technical Reports Server

September 1969
	
Brief 69-10431 
•: $ticy
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AEC-NASA Tech Briefs describe innovations resulting from the research and development program 
of the U.S. AEC or from AEC-NASA interagency efforts. They are issued to encourage commercial 
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the Clearinghouse for Federal Scientific and Technical Information, Springfield, Virginia 22151. 
Energy-Storage of a Prescribed Impedance 
A mathematical method for deriving the minimum 
storage of impedance networks is presented (I). The 
problem of inferring the total average energy-storage 
of a passive, linear, electrical impedance, from its 
observed or specified terminal behavior alone, is dis-
cussed. This electromagnetic energy-storage is de-
termined at different excitation frequencies through 
analysis of the network's terminal and reactance 
characteristics. In only a few cases is the energy-
storage (for sinusoidal external excitation) uniquely 
determinable. 
A general expression for the energy-storage is 
derived that involves, in addition to terminal proper-
ties, properties of a set of functions describing the 
separate dissipative processes. This expression is 
used for finding a new minimum-energy storage, for 
a lumped-element network, that all realizations of 
the impedance must equal or exceed. 
Results show that the average energy-storage of a 
sinusoidally excited, generally linear, passive net-
work is not determined by terminal behavior alone; 
that is, equivalent circuits in the usual terminology 
do not store the same energy for the same excitation. 
The class of networks related by Cauer's equivalence 
transformation do, however. 
The energy-storage is uniquely determined by the 
terminal behavior whenever the system dissipates no 
energy, when it is an RL or RC network, or when the 
dependence of the terminal behavior on the individual 
dissipative processes is known. For reciprocal-element 
networks,' a simple lower bound exists for the energy-
storage at each frequency. This bound is attained at 
all frequencies by RL or RC networks. 
The general expression found for energy-storage 
shows that for any linear, passive network there is 
a minimum possible energy-storage corresponding
to a prescribed impedance. Furthermore this bound is 
attainable at all frequencies as a minimum-energy 
synthesis. The minimum-energy synthesis is a mini-
mum-phase-shift Darlington synthesis, and normally 
nonreciprocal elements (gyrators) are required for its 
realization. 
The methods used in the development are not 
intrinsically restricted to lumped-elements systems, 
so that, with perhaps some qualifications, the results 
could extend to distributed systems with nonrational 
Z0 (p). Also, the results are not restricted to electri-
cal impedances; for example, ,
 the same arguments 
apply for an acoustic impedance. An extension to 
multiport systems also may be possible. That non-
reciprocal elements may sometimes be necessary, 
for the general case to attain the minimum energy, 
can be expected intuitively from relations between 
group time-delay and energy-storage. 
Reference: 
I. W. E. Smith, "The energy storage of a prescribed 
impedance" (Argonne National Laboratory, Nov. 
1967). 
Notes: 
I. This information may interest designers or manu-
facturers of microwave-antenna systems. 
2. Inquiries concerning this innovation may be di-
rected to:
Office of Industrial Cooperation 
Argonne National Laboratory 
9700 South Cass Avenue 
Argonne, Illinois 60439 
Reference: B69-1043 I
Source: W. E. Smith
Applied Mathematics Division 
(ARG- 10428) 
(continued ,verleul) 
This document was prepared under the sponsorship of the Atomic 
Energy Commission and/or the National Aeronautics and Space 
Administration. Neither the United States Government nor any per- 
son acting on behalf of the United States Government assumes any
liability resulting from the use of the information contained in this 
document, or warrants that the use of any information, apparatus, 
method, or process disclosed in this document may not infringe pri-
vately owned rights.
https://ntrs.nasa.gov/search.jsp?R=19690000422 2020-03-16T18:01:38+00:00Z
Patent status: 
Inquiries concerning rights for commercial use of 
this innovation may be made to: 
Mr. George H. Lee, Chief. 
Chicago Patent Group 
U.S. Atomic Energy Commission 
Chicago Operations Office 
9800 South Cass Avenue 
Argonne, Illinois 60439 
Brief 69-10431 Category 02, 01


Energy-storage of a prescribed impedance

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