§ 16-5 Tellegen’s(特勒根 ) theorem
Tellegen’s theorem 1:
Consider a lumped network n consisting of N+1 nodes and B
branches,let, Where
TbBbbbTbBbbb iiii ][,][ 2121 ?? ??
??
????
denote the voltage across and current through
branch bk for k = 1,2…B,Thenbkbk iand?
00
1
?? ?
????
bk
B
k
bkb
T
b iori ??
Proof:
???
? nTb A ???
??? ?? A
T
n
T
b ??
?????
?? bTnbTb iAi ?? ori
b
T
b ??? ?? 0?
0
1
??
?
bk
B
k
bk i?
The sum of the powers delivered to or absorbed by all the
branches of a given lumped network is equal to zero.
?0
Proof,for the n
???
? nTb A ???
??? ? Aor
T
n
T
b ?? ????? ?? b
T
nb
T
b iAi ??
^ ^
For the n^
??? ? 0biA
^^
?? ?AA
^
??? ? 0biA
^
We obtain ^
.0 tallfo ri bTb ?
??
??
^
.0 ta l lfo ria n d bb ?
??
??
T
?0
.00; tallf o ria n diT h en bbbTb ?
?????
?? ??
Consider two networks n and made up of lumped two-
terminal elements whose graphs are identical,Denote their
corresponding branch voltage and current vectors
n^
?? bb
iand,?
^ ^ ^T^ ?? bb i,?
Tellegen’s theorem 2:
Note that the conclusion cannot be interpreted as a
conservation of power; it is merely a mathematical
relationship that exists between the branch voltages of
one circuit and the branch currents of another circuit
with the same topology(拓扑 ).
^
6i
^
2i
^
1i1?? 6??5??
^
3i3??2?? 4??
^
4i
0?
^
5i
?? 6?
?? 4??? 2?
?
?
1?
?
?
3?
?
?
5?
6i
^
1i
^
5i
^4i^2i^
3i
^
Example:
..3,9,8.0,4.1;2,2,8,2
2121
2121
??
??
F i n dAiVRRf or
VAiVifRRF or
s
s
????
????? ^ ^^
Solution,AiVAiV 1,2;2,4228
2211 ??????? ??
8.0/?,;3,8.44.139 22211 ??? ??????? iAiV
^ ^ ^ ^ ^
??
??
???????
B
k
kkk
B
k
kkk iiRiiiiRii
3
2211
3
2211 ????
^ ^^ ^^^
??
??
???????
B
k
kk
B
k
kk iiiiii
3
2211
3
2211 ??????
^^ ^^^ ^
128.48.0/234 22 ????????? ??
^^ ^
2?
=1.6V
?R
1R
2Rs?
?
?
2?
2i1i
?
?
1?