Tellegen's theorem
2
Tellegen’s Theorem Circuit: v 0 v 1 v 2 v 3 v 4 v 5 + – + – + – + – + – + – φ 0 i 4 i 5 i 1 i 0 i 2 i 3 φ 1 φ 2 φ 3 v 0 = φ 0 – φ 3 , v 1 = φ 1 – φ 0 , v 2 = φ 0 – φ 2 , v 3 = φ 3 – φ 2 , v 4 = φ 1 – φ 3 , v 5 = φ 3 – φ 2 . Incidence Matrix: = A node: 0 1 2 3 element: 0 1 2 3 4 5 +1 0 0 –1 –1 +1 0 0 +1 0 –1 0 0 +1 –1 0 0 +1 0 –1 0 0 –1 +1 = A T element: 0 1 2 3 4 5 node: 0 1 2 3 +1 –1 +1 0 0 0 0 +1 0 +1 +1 0 0 0 –1 –1 0 –1 –1 0 0 0 –1 +1 Potential, Voltage, and Current Vectors φ 0 φ 1 φ 2 φ 3 φ = v = v 0 v 1 v 2 v 3 v 4 v 5 i = i 0 i 1 i 2 i 3 i 4 i 5 Then, v = Aφ, i·Aφ = i·v = 5 0 n n n iv = ∑ . Also, by KCL A T i = 0 0 0 0 ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , φ·A T i = 0.
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Tellegen's theorem
Transcript of Tellegen's theorem
Tellegen’s Theorem Circuit:
v0
v1 v2
v3
v4 v5
+ –
+ –
+ –
+
–
+
–
+
–
φ 0
i4
i5
i1
i0
i2
i3φ 1 φ 2
φ 3 v0 = φ0 – φ3, v1 = φ1 – φ0, v2 = φ0 – φ2, v3 = φ3 – φ2, v4 = φ1 – φ3, v5 = φ3 – φ2.
Incidence Matrix:
= A
node: 0 1 2 3
element: 012345
+1 0 0 –1–1 +1 0 0+1 0 –1 0 0 +1 –1 0 0 +1 0 –1 0 0 –1 +1
= AT
element: 0 1 2 3 4 5
node: 0123
+1 –1 +1 0 0 0 0 +1 0 +1 +1 0 0 0 –1 –1 0 –1–1 0 0 0 –1 +1
Potential, Voltage, and Current Vectors
φ0
φ1
φ2
φ3
φ = v =
v0
v1
v2
v3
v4
v5
i =
i0
i1
i2
i3
i4
i5 Then,
v = Aφ, i·Aφ = i·v = 5
0n n
n
i v=
∑ .
Also, by KCL
ATi =
0000
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
, φ·ATi = 0.
But i·Aφ ≡ φ·ATi,
leading to Tellegen’s Theorem 5
0n n
n
i v=
∑ = 0.
The only requirement is that all the in be for one set a of elements in the circuit so that KCL holds, and all the vn be for another set b of elements in the circuit so that KVL holds (a set of potentials φ can be assigned). When set a is the same as set b, the result is simply the conservation of power. But Tellegen’s Theorem is more general and leads to many other results such as reciprocity theorems. See Tellegen's Theorem and Electrical Networks (MIT research monograph no. 58) by Paul Penfield, Robert Spencer, S. Duinker. Example of Using Tellegen’s Theorem Consider two networks with the same topology and, inside their respective two-port boxes, the same set of elements—passive complex impedances zn(s). The outside elements differ—an open circuit at port 0 and a source at port 1 in one case, and a source at port 0 and a short circuit at port 1 in the other case.
+
–
zn(s)
n = 2 ... Nva0 va1
ia0 ia1
+
–
zn(s)
n = 2 ... Nvb0 vb1
ib0ib1
+–
+
–
+
–
–ib0
+–
Choosing the currents from network a and the voltages from network b for Tellegen’s Theorem,
0 0 1 1 0 10 2 2 2
2
0 0 0
( ).
N N N N
an bn a b a b an bn b a an bn an bnn n n n
N
an bn nn
i v i v i v i v v i i v i v
i i z s
= = = =
=
= = + + = ⋅ + ⋅ + =
=
∑ ∑ ∑ ∑
∑
Choosing the voltages from network a and the currents from network b for Tellegen’s Theorem,
0 0 1 1 0 0 1 1 0 0 1 10 2 2 2
0 0 1 1
0 ( ) ( )
.
N N N N
an bn a b a b an bn a b a b an n bn a b a b an bn nn n n n
a b a b
v i v i v i v i v i v i i z s i v i v i i i z s
v i v i= = = =
= = + + = + + = + +
= +
∑ ∑ ∑ ∑
Therefore we have a reciprocity of the reverse open-circuit voltage transfer equaling the forward short-circuit current transfer:
0 1
1 0.a b
a b
v iv i
=−
