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Tellegen’s Theorem

Circuit:

φ0

i0 + v0 – i1

+

i2

+

v1

–

v2

–

φ1

i4 v4

i3

+ v3 –

φ2

v5

φ3

i5

v0 = φ0 – φ3, v1 = φ1 – φ0, v2 = φ0 – φ2, v3 = φ3 – φ2, v4 = φ1 – φ3, v5 = φ3 – φ2. Incidence Matrix:

node: 0 element: 0 1 2 3 4 5 1 2 3 element: 0 node: 0 1 2 3 4 5

+1 0 –1 +1

0 –1 0 0 = A

+1 –1 +1

+

–

0

+ –

+1 0 –1 0 0 +1 –1 0 0 +1 0 –1 0 0 –1 +1

1 2 3

0

0

0 +1 0 +1 +1 0 = AT 0 0 –1 –1 0 –1 –1 0 0 0 –1 +1

Potential, Voltage, and Current Vectors

v0 i0 i1 i = i2 i3 i4 i5

φ0

φ =

v1 v = v2 v3 v4 v5

φ1 φ2 φ3

Then, v = Aφ, i·Aφ = i·v = Also, by KCL

∑i v

n =0

5

n n

.

⎡0 ⎤ ⎢0 ⎥ ⎢ ⎥ ATi = ⎢0⎥ , φ·ATi = 0. ⎢ ⎥ ⎣0 ⎦

But

i·Aφ ≡ φ·ATi,

leading to Tellegen’s Theorem

∑i v

n =0

5

n n

= 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. ib1 ia1 ia0 ib0

+

va0

–

zn(s) n = 2 ... N

+ v a1 – –

+

–ib0 vb0 +

– – +

zn(s) n = 2 ... N

+

vb1

–

Choosing the currents from network a and the voltages from network b for Tellegen’s Theorem,

0= =

∑

n =0 N n=2

N

ian vbn = ia 0 vb 0 + ia1vb1 +

an ibn zn ( s )....

Circuit:

φ0

i0 + v0 – i1

+

i2

+

v1

–

v2

–

φ1

i4 v4

i3

+ v3 –

φ2

v5

φ3

i5

v0 = φ0 – φ3, v1 = φ1 – φ0, v2 = φ0 – φ2, v3 = φ3 – φ2, v4 = φ1 – φ3, v5 = φ3 – φ2. Incidence Matrix:

node: 0 element: 0 1 2 3 4 5 1 2 3 element: 0 node: 0 1 2 3 4 5

+1 0 –1 +1

0 –1 0 0 = A

+1 –1 +1

+

–

0

+ –

+1 0 –1 0 0 +1 –1 0 0 +1 0 –1 0 0 –1 +1

1 2 3

0

0

0 +1 0 +1 +1 0 = AT 0 0 –1 –1 0 –1 –1 0 0 0 –1 +1

Potential, Voltage, and Current Vectors

v0 i0 i1 i = i2 i3 i4 i5

φ0

φ =

v1 v = v2 v3 v4 v5

φ1 φ2 φ3

Then, v = Aφ, i·Aφ = i·v = Also, by KCL

∑i v

n =0

5

n n

.

⎡0 ⎤ ⎢0 ⎥ ⎢ ⎥ ATi = ⎢0⎥ , φ·ATi = 0. ⎢ ⎥ ⎣0 ⎦

But

i·Aφ ≡ φ·ATi,

leading to Tellegen’s Theorem

∑i v

n =0

5

n n

= 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. ib1 ia1 ia0 ib0

+

va0

–

zn(s) n = 2 ... N

+ v a1 – –

+

–ib0 vb0 +

– – +

zn(s) n = 2 ... N

+

vb1

–

Choosing the currents from network a and the voltages from network b for Tellegen’s Theorem,

0= =

∑

n =0 N n=2

N

ian vbn = ia 0 vb 0 + ia1vb1 +

an ibn zn ( s )....

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