Tellegen’s theorem states:
In an arbitrarily lumped network subject to KVL and KCL constraints, with reference directions of the branch currents and branch voltages associated with the KVL and KCL constraints, the product of all branch currents and branch voltages must equal zero.
Tellegen’s theorem may be summarized by the equation
where the lower case letters v and j represent instantaneous values of the branch voltages and branch currents, respectively, and where b is the total number of branches. A matrix representation employing the branch current and branch voltage vectors also exists. Because V and J are column vectors
V · J = VT J = J T V
The prerequisite concerning the KVL and KCL constraints in the statement of Tellegen’s theorem is of crucial importance.
Example 3.3. Figure 3.16 displays an oriented graph of a particular network in which there are six branches labeled with numbers within parentheses and four nodes labeled by numbers within circles. Several known branch currents and branch voltages are indicated.
Because the type of elements or their values is not germane to the construction of the graph, the other branch currents and branch voltages may be evaluated from repeated applications of KCL and KVL. KCL may be used first at the various nodes.
node 3: j2 = j6 – j4 = 4 – 2 = 2 A
node 1: j3 = –j1 – j2 = –8 – 2 = –10 A
node 2: j5 = j3 – j4 = –10 – 2 = –12 A
Then KVL gives
v3 = v2 – v4 = 8 – 6 = 2 V
v6 = v5 – v4 = –10 – 6 = –16 V
v1 = v2 + v6 = 8 – 16 = –8 V
The transpose of the branch voltage and current vectors are
VT = [–8 8 2 6 –10 –16] V
and
JT = [8 2 –10 2 –12 4] V
The scalar product of V and J gives
–8(8) + 8(2) + 2(–10) + 6(2) + (–10)(–12) + (–16)(4) = –148 + 148 = 0
and Tellegen’s theorem is confirmed.
In an arbitrarily lumped network subject to KVL and KCL constraints, with reference directions of the branch currents and branch voltages associated with the KVL and KCL constraints, the product of all branch currents and branch voltages must equal zero.
Tellegen’s theorem may be summarized by the equation
where the lower case letters v and j represent instantaneous values of the branch voltages and branch currents, respectively, and where b is the total number of branches. A matrix representation employing the branch current and branch voltage vectors also exists. Because V and J are column vectors
V · J = VT J = J T V
The prerequisite concerning the KVL and KCL constraints in the statement of Tellegen’s theorem is of crucial importance.
Example 3.3. Figure 3.16 displays an oriented graph of a particular network in which there are six branches labeled with numbers within parentheses and four nodes labeled by numbers within circles. Several known branch currents and branch voltages are indicated.
Because the type of elements or their values is not germane to the construction of the graph, the other branch currents and branch voltages may be evaluated from repeated applications of KCL and KVL. KCL may be used first at the various nodes.
node 3: j2 = j6 – j4 = 4 – 2 = 2 A
node 1: j3 = –j1 – j2 = –8 – 2 = –10 A
node 2: j5 = j3 – j4 = –10 – 2 = –12 A
Then KVL gives
v3 = v2 – v4 = 8 – 6 = 2 V
v6 = v5 – v4 = –10 – 6 = –16 V
v1 = v2 + v6 = 8 – 16 = –8 V
The transpose of the branch voltage and current vectors are
VT = [–8 8 2 6 –10 –16] V
and
JT = [8 2 –10 2 –12 4] V
The scalar product of V and J gives
–8(8) + 8(2) + 2(–10) + 6(2) + (–10)(–12) + (–16)(4) = –148 + 148 = 0
and Tellegen’s theorem is confirmed.
No comments:
Post a Comment