Understanding the Telegan Law: The Role of Double Summations in Power Analysis

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In summary, the Telegan Law states that the total sum of power is zero. A node is chosen and designated with a voltage of zero, while the other nodes are designated with the voltages e_k. J_k represents the current and is used to calculate the total power at each node. The summation of the products of voltage and current at all nodes is equal to zero, as per Kirchhoff's Current Law. The notation \sum\sum represents the sum of powers at all nodes, not multiplication. This is shown by summing for all possible node combinations.
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lom
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the telegan law basically states that the total sum of power is zero.
my prof proved lik this:

we choose a node (a point where more then one currents come together)

and decide that the voltage on that node to be zero.
we designate the voltages on the nodes to be [tex]e_k[/tex]
[tex]J_k[/tex] is the current.

[tex]v_kJ_k=(e_a-e_b)J_{ab}[/tex]
[tex]v_kJ_k=\frac{1}{2}[(e_b-e_a)J_{ab}+(e_a-e_b)J_{ab}][/tex]
[tex]n_t[/tex] is the number of nodes.[/tex]
[tex]B[/tex] is the number of branches.[/tex]
[tex]\sum_{k}^{B}v_kJ_k=\frac{1}{2}\sum_{a=1}^{n_t}\sum_{b=1}^{n_t}(e_a-e_b)J_{ab}[/tex]
each J that does not exist in the graph will be zero.
[tex]\sum_{k}^{B}v_kJ_k=\frac{1}{2}\sum_{a=1}^{n_t}e_a\sum_{b=1}^{n_t}J_{ab}-\frac{1}{2}\sum_{a=1}^{n_t}e_b\sum_{b=1}^{n_t}J_{ab}=0[/tex]

because by kcl
[tex]\sum_{b=1}^{n_t}J_{ab}=0[/tex]

my problem iswhen he sums for all nodes
he uses
[tex]\sum\sum[/tex] sign which by me represents multiplication
of the sums

why not [tex]\sum+\sum[/tex],thus we can know that ist the sum of many similar equations.

but how he did it doesn't represent a sum
 
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  • #2
[tex]\sum_{a=1}^{n_t}\sum_{b=1}^{n_t}[/tex]

is not multiplication of summations in this context. It means that you take the sum of the powers at all nodes. For example say you have 2 total nodes, so the summation would look like

[tex]\frac{1}{2}\sum_{a=1}^{2}\sum_{b=1}^{2}(e_a-e_b)J_{ab} = \frac{1}{2}[(e_1-e_1)J_{11}+(e_1-e_2)J_{12}+(e_2-e_1)J_{21}+(e_2-e_2)J_{22}].[/tex]

You'll say that it's multiplication when there is already something in between the two summation terms as in the second summation equation.

(An analogy is by considering a loop with in a loop in programming. The outer loop will start say from 1, so while the first loop is at 1, the inner loop will continue looping until it satisfies a certain condition. After the inner loop stopped, the outer loop will proceed to the 2nd iteration and so the inner loop will loop again and so on... until both conditions in the inner and outer loop is satisfied.)
 
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1. What is a double sum sign?

A double sum sign is a mathematical symbol that represents the summation of two variables. It is usually denoted by a symbol with two sigma signs, one on top of the other.

2. How do you solve a double sum?

To solve a double sum, you need to first identify the pattern and the limits of the summation. Then, you can use the formula for the sum of a finite arithmetic series or geometric series to calculate the total value of the double sum.

3. What is the purpose of using a double sum in mathematical equations?

A double sum is used to simplify complex mathematical expressions by representing the summation of two variables in a single symbol. It is also used to calculate the total value of a series of numbers.

4. Can a double sum have more than two variables?

Yes, a double sum can have more than two variables. In fact, there can be triple sums, quadruple sums, and so on, depending on the number of variables involved in the summation.

5. How is a double sum sign different from a single sum sign?

A double sum sign represents the summation of two variables, while a single sum sign represents the summation of only one variable. Additionally, a double sum sign has two sigma signs, while a single sum sign has only one.

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