- #1
lom
- 29
- 0
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
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