# Basic Circuit Theory Question, independent loop definition wrong?

Discussion in 'Electronic Basics' started by ShamShoon, Jun 5, 2005.

1. ### ShamShoonGuest

Hello there,
In the book "Fundamentals of Electric Circuits" by Alexander/Sadiku,
the text says that "A loop is said to be independent if it contains at
least one branch which is not part of any other independent loop", I
find this definition misleading, since in the example on the previous
page there's a circuit that has a voltage source and a resistance in
series and those in parallel with a resistor, in parallel with another
resistor, and then in parallel with a current source.
If we apply the definition to the current source and voltage source, we
get two independent loops one on the right and one on the left, however
the middle loop shares its two elements (the two parallel resistors),
with both loops, yet it's still independent, which shows that the
definition is not accurate. I found it hard to find a definition of an
independent loop.

Any ideas? Am i missing something?

2. ### John BokmaGuest

1
+-----+-----+-----+
| | | |
V R R I
| | | |
R | | |
| | | |
+-----+-----+-----+
2
?

Has been ages and ages, but I would say that 1 and 2 are both branches
that don't belong to the left and the right loops.

3. ### Larry BrasfieldGuest

That is a commonly accepted definition.
That description mentions three resistances. Yet below
you mention "the two parallel resistors". I get all three
in parallel from the above description. For discussion, I
propose the following diagram. (View with fixed-width font.)

B1 B12 B2
.--------o--------o-------o-------.
| | | | |
/+\ .-. .-. .-. / \
( ) | | | | | | ( | )
Es \-/ R1| | R3| | R2| | Is \^/
| '-' '-' '-' |
| | | | |
'--------o--------o-------o-------'
Using the above diagram so we can refer to resistors easily,
and pretending R3 does not exist, one set of independed loops is:
(Es B1 R1), (R1 B12 R2), (R2 B2 Is).
Another set would be:
(Es B1 B12 R2), (Is B2 B12 R1), (R1 B12 R2).
Another set would be:
(Es B1 B12 B2 Is), (Es B1 R1), (Is B2 R2).
There are a few more. (factorial(3) altogether)

Let's consider just the first set, which is the one you
appear to find confusing.
The (Es B1 R1) loop is at least one independent loop.
The (R1 B12 R2) loop is independent because it contains at
least one branch (B12 or R2) which is not part of any other
independent loop.
The (R2 B2 Is) loop is independent because it contains at
least one branch (B2 or Is) which is not part of any other
independent loop.
There are no more because all the possible branches
have been included in the loops already listed.

Let's consider the members in the order that leaves
The (Es B1 R1) loop is at least one independent loop.
The (R2 B2 Is) loop is independent because it contains at
least one branch (Is, B2 or R2) which is not part of any
other independent loop.
The (R1 B12 R2) loop is independent because it contains
at least one branch (B12) which is not part of any other
independent loop.
You found the definition you quoted.
Apparently, you have mssed the presence of branch
B12. Branches B1 and B2 are redundant with Es
and Is, so you can eliminate them from consideration.
But branch B12 is not uniquely associated with any
circuit element, so you can eliminate it only if you are
willing to forget what a branch is.

Another clue to your difficulty is this: The order I
first elaborated satisfies the definition even if the B?
branches are left out. Yet the order you mentioned
does not when the B? branches are left out. So it
must be the case that branches and circuit elements
are not the same thing.

4. ### John BokmaGuest

Yup, the 1st one is "voltage source and a resistance in series"

e.g.:

+-----+-----+-----+
| | | |
V R R I
| | | |
R | | |
| | | |
+-----+-----+-----+

5. ### Larry BrasfieldGuest

That's certainly a more sensible way to interpret those words.

6. ### John BokmaGuest

I had to read it a few times, and draw it on paper

7. ### Don KellyGuest

The problem is that you have a current source on one side. You only have two
independent equations to solve for the two unknown currents. The third
current is known. One approach is to replace the current source with a
voltage source equivalent then write the loop equations - you will then have
2 independent loops with two known voltage sources and two unknown currents.
This is the easy way
-----R1-----o--------R2-------o
| -->I1 | -->I3 |
V1 R3 V2 R2 in parallel
with current source Is becomes
|-------------|-------------------| V2 =R2*Is in series with
R2

V1=(R1+R3)I1 -R3*I3
-V2= -R3*I1 +(R2+R3)I3 or V2=R3*I1 -(R2+R3)I3

It can be solved using the current source but the "Ideal " current source is
known -eliminating the 3rd loop equation.
Put it this way with the original circuit as I see it.
-----R1-----o----------o---------
| -->I1 | --> | |
V1 R3 I3 R2 Is ^
|-------------|------------|---------|

Left loop V1=(R1+R3)I1 -R3*I3 where I3 is the loop current circulating
between the parallel resistors.
Center loop 0 =-R3I1 +(R3+R2)I3 +R2*Is
Right loop: V2 =R2*Is +R2*I3
V2 is unknown and Is is known. so the right loop equation simply gives the
unknown voltage V2 in terms of I3 and Is This leaves you with the two
equations, left and center in terms of the two unknown currents. . Knowing
I2 then allows you to solve the left and center loop equations
simultaneously- that is - there are only two independent loop equations in
fact as only two of the 3 current variables are unknown. The right side
equation can be used to solve for the voltage V2 but it isn't necessary once
you solve for I1, I3
Note that these become
center: R2*Is =V2 =R3*I1 -(R3+R2)I3 which is the same as above
left equation is unchanged from above.

Anytime you have a current source and are using loop equations, the known
current source eliminates a variable and hence eliminates a simultaneous
equation (making life easier)

8. ### John BokmaGuest

Use mono spaced fonts for this. Easier, and if someone uses a proportional
font, he/she can fix it by copy pasting it to an editor. The other way