1. Theory of circuit
analysis
1.1. Basic laws
The
methods of the circuit analysis are based on Ohm's law and Kirchhoff's
laws. These laws were originally based on experiments, but they can be
derived from Maxwell's equations, too.
1.1.1. Ohm's law
Ohm's
law states that a current ( I ) flowing in a circuit is proportional to
the voltage ( U or V. In this webpage U is used for Voltage)
R is a
constant called Resistance ( 'R' ), its measurement unit is 'volt per
ampere', or 'ohm' [Ω]
1.1.2. Kirchhoff's circuit laws
Kirchhoff's
first law or Kirchhoff's junction rule expresses the conservation of
the electric charge: the sum of charges (or currents) flowing into a
node or any part of a circuit is equal to the sum of charges (or
currents) flowing out of it. If we declare every current directions
into the node as positive, then Kirchhoff's first law has this formula:
Kirchhoff'
second law or Kirchhoff's loop rule expresses the voltage equilibrium
of a loop: it states that the directed sum of electrical potential
differences (voltages) in a closed circuit is zero:
1.2 Methods of circuit analysis
1.2.1. Combined method
In
this method we use Kirchhoff's circuit laws for analyzing the circuit.
When using this method, the currents in the branches are unknown, and
we write the equations using junction rule and loop rule. Let's take
this example circuit:
The
number of independent equations is the number of nodes minus one. On
the above picture the equations based on Kirchhoff's first law
in
the A, B, C nodes:
A:
B:
C:
The
equations based on Kirchhoff's second law and Ohm's law in the I, II,
II loops:
I:
II:
III:
These six
equations are independent linear ones and can be solved by using
mathematical methods.
1.2.2.
Other methods
The
main disadvantage of the combined method is that to get the results you
have to solve many equations, which is very time-consuming, and
without the help of a computer it's very easy to make a mistake.
Because of this disadvantage many alternative methods have been
developed, which are based on Kirchhoff's laws and the combined method,
but require fewer equations.
Some of these methods are
nodal analysis, mesh analysis, or equivalent transformations. These
methods can be found in the books, or on other relevant internet pages.
This program doesn't use these methods, because in most cases they
require human intuition, which is hard to put in a computer programme.
The other reason is, that these methods require some prerequisites,
which cannot be fulfilled in every case.
1.2.3.
Method of this programme
In
the programme I used the combined method with a little modification. I
use this modification because current generators are like open
circuits so the branch that contains it cannot be part of a loop
equation. In this case the used equation is the equation, tat the
branch current equals the current of the current generator.
2. Storing
the circuit's data
The
"Circuit" class (class in the meaning of class of
object-orientated programming languages, like C++ or JAVA)
stores
the data of the circuit, and is also responsible for changing it, and
for displaying it on the screen.
The circuit is made of
display nodes (points) and components (resistance, voltage generator,
current generator, wire). Circuit nodes can be placed only to the nodes
of the screen, and components can be placed only between two
nodes. Because the graphical display should be unambiguous, there can
not be a node on the line connecting the two ends of a component.
The
"Point" class stores the place of the points (nodes of the display grid,
but not necessary nodes of the circuit) and the information and
procedures needed to display it.
As excluding the analysis,
the exact drawing and the dimensions, there's no difference among
components, every component classes ("Wire", "Resistance",
"VoltageGenerator", "CurrentGenerator") have the same base class
"Section".
The "Section" class stores, which are the end
points of a section, what is its numeric value, dimension, its name and
direction. It also stores information on graphical display, and does
part of it. (which is common to every component, like name) The exact
drawing of a component is done by the child classes ("Wire",
"Resistance", "VoltageGenerator", "CurrentGenerator")
The
"Circuit" class has a "points" and a "Sections" array. It has
procedures for adding and deleting points and components. It handles
the commands made by the mouse, and after giving a proper command it
fulfils the adding or deleting procedure. The actual function of the
mouse can be determined by using the control panel.
3. Analysis
The process of the analysis is as follows:
3.1. Determining the nodes
That
display node (point) can be considered as a circuit node, which connects
at least 3 sections. Which connect only 2 is nothing else
but a
connection of 2 components, and creates a single branch. The
case
when a point has only 0 or 1 connection is an error. If there
aren't at least 2 nodes, that's an error, too.
3.2 Determining the branches
After determining the nodes, the branches have to be found. A branch is
always between two nodes.
As a first step, we make a copy of the circuit. Then we pick one point,
which is a node in the original circuit.
We
pick one section starting from the point, and if it isn't a node, we
pick the continuing section, etc. We continue doing this, and
soon we get to a point which is a node. These sections
constitute
a branch.
At this point we delete the whole branch from the
temporary copy or the circuit, and continue this process. At the end
the remaining circuit is empty, and we have all the branches we wanted.
3.3 Determining the loops
For
the loop-equations we have to determine the loops. The first thing to do
is to copy the circuit to a temporary copy, and remove the current
generators, as they cannot be part of a loop-equation.
Then
we remove those remaining components, which now, as we removed the
current generators has only one connection to the circuit. We pick
a section, then an other one, which connects to this one, and we
continue this process. We stop when we reach to a point, which has
already been selected before. This isn't necessary the first point, at
this figure below shows: here we started form point "A", but the
"circle" ends in point "C".
We
skip those sections, which aren't part of the loop (on the figure AB
and AC). The loop contains original circuit nodes, so we can
determine which branches make the loop.
The next step is to
remove one of the branches from the temporary circuit, and ten remove
those sections, which aren't connected to the circuit with their both
sides. We continue this process to the end. As we always removed a
branch, which was part of the actual loop, this way we ensure, that the
selected loops are independent.
3.4 Determining and solving the equations
As
now we have the nodes and the loops, we can set the equations. To these
we add those equations, which state, that the current in a branch with a
current generator equals with the generator's current
In this program we write the equations in a matrix and solve it
using
determinants.
