# Circuit Theory/Kirchhoff's Law

Kirchhoff\'s law is a method of finding the voltage across every device
and the current through every device. It takes into account the circuit
topology (series/parallel), multiple sources, sources of different
types, and components of different types.
Technician courses spend a lot of time
developing an intuition that will eventually lead to an expertise that
is beyond the \"design, theory, science\" scope of this course.

Kirchhoff\'s law always works. It is the basis of all simulation
software. The goal here is to be able to check, to develop estimations,
and build up one\'s trust/understanding of simulation
software.

Kirchhoff\'s law starts with a drawing. The drawing is of a circuit that
is labeled with voltage polarities and current directions, with loops
and junctions as described earlier. Kirchhoff\'s circuit analysis can
not start without this drawing.

The circuit may be an existing one that is reversed engineered. The
circuit may be a thought experiment that is exploring alternatives.
Resistance, Capacitance, Inductance, and any voltage or current, any
source voltage or current may be an unknown.

For every unknown, there needs to be an equation according to linear
algebra. Kirchhoff\'s laws find equations in
three places:

-   Terminal Relations
-   Loops
-   Junctions

### Counting knowns and unknowns

Look at the drawing and write down the symbols for each known and
unknown. If an unknown write $= ?$ it. If a known, write
$= value {       ..} units$ next to it.

Power supply values should be assigned symbols such as
$V_{s1} V_{s2} I_{s1} I_{s2}$ and then given values if they are known.

Count the number of unknowns. A component value (resistance,
capacitance, inductance), a voltage or a current can be an unknown. This
is how many equations there need to be.

Count the number of components. Add the number of loops and junctions.
This is how many equations there are. If the number of unknowns doesn\'t
match, there is a problem.

If there are too many equations, there are two possibilities:

-   problem is over constrained (a fixed numerical constant needs to be
    made variable, a requirement needs to be renegotiated in the design
    process)
-   there is a dependent equation (much more difficult, this is why
    there are all the rules for sign, direction, counting loops and
    junctions)

If there are too many unknowns, these are the possibilities.

-   The problem is under constrained. Solve with symbols, make list of
    the independent variables that are going to be symbols in the
    answers and dependent variables that are going to equal something.
    Sometimes the entire problem is to be solved with symbols and there
    are no numbers.
-   If the problem is out of a text book, then re-read the problem
    perhaps there are other facts that can be turned into equations.
-   Make assumptions. Make them very clearly and loudly .. NOW at the
    beginning of the problem.

### Terminal Equations

The terminal equations were covered earlier. Here are some additional
points:

-   If the current is not going into the + voltage side, then add a
    negative sign.
-   Write the differential form, not the integral form of the terminal
    relation.
-   There are no terminal relations for independent supplies, dependent
    supplies do have a terminal relation.
-   Don\'t try to solve the equations that include differentials right
    now just write them.

### Loop Equations

The principle of energy conservation implies that the sum of the
electrical potential differences (voltage) around any closed loop is
zero:

$$\sum_{k=1}^n V_k = 0$$

Go around the loop in the direction labeled with your finger. If your
finger enters the plus side first, the voltage (in the formula, not
perhaps in the solution) is positive. If your finger enters the negative
side first, the voltage is negative. There are two cases where the
voltage will be negative:

-   a voltage source arbitrarily ended up negative
-   a previous loop labeled the voltages first and it happened to be in
    the opposite direction

The current direction does not matter in the loop equations.

Write an equation for each loop.

The goal is not to guess the + or - voltage of the answer (a
technician\'s goal). The goal is to get the + or - in the equation to
correctly match the circuit. When finished, the last step is to look
through the answers and see if they make sense from an intuitive point
of view. That\'s it.

### Junction Equations

The principle of electric charge conservation implies that at any
\"junction\", the sum of the currents flowing into the note is zero.

$$\sum_{k=1}^n {I}_k = 0$$

*n* is the total number of branches with currents attached to the
\"junction.\"

If the current is flowing into the node, make it positive. If current is
flowing out of the node, make it negative.

Check series sections for two different current symbols. Current should
be the same in a series section.

The goal of the + and - signs is to capture the circuit design in the
equations. This has nothing to do with whether the ultimate current is
flowing in or out of the node. Do not try to guess current direction
either.

### What\'s Next

Nothing.

Kirchhoff\'s analysis falls apart because the number of equations
increases dramatically as a circuit becomes more complex. This is not a
problem for computers, but it is a problem for engineers and techs. The
rest of this course is full of math, simplifications, quirks, tricks,
tips, and shortcuts. All have limitations. All fail at some point. The
goal is to go through them, observe conditions when they succeed and
fail, and then add them to our design tool belt and our intuition.

