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: The current direction does not matter in the loop equations.
 * a voltage source arbitrarily ended up negative
 * a previous loop labeled the voltages first and it happened to be in the opposite direction

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.