Refresher on Circuit Analysis Laws

Circuit analysis can become quite complex in nature when dealing with multiple components, but there are methods that make it easier built around the laws that define circuit analysis. This post will be about reviewing the laws that define these methods and providing advice to make it less confusing. The post will also only involve resistance and simple DC Voltage sources.

Ohm’s Law

The first equation to mention is Ohm’s Law. Ohm’s law states that Voltage equals Current times resistance or V=I*R. Current is measured in amps and resistance is measured in Ohm’s. Here is more context on Ohm’s law see this post: Ohm's law calculations in a simple DC Circuit This formula is key to understanding most analysis methods in DC analysis (works in AC analysis too when dealing with the frequency domain).

Kirchhoff’s Current Law

The second law that should be mentioned is Kirchhoff’s Current Law(KCL). This law states that current coming into a node has to equal what current comes out of a node. Here are some visual scenarios to help explain this law:

These scenarios are all valid for current flow, but not all scenarios will appear in analysis nor will they be practical. It is also possible to have way more wires coming from a node, the pictures above are examples of the least amount of current splitting. The fourth scenario where all current is coming into a node and nothing comes out is particularly impractical. There are more scenarios than the ones drawn because current can either be coming into the node or coming out of the node for all three lines. Deciding which way current goes on a line is referred to as an assumption. We don’t really know which way current will really flow on more complex circuits, so it is just better to assume a direction. If the answer is negative, the assumption was wrong and current travels in the opposite direction at the same magnitude calculated. I recommend sticking with one assumption scenario for all lines of current for each node. I usually use the first scenario where current “A” comes into a node and splits into currents “B” and “C” coming out of the node. If new nodes appear later on in a circuit, I will just assume current is splitting out of that node from what is going in.

It is important to note that new nodes require new current names that are unique. If current “B” or “C” gets split at some further point in the circuit, new names like “D” and “E” will have to be used.

Kirchhoff’s Voltage Law

The third law that should be remembered is Kirchhoff’s Voltage Law(KVL). This law states that all voltages added up in any complete circuit loop must equal zero. A valid loop starts and stops at the same point. Any loop that does not meet up with the starting point is invalid. Loops that are interrupted by a break in a wire also do not count because current cannot travel on a non-existent path.

For example, the following diagram shows all the valid loops for 3 parallel resistors.

Yes, there are 7 valid circuit loops in this simple circuit, however, I would advise not drawing this for a few reasons.

  1. This is confusing when starting out.
  2. This is more than enough information to solve. It is over-defined.

It is also valid to draw loops in any direction (clockwise or counterclockwise) because there may be a preference. I would recommend using one direction instead of using multiple ones to avoid confusion. These loops will be important in a different method of analysis.

So which loops are best to be drawn if drawing all the valid ones is confusing? Draw the most obvious loops that don’t intersect other loops. The example above, the most obvious loops, in my opinion, are the black, red, and green loops.

The loops above are sufficient enough when considering any analysis technique and minimize confusion. The second part of this law is adding voltages and setting it equal to zero.

Steps for Solving Using These Laws

Here are the steps using Ohm’s law specifically for analysis.

  1. *Draw polarities for all resistances, use the following suggestion: If the resistor is vertical draw a plus at the top and a minus at the bottom. If the resistor is horizontal draw a plus at the left and a minus at the right. *NOTE: Resistors are not actually polarized, but this is a detail important to just using Ohm's Law, Kirchhoff's Voltage Law, and Kirchhoff's Current Law for analysis.

  2. Draw current lines and label lines with unique names for the current names based on the desired assumption based on Kirchoff's Current Law. Write the equations for KCL for each node. In this case, it is A=B+C and C=D+E.

  3. Pick a starting point to begin writing an equation for one loop, the easiest is the bottom left of the loop if going clockwise.

  4. Follow the loop line until a component is met (including the source)
    1. If the component is the source, write which voltage is seen first, in this case, it is negative V (-V).
    2. If the component is a resistor, write the Ohm's law equivalent (V=I*R). If the plus sign is met first when following the loop don't change the equation. If the negative sign is met first when following the loop, add a negative to "I" (-I*R).
  5. Repeat step 4 until the starting point is reached again when following the loop direction.
  6. Set the equation equal to 0 based on KVL
  7. Repeat steps 3-6 for the remaining loops and solve the system of equations on paper or via a graphing calculator. (I prefer the latter)
Here is the system of equations this circuit follows based on the steps:

This might seem impossible because there are 5 equations and 9 unknowns, however, this can be solved based on preference or desired design. Let’s say we want the current to be exact values for B, D, and E. We can choose values for B, D, and E because those three values will directly affect what A and C will be and reduce the number of unknowns to 6. This leaves one more value that can easily be determined: voltage. Voltage is simply whatever is being supplied given by data sheets, measured via digital multimeter, read off a battery, or even made up at random for experimenting.

For the set of equations above, let’s assume V = 12 Volts and we want B to be 1 Amp, D to be 5 Amps, and E to be 0.5 Amps. Substitute the values and solve for R1, R2, R3, A, and C.

R1 = 12 Ω's
R2 = 2.4 Ω's
R3=24 Ω's

In general, circuit analysis is easier to understand if consistency is applied. Be consistent with arrow direction for current for KCL, loop direction for KVL, and plus and minus signs for using Ohm’s Law, KVL, and KCL. When I started learning along with other students, inconsistency caused the most amount of confusion. I also recommend trying out the problem in simulation software like LTSpice or even transferring the circuit to a breadboard and measuring values using a digital multi-meter. If anything is unclear, please feel free to ask questions.