Kirchhoff’s Laws: How to solve EVERY circuit problem
These two laws form the basis of circuit analysis and they are truly all that is needed to solve any circuit problem.
Circuits here, there and everywhere
I think many people will share my pain of having to work out the parallel and series parts of a circuit diagram such as the one below:
Then, the question would probably ask what the current through some point is or the potential difference across some two points.
But we are not going to bother with each individual calculation. We are going to find the current through and potential difference across ANY points in the circuit.
And Kirchhoff’s laws are what will enable us to do that.
Kirchhoff’s Fantastic Laws and Where to Find Them
1. Current Law
Kirchhoff’s current law (also less usefully named the 1st law) states:
The algebraic sum of currents in a network of conductors meeting at a point is zero.
What it really means is we take the currents going into the point as positive and the currents going out as negative. Then we add them to always get zero. More intuitively, we can restate it as:
“The total current going into a point equals the total current coming out.”
In this simple example, Kirchhoff’s current law tells us that:
i₂ + i₃ = i₁ + i₄
Based on Conservation of Charge
Let’s say we wait for some time interval, Δt, and measure the total charge that went into point P and the total charge that went out from it during that time.
Conservation of charge tells us that the total charge in an isolated system stays the same. So,
total charge in = total charge out
Dividing by Δt, we get the following relationship:
As Δt becomes smaller and smaller, ΔQ / Δt will approach dQ/dt, which is the current, I.
So Kirchhoff’s Law is a result of conservation of charge.
2. Voltage Law
Kirchhoff’s voltage law (conveniently the second and the last) states:
The directed sums of the potential differences (voltages) around any closed loop is zero.
“Directed” means the change in potential in the direction we choose to go around the loop. For example, going from the (-) to (+) terminal of a 12V battery, the directed potential difference is +12V. If we choose to go from (+) to (-) terminal, the directed p.d is -12V.
This is the relationship we can get from the voltage law.
Actually, you might like to come back to this diagram at the end and try finding V₁ and V₂ using the current law.
Bringing it all together
Let’s revisit that weird circuit from the start.
Yes, it does look quite messy, but remember we are trying to solve the entire circuit. And we can! We just need to have some patience in applying Kirchhoff’s laws.
But first off, stating the obvious with Oh’s Law, V = IR,
We can use the voltage law to relate the voltages to each other around each of the three red loops.
Our final set of equations will come from the current law applied at each junction in the circuit.
DONE! We have effectively worked out the equations to find the current and voltage at any point, given we know the resistances.
We could painstakingly solve it manually. But it is probably more efficient to plug these equations into a computer algebra system and set the resistance values.
In most cases, you will only be asked for the current or voltage at one point so you only need to use a few of the equations.
