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Kirchhoff's Voltage Law

Intro​

In part 4, we learned from Ohm's law that resistance reduces the amount of current that flows through a circuit, but it turns out, voltage is also reduced.

Kirchhoff's voltage law states that the sum of all voltage drops around a circuit loop is equal to the sum of all voltage sources. In other words, if we had a circuit loop as in following illustration, with a 5V voltage source, the total drop in voltage across all the resistors would be 5V:

Diagram of a circuit with a voltage source connected to a series of resistors, starting at point A before resistor R1 with voltage drop V1, then to point B before resistor R2 with voltage drop V2, then to point C before resistor R3 with voltage drop V3, before returning to the voltage negative terminal.

Algebraically, we can express this as:

Vs = V1 + V2 + V3

Where V1 represents the voltage drop at R1, and so forth.

This means that at point A, relative to ground, the voltage would be 5V, and at point D, the voltage is OV. It turns out, that the voltage through each resistor is proportionate to that resistor's part of the overall resistance. So if that resistance represents one half of the total resistance, then the voltage drop through that resistor will also be half of the total voltage drop, or 2.5V, since 5V * .5 = 2.5V.

Deriving from Ohm's Law​

We can calculate Kirchhoff's voltage law from Ohm's law.

Recall that in a series resistance, each resistor experiences the same amount of current:

Diagram of a circuit with a 5-volt source and three resistors in series with a current of 0.25 amps: 12 Ohms, 6 Ohms, and 2 Ohms.

In this case, the total resistance (series resistances are additive), is 20Ω, therefore the current is:

Given:
I = V / R

Therefore:
I = 5V / 20Ω = 0.25A = 250mA

So each resistor sees 0.25A of current.

Now, recalling Ohm's law solved for voltage (V = I * R), we can calculate the voltage drop for that current at each resistor:

R1 Voltage = 0.25A * 2Ω = 0.5V
R2 Voltage = 0.25A * 6Ω = 1.5V
R3 Voltage = 0.25A * 12Ω = 3V

If we add each of these voltage drops up:

Total voltage drops = 0.5V + 1.5V + 3V = 5V = Voltage Source

The total amount of voltage drop is the exact same amount as the voltage source!

Voltage Drop​

Revisiting the previous circuit, if we put a voltmeter between points B, C, D, and ground, we'd get 2V, .5V, and 0V, respectively:

Diagram of a circuit with a 5-volt source and with three resistors in series: 12 Ohms, 6 Ohms, and 2 Ohms. There are voltage measurements between each resistor and ground showing voltage drops from 5 volts to 2 volts to 0.5 volts and then to zero volts.

This is because the voltage drop is removed from the source voltage to get the output voltage after each resistor:

B -> Common/Ground = 5V - 3V = 2V
C -> Common = 2V - 1.5V = .5V
D -> Common = .5V - .5V = 0V

Forward Voltage​

Voltage drop is often referred to as forward voltage (Vf). In many components, their voltage drop is actually the same amount of voltage that they require to work. For example, lights such as LEDs require a particular forward voltage to be met in order for them to have enough energy to work.

Simplifying Series Resistance​

Since we know that resistors in series are additive, if we want to know the voltage drop at any point between resistors in series (as in the above diagram), we can simplify the previous circuit by considering any series resistances as a single resistor by adding them up, as illustrated in the following circuit diagram:

Diagram of a circuit with a voltage source connected to three resistors in series. The voltage measured at some point calculated from the sum of all resistors prior to the desired point.

Note that the sigma (∑) symbol means mathematical sum, and R...Rn means R through Rn.

Simplifying circuits by combining resistances is a common way to analyze circuits because it makes them more practical to deal with. In fact, when dealing with circuits, we often simplify them into single values that describe their overall resistance and capacitance, which we'll explore later. By simplifying circuits, we can more effectively design complex systems that incorporate multiple circuits into a larger design.

Using the same numbers from before, calculate the forward voltage (voltage drop) at B:

Given:
Vf = I * R1

Therefore:
Vf = 0.25A * 18Ω = 4.5V

That means that there is only 0.5V from the original 5V of electromotive force at point B, since we have to remove the drop from the source voltage:

V (at point B) = 5V - 4.5 = 0.5V

Voltage Drop (Vf) of R2 is Vout​

By definition, 0.5V is also the voltage drop from B to C. This becomes a little clearer if we simplify the circuit diagram even further:

Diagram of a circuit with a voltage source connected to two resistors in series where the voltage drop across the second resistor is labeled.

In this case, we see that Vout = VS - R1Vf and R2Vf = VS - R1Vf, which means that Vout = R2Vf. Therefore, if we want to calculate Vout, we can simplify things by calculating the voltage drop of R2.

Voltage Drop is Proportionate​

Because Ohm's law is proportionate, the ratio of resistances determines how much the voltage is reduced at each interval. We can prove this by starting with Ohm's law, solved for I:

Given:
I = V / R

Therefore
I = Vs / (R1 + R2)

And because Vout = R2Vf, we solve for the voltage drop (Vf) of R2 and substitute Vs / (R1 + R2) in for I:

Given:
R2Vf = I * R2

And:
I = Vs / (R1 + R2)

Therefore:
Vout = R2VF = (Vs / (R1 + R2)) * R2

We can test this using the values from before:

R2Vf = (5v / (18Ω + 2Ω)) * 2Ω = 0.5V

Next - Voltage Division​