How voltage and current correlates

(C) Gustaf Alhäll, released under CC BY 4.0
Original document can be found at hanicef.me
License can be found at creativecommons.org

Ohm's law

Perhaps the most basic and fundamental law in electronics, Ohm's law defines how voltage, resistance and current affects each other, and is based on this simple equation:

V = I * R

Here, V is voltage, I is current and R is resistance. Using basic algebra, we can also rearrange the equation if we need to figure out any other value:

    V
I = ⎯
    R
    V
R = ⎯
    I

To illustrate how this works, consider this circuit:

/schematics/ohm.png

Here, we have 5V input connected to a 10Ω resistor. Since we know the resistance and voltage, we can easily figure out the current using this formula:

V    5
⎯ = ⎯⎯ = 0.5
R   10

Thus, the current through this circuit is 0.5A.

Resistors in parallel

Ohm's law still applies even when resistors are connected in parallel, but it requires calculating the resistance differently from when resistors are connected in sequence. For resistors in sequence, all you have to do is to just add the resistance of each resistor together and you end up with the total resistance. However, resistors in parallel is more tricky.

Since resistors in parallel allows for more paths than resistors in sequence, the current increases when a resistor is added in parallel than when one is added in sequence. This effectively means that the resistance decreases despite the added resistor. The way this is calculated is through the inverse of the resistance:

1   1   1   1         1
⎯ = ⎯ + ⎯ + ⎯ + ... + ⎯
R   R₁  R₂  R₃        Rₙ

Here, R is the total resistance, and Rₙ is the n'th resistor in parallel. To calculate R, we can apply some algebra:

1   1   1   1         1
⎯ = ⎯ + ⎯ + ⎯ + ... + ⎯
R   R₁  R₂  R₃        Rₙ

     1   1   1         1
1 = (⎯ + ⎯ + ⎯ + ... + ⎯ ) * R
     R₁  R₂  R₃        Rₙ

         1
⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
1   1   1         1 = R
⎯ + ⎯ + ⎯ + ... + ⎯
R₁  R₂  R₃        Rₙ

So, by using this equation, we can calculate the total resistance of parallel resistors. To get a feel for this, let's try calculating two resistors in parallel: 5Ω and 10Ω.

  1
⎯⎯⎯⎯⎯⎯
1    1 = R
⎯ + ⎯⎯ 
5   10

    1
⎯⎯⎯⎯⎯⎯⎯⎯⎯ = R
0.2 + 0.1

 1
⎯⎯⎯ = R = 3.333...
0.3

Thus, the resistance of 5Ω and 10Ω in parallel is 3.333Ω

Kirchhoff's current law

Kirchhoff's current law states that the sum of currents that connects to a junction is zero. A different way of looking at it is that the current entering a junction is equal to the current leaving a junction. To illustrate, consider this circuit:

/schematics/kirchhoff-current.png

Here, we have a 12V input branching off into two resistors, one at 2Ω labeled R1 and one at 4Ω labeled R2. These are then connected to a junction that splits in two paths, one at 3Ω labeled R3 resistor and one at 6Ω labeled R4. To start, we first need to figure out the total resistance through the circuit, which we can do by adding the resistance together as we did above:

  1
⎯⎯⎯⎯⎯       1         1
1   1 = ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ = ⎯⎯⎯⎯ = 1.333...
⎯ + ⎯   0.5 + 0.25   0.75
2   4

  1
⎯⎯⎯⎯⎯            1              1
1   1 = ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ = ⎯⎯⎯ = 2
⎯ + ⎯   0.333... + 0.1666...   0.5
3   6

1.333... + 2 = 3.333...

(If you have a keen eye, you might've noticed that 0.333... + 0.1666... = 0.5; as weird as this may seem, it is mathematically correct since 0.999... = 1).

This gives us a resistance of 3.333Ω. Now, we can just apply Ohm's law to get the current:

12 / 3.333... = 3.6

So, the total current going through the circuit is 3.6A. With this in place, we can now calculate the amount of current that goes through each resistor.

To understand why we have to know all this, consider this: Kirchhoff's current law says that the current into a junction is the same as the current out of the junction. However, since the resistance is different, the voltage must also change for the current to remain the same because of Ohm's law. By knowing the current through the entire circuit, we also know the current through all resistors into the junction and all resistors out of the junction, but not the current through each resistor individually.

With all of this in mind, we first need to know the voltage going into and out of the junction. This can be calculated using Ohm's law:

3.6 * 1.333... = 4.8
3.6 * 2 = 7.2

So, the input voltage is 4.8V and the output voltage is 7.2V. Now, we can finally calculate the current through each resistor by once again using Ohm's law:

R₁ = 4.8 / 2 = 2.4
R₂ = 4.8 / 4 = 1.2
R₃ = 7.2 / 3 = 2.4
R₄ = 7.2 / 6 = 1.2

And so, we know that the current through R1 is 2.4A, R2 is 1.2A, R3 is 2.4A and R4 is 1.2A. This also adds up as R1 + R2 = R3 + R4, thus adhering to the law.

Kirchhoff's voltage law

Kirchhoff's voltage law states that the sum of voltage around a closed loop is zero. This effectively means that if you measure the voltage in a circuit in sections, all these voltages add up to the total voltage in a circuit. To illustrate, consider this circuit:

/schematics/kirchhoff-voltage.png

Here, we have a 6V input going through 3 resistors, one at 5Ω labeled R1, one at 10Ω labeled R2 and one at 15Ω labeled R3. Since all the voltages must add up to 6V, the voltage going through each resistor can be treated as a precentage of the total voltage. Thus, we can simply divide the resistance of the resistor we want the voltage of with the total amount of the resistance. This allows us to calculate it like this:

R = 5 + 10 + 15 = 30
Vₙ = 6 * Rₙ / R
V₁ = 6 * 5 / 30 = 1
V₂ = 6 * 10 / 30 = 2
V₃ = 6 * 15 / 30 = 3

Thus, the voltage going through R1 is 1V, the voltage through R2 is 2V and the voltage through R3 is 3V.