Most of the people know that the chord C-E-G (from the bottom up) is a root position chord. It has two inversions: E-G-C (a 6th chord) and G-C-E (a 64th chord). The theory behind this concept is very simple.
In root position chord, there are no intervals of the fourth, only two thirds. In the first inversion chord, we see a third and a fourth and in the second inversion chord, a fourth and a third.
The most dificcult part in determining the inversion of the chord is when the chord is presented in four-part harmony notation. In this case, one note of the three-note chord has to be doubled. In root position chord, we usually double the root, in first inversion chord - the root or the fifth, and in the second inversion chord - the fifth.
But we must not forget that the chords could be written in open or closed (or even mixed) position, too. In the closed position, the distance between the three upper voices is no more than a fourth. In open position, from a fifth up to an octave. If some intervals are a fourth or less and some - more than a fifth, then such a chord is written in mixed position. This normally applies to first inversion chords.
Depending on which chordal tone is in the soprano, the melodic position could be root, third or a fifth.
So you can see that even in root position chords there are 6 different versions: 3 in closed position (C-E-G-C, C-G-C-E, and C-C-E-G) and 3 in open position (C-G-E-C, C-C-G-E, and C-E-C-G).
So the key steps in determining the inversion of the three-note chord are these:
1. Take the bass note of the chord and build the three-note chord without any doubling.
2. Then you will see, if the chord has only two thirds (a root position chord), a fourth on the top (a first inversion chord) or a fourth on the bottom (a second inversion chord).
That's it. It's really that simple. If you apply the above steps, you will have no trouble in determining the inversion of the three-note chord.