From this answer:
The Fourier series:
\$ V_t = \dfrac{a_0}{2} + \displaystyle \sum_{i=1}^{\infty}[a_i sin(i \omega_0 t) + b_i cos(i \omega_0 t) ] \$
Why is the DC level written as \$\dfrac{a_0}{2}\$, and not \$a_0\$? stevenvh says it's convention, but there has to be an explanation. Where does it come from?
This stumped me for a while as well but it is actually quite simple.
The general Fourier series:
\$ g(x) = a_0 + \displaystyle \sum_{n=1}^{\infty}[a_n sin(n x) + b_n cos(n x) ] \$
I am not going to do all the math but if you use the orthogonal signal space \$ \left\{{1, cos(nx),sin(nx)}\right\} n\in N \$ you can derive the fourier series terms as:
\$ a_n = \dfrac{1}{\pi} \displaystyle \int_{-\pi}^{\pi} g(x)cos(n x)\,\mathrm{d}x\$
\$ b_n = \dfrac{1}{\pi} \displaystyle \int_{-\pi}^{\pi} g(x)sin(n x)\,\mathrm{d}x\$
And:
\$ a_0 = \dfrac{1}{2\pi} \displaystyle \int_{-\pi}^{\pi} g(x)\,\mathrm{d}x\$
The "regular" Fourier series
\$ g(x) = \dfrac{a_0}{2} + \displaystyle \sum_{n=1}^{\infty}[a_n sin(n x) + b_n cos(n x)]\$
For the sake of symmetry the \$a_0\$ term was redefined as:
\$ a_0 = \dfrac{1}{\pi} \displaystyle \int_{-\pi}^{\pi} g(x)\,\mathrm{d}x\$
to be similar to \$a_n\$ and \$b_n\$. Therefore the \$\dfrac{1}{2}\$ term was added to the expression of the Fourier series.
Reference:
That is one form the fourier series can be written. Another one is $$V_t = \sum\limits_{i=-\infty}^\infty c_ie^{ji\omega t}$$ with the imaginary unit j.
When you apply Euler's formula \$e^{j\phi}=\cos\phi+j \sin\phi\$ you get the form you already know. The coefficients are \$a_i = c_i + c_{-i}\$, \$b_i = j(c_i + c_{-i})\$.
For the special case \$i = 0\$ you have now two options that looks feasible
What you choose seems to be a matter of taste.
Try another perspective to Pet's one.
This is not formal and may be just wrong but it seems to work and make sense:
Deal with ai and sine terms. Same argument applies to bi and cos terms.
For any ith term with i > 0 if ai = 1 you get a sinusoid that extends from -1 to +1. ie
Peak to peak magnitude of any sine term = 2ai.
RMS magnitude of any single sine term is sqrt(2).ai
Similarly for bi and cos terms.
If we apply the same convention for DC, for DC to have Vdc = a0 implies a DC peak to peak value of 2a0 (as is the case with ai sine terms).
To maintain a consistent convention we set
Vdc = 1/2.a0
The above arguably arises because an AC value is measured about its mean value = about 0 and so implies that an equal and opposite peak value exists.
For a sine wave it does.
For DC it doesn't.
We are trying to fit a single side signal in one case only into an otherwise bipolar system.
In this useful 28 page PDF Introduction to the Fourier series ewuation (3) on page 4 uses a0 and not a0/2, thus differing from your example, and violating the consistent convention as mentioned above.
This wikipedia page follows your convention without comment.
See 2nd kjsingh123 post on this page for a0 derivation
