# What is the formal boundary between AC and DC?

This question grew out of an exchange here: Using DC voltage in Transformers

Typically when we refer to more complicated currents than pure DC or sinusoidal AC, we refer to them as having a "DC component" and AC components of various frequencies. As far as I know, the DC component refers to the zero-frequency component of the signal; hence, if a signal has no zero-frequency component, it should be called pure AC.

But I've also heard the following argument: AC, in order to be "alternating" in any real sense, must change direction. Rectified AC does not change direction (the current is either zero or positive), so rectified current cannot be AC; as such, it should be referred to as "non-constant DC."

So we now have two conflicting definitions: rectified AC is either pure AC, because it has no zero-frequency component, or is non-constant DC, because it is always the same sign.

Which of these definitions is more commonly used? And generally, what is the formal difference between an AC signal and a DC signal?

• Please add a link to the question that sparked this one, for context. – DanielSank Mar 23 '18 at 22:50
• Though I believe this question is conceptually separate from the one you're referring to, I will do so. – probably_someone Mar 23 '18 at 22:51
• It's conceptually different, but the quotation about "non-constant dc", which is an important part of this post, came from that other question. It's important to cite the context because that term "non-constant dc" isn't standard at all. – DanielSank Mar 23 '18 at 22:53
• Ah, ok, I didn't realize that was non-standard terminology. In that case, yeah, I can definitely see why the other question is needed. – probably_someone Mar 23 '18 at 22:53
• I think the big point is that the voltage crosses through zero twice per cycle for AC. – zeta-band Mar 23 '18 at 22:55

Typically when we refer to more complicated currents than pure DC or sinusoidal AC, we refer to them as having a "DC component" and AC components of various frequencies. As far as I know, the DC component refers to the zero-frequency component of the signal; hence, if a signal has no zero-frequency component, it should be called pure AC.

Yes. This is completely correct.

But I've also heard the following argument: AC, in order to be "alternating" in any real sense, must change direction. Rectified AC does not change direction (the current is either zero or positive), so rectified current cannot be AC; as such, it should be referred to as "non-constant DC."

First of all, the term "non-constant dc" is intuitively clear, but not standard. Still, there's a good question here about what you'd call a rectified sinusoid. It all depends on the context.

## Power supply

Suppose I build a power supply by rectifying the power line. The result of the rectification might have the mathematical form $$V_0 (1 + \cos(\omega t)) \, .$$ This has, as noted in the OP, a dc component and an ac component, and the language "dc component" and "ac component" is completely standard and universally understood. Now, in a power supply, I probably filter this signal, which reduces the amplitude of the cosine part, giving e.g. $$V_0 + V_\text{ripple} \cos(\omega t)$$ where $V_\text{ripple}$ represents the amplitude of any remaining ac component.$^{[a]}$ In common discussions, a signal like this is usually just refereed to as a "dc voltage", because the intended purpose of this signal is as a dc voltage source and because in many applications the $V_0$ is important and the $V_\text{ripple}$ is small enough to not matter. The fact that the language depends on the context and intended use may be kind of unsatisfying and annoying, but that's how it is. Language is hard.

Now suppose we're really engineering this power supply and I want to make it better, i.e. I want to reduce $V_\text{ripple}$. In this case, I'm focusing on the ac component of the signal, and in communicating with my colleagues (and in my own head) I am definitely going to think about the power supply's output as having a "dc component" and an "ac component".

Now suppose we have an AM radio signal $$m(t) \cos(\omega t)$$ where $m(t)$ is the amplitude modulation we want to send. On the receiver end, we rectify and filter. As long as $m(t) \ll 1$ and the bandwidth of $m(t)$ is low compared to the filter, the resulting signal is $m(t)$ but with a dc offset, e.g. $$m(t) + V_\text{offset} \, .$$ Do we call this dc, ac, or something else? Well, it's certainly correct to say that "the signal has a dc component and an ac component". Would I call this "non-constant dc"? Absolutely not. The phrase "non constant dc" isn't standard, and more to the point, it fails to capture the most important/relevant part of the signal, $m(t)$, which is an ac signal. In most cases, I'd call this an "ac signal with a dc offset".
$[a]$: In a power supply, you usually want to design for a small value of $V_\text{ripple}$.
The identity $\frac{1+\cos\omega t}{2}=\cos^2\frac{\omega t}{2}$ relates two notions of AC, with the $\cos$ term on the left unrectified while the $\cos^2$ term on the right is rectified. (We don't need to separately consider $\sin$, because a translation of $t$ can shift the $\cos$ argument by a constant, allowing us to absorb any $\sin$ term with a compound-angle formula.) You can thus write $A+B\cos\omega t = A-B+2B\cos^2\frac{\omega t}{2}$ as a linear combination of DC with either type of AC, but the AC type chosen changes how much DC you get.