Jacob Kiers - electronicsZola2023-01-10T00:00:00+00:00https://jacobkiers.net/tags/electronics/atom.xmlAlternating current and its relation to voltage2023-01-05T00:00:00+00:002023-01-10T00:00:00+00:00https://jacobkiers.net/post/alternating-current-and-relation-to-voltage/<h2 id="intro">Intro</h2>
<p>For a while now I am interested in electronics. So after some false starts,
I started following a long course on it: <a rel="nofollow noreferrer" href="https://www.udemy.com/course/crash-course-electronics-and-pcb-design/">Crash Course Electronics and PCB Design</a>.</p>
<p>Lecture 9 was about alternating current (AC), the kind of current that
flows from a wall socket. I struggled to understand how it's voltage is
determined, so I took a lot of notes. This is the result.</p>
<p>Warning: algebra ahead. Also: everything here may be wildly inaccurate.
Did I mention that I am just learning this stuff?</p>
<h2 id="what-is-alternating-current">What is alternating current</h2>
<p>Alternating current measn that the supply source voltage changes as a
function of time. In general, this is a sine wave, with positive high
peak voltages and negative low peak voltages.</p>
<p>A negative voltage means that the current flows in the opposite direction,
from minus (-) to plus (+). In the span of a full sine wave, the current
changes direction twice. So with a standard 60Hz alternating current, the
current changes direction 120 times per second.</p>
<h2 id="properties-of-alternating-current">Properties of alternating current</h2>
<p>Alternating current has a few commonly-used properties:</p>
<ul>
<li>Peak voltage: the peak voltages on both the upper peak ($V_{pmax}$) and lower peak ($V_{pmin}$).</li>
<li>Peak to peak voltage: the voltage difference between the two peaks: $V_{pp} = V_{pmax} - V_{pmin}$.</li>
<li>Frequency: the number of complete cycles (of $V_{pmax}$ and $V_{pmin}$) per second.</li>
<li>Root-mean-square voltage: the voltage calculation usually associated with alternating current, e.g. 230V or 120V.</li>
</ul>
<p>In the rest of this note, we will work with an AC voltage with the following properties:</p>
<h2 id="root-mean-square-voltage">Root Mean Square Voltage</h2>
<p>Root Mean Square (RMS, <a rel="nofollow noreferrer" href="https://en.wikipedia.org/wiki/Root_mean_square">wikipedia</a>) is a formula that is often used
whencalculating the voltage over time for an alternating current circuit.</p>
<p>Its definition is as follows:</p>
<p>$$
f(rms) = \lim_{T \to \infty} \sqrt{\frac{1}{T} \int_0^T [f(t)]^2 dt}
$$</p>
<p>For a pure sine wave, which is common, the formula for the peak-to-peak
amplitude can be simplified as:</p>
<p>$$ V_{pp} = 2\sqrt{2} \cdot RMS \approx 2.828 \cdot RMS$$</p>
<h3 id="alternate-ways-of-writing">Alternate ways of writing</h3>
<p>That also means that</p>
<ul>
<li>$V_{rms} = V_{pp} \div 2\sqrt{2} = \frac{V_{pp}}{2\sqrt{2}}$</li>
<li>Written differently: $V_{rms} = V_{pp} \cdot \frac{1}{2\sqrt{2}} = V_{pp} \cdot \frac{1}{2} \cdot \frac{1}{\sqrt{2}}$</li>
<li>This works, because we can pull out the $\frac{1}{2}$ out of $\frac{1}{2\sqrt{2}}$,
as long as we then <em>multiply</em> by the remainder ($\frac{1}{\sqrt{2}}$), which, again,
then gives in full: $V_{rms} = V_{pp} \cdot \frac{1}{2} \cdot \frac{1}{\sqrt{2}}$.</li>
</ul>
<p>I know, this is high school level algebra, but it's been a long time and I
struggled to get there. So I wrote it out in full.</p>
<p>So, when calculating, one then gets:</p>
<ul>
<li>$V_{rms} = V_{pp} \cdot \frac{1}{2} \cdot \frac{1}{\sqrt{2}}$</li>
<li>$V_{rms} \approx V_{pp} \cdot 0.5 \cdot \frac{1}{1.414}$</li>
<li>$V_{rms} \approx V_{pp} \cdot 0.5 \cdot 0.707$</li>
<li>$V_{rms} \approx V_{pp} \cdot 0.354$</li>
</ul>
<h3 id="example-calculations">Example calculations</h3>
<p>With wo common voltages 120V RMS and 230V RMS, we get:</p>
<ul>
<li>For 120V RMS: $V_{pp} \approx 2.828 \cdot 120 \approx 339V$</li>
<li>For 230V RMS: $V_{pp} \approx 2.828 \cdot 230 \approx 650V$</li>
</ul>
<p>And when we go back:</p>
<ul>
<li>For 120V RMS: $V_{rms} \approx V_{pp} \cdot .354 \approx 339 \cdot .354 \approx 120V$</li>
<li>For 230V RMS: $V_{rms} \approx V_{pp} \cdot .354 \approx 650 \cdot .354 \approx 230V$</li>
</ul>
<p>Given a symmetrical sine wave with the reference point as 0V, that gives also:</p>
<ul>
<li>For 120V RMS: $V_{pp} = V_{phigh} - V_{plow} \approx 169V - (-169V) $</li>
<li>For 230V RMS: $V_{pp} = V_{phigh} - V_{plow} \approx 325V - (-325V) $</li>
</ul>
<h2 id="fin">Fin</h2>
<p>That was it for today. If you found this useful, I highly recommend
following <a rel="nofollow noreferrer" href="https://www.udemy.com/course/crash-course-electronics-and-pcb-design/">the course</a>.</p>