Foundations — Analyze simple AC circuits with reactance

This page assumes nothing. Every squiggle, letter, and arrow the parent note throws at you gets built here from the ground up. Read it top to bottom — each block earns the symbol used in the next.

1. Direct current vs alternating current — the picture first

Everything in this topic lives in the AC world. So the very first thing to see is that wave.

The wave repeats. One full up-then-down-then-back is called one cycle. Hold that picture — every symbol below is a way of describing this wave precisely.

2. $t$ and $V_m$ — reading the height of the wave

Why a cosine, $\cos$? A cosine is nature's smoothest back-and-forth. Picture a dot going around a circle at steady speed: its horizontal shadow traces a cosine as time passes. AC voltage behaves exactly like that shadow. We are nearly ready to write the formula — but it needs one more symbol, the spin-speed, which we build in §3. For now just hold the shape in mind.

3. $f$ and $\omega$ — how fast the wiggle goes, and the wave formula at last

Why do we need $\omega$ when we already have $f$? Because the cosine "thinks" in angle, not in cycles. One full trip around the circle is $2\pi$ radians (a radian is just a way to measure angle where a full circle $=2\pi \approx 6.28$). So each second the dot sweeps $2\pi$ radians per cycle, $f$ cycles' worth:

Now — with $V_m$, $t$, and $\omega$ all defined — we can finally write the AC voltage wave:

4. Angle and phase, $\varphi$ — the "how late" number

Two waves of the same speed can still be shifted — one crests while the other is still climbing. That shift is a phase.

• Lag = arrives later (behind). "Current lags" → current crests after voltage does.
• Lead = arrives earlier (ahead). "Current leads" → current crests before voltage does.

This picture is the entire reason the topic needs complex numbers: we must carry both a height and a shift. Keep it in mind for §7.

5. $R$, $I$, and Ohm's Law — the DC bedrock

See Ohm's Law for the deep version. This topic's whole plan is: keep this law, but replace the plain fight $R$ with a smarter fight $Z$ that also remembers timing.

6. Rate of change — $\frac{di}{dt}$ and the integral $\int$

The parent note's derivations use two calculus tools. Here's each from zero.

Why we need it: an inductor's whole personality is "I fight changes in current," so we literally need a symbol for "rate of change of current." That's $\frac{di}{dt}$.

Why we need it: if we know a voltage is forcing a certain rate of current change, undoing that (integrating) tells us the current itself. That's the step $i(t)=\frac{1}{L}\int V_m\cos (\omega t)dt$.

Two facts we'll use: the slope of $\cos$ is $-\sin$, and adding up (integrating) $\cos$ gives $+\frac{1}{\omega }\sin$. The $\frac{1}{\omega }$ appears because each wiggle is $\omega$ times faster, so its total per push is $\omega$ times smaller — this $\frac{1}{\omega }$ is where reactance's frequency-dependence is born.

7. The two components — their laws, and where $X_L$, $X_C$ come from

We now have every tool to derive (not just quote) the reactance formulas. See Inductors and Capacitors for the physics behind each law.

Inductor — deriving $X_L=\omega L$

Drive it with $v(t)=V_m\cos (\omega t)$ and find the current. Since $v_L=L\frac{di}{dt}$, we rearrange to $\frac{di}{dt}=\frac{V_m}{L}\cos (\omega t)$ and integrate (undo the slope, §6): $i(t)=\frac{1}{L}\int V_m\cos (\omega t)dt=\frac{V_m}{\omega L}\sin (\omega t)$ What we did: used integrating-$\cos$-gives-$\frac{1}{\omega }\sin$. Why: to turn "voltage forces a rate" into the current itself. The peak current is $I_m=\frac{V_m}{\omega L}$. Compare with Ohm's Law $I_m=V_m/X$ — the "fight" is:

Capacitor — deriving $X_C=\frac{1}{\omega C}$ (and its minus sign)

Drive it with $v(t)=V_m\cos (\omega t)$ and take the slope (§6, slope of $\cos$ is $-\sin$): $i(t)=C\frac{d}{dt}\left(V_m\cos \omega t\right)=-\omega C{V}_m\sin (\omega t)$ The peak current is $I_m=\omega C{V}_m$, so $V_m/I_m=\frac{1}{\omega C}$:

8. $j$ and complex numbers — turning "a quarter-beat off" into arithmetic

From §4 we need a number that carries two facts (height and shift) at once. A plain number can't. A complex number can.

The concrete "why" — how $j$, $\frac{1}{\omega }$, $L$ and $C$ combine into impedance. Instead of carrying $\sin$ and $\cos$ around, we let the driving voltage be a rotating arrow and ask what "fight" each component applies to it. The key rule is that taking a slope ($\frac{d}{dt}$) of such a rotating arrow is the same as multiplying by $j\omega$ (it speeds up by $\omega$ and turns the timing by $90°$, which is exactly what $j$ does):

• Inductor: $v=L\frac{di}{dt}⟶$ "fight" $=\frac{v}{i}=j\omega L$. Its size is $\omega L=X_L$ (matching §7), and the $j$ says "$+90°$, current lags."
• Capacitor: $i=C\frac{dv}{dt}⟶$ "fight" $=\frac{v}{i}=\frac{1}{j\omega C}=-\frac{j}{\omega C}$. Its size is $\frac{1}{\omega C}=X_C$, and the $-j$ says "$-90°$, current leads." (Here $\frac{1}{j}=-j$ because $j\cdot (-j)=-j^2=1$ — this is precisely where the capacitor's minus comes from.)

A complex number is written $a+jb$: $a$ is the real part (in-step, horizontal axis) and $b$ is the imaginary part ($90°$-shifted, vertical axis). Draw it as an arrow from the origin — its length is the size, its angle is the timing. See Complex Numbers and Phasors.

9. Pythagoras, ${\tan }^{-1}$ / atan2, and the sign of $\varphi$

Combining the components in series adds the fights: $Z=R+j(X_L-X_C)$. This is an arrow with horizontal leg $R$ and vertical leg $X=X_L-X_C$. We need its length and angle.

Why these two and nothing else? Because $R$ points horizontally and $X$ points vertically — they meet at exactly $90°$, forming a right triangle. Right triangles are precisely what Pythagoras (length) and ${\tan }^{-1}$ (angle) were built for.

So the parent's headline results are just "length of the arrow" and "angle of the arrow": $|Z|=\sqrt{R^2+(X_L-X_C{)}^2},\varphi ={\tan }^{-1}\frac{X_L-X_C}{R}$ Here $|Z|$ (bars mean length/magnitude) is the total opposition; $\varphi$ is the phase shift of §4.

10. Putting the new symbols in one table

Every one of these is now built from a symbol you met earlier on this page.

Prerequisite map

Equipment checklist

Cover the right side and test yourself. If any answer is fuzzy, reread that section before the parent note.

Connections

• Ohm's Law — the law we extend from $R$ to $Z$.
• Complex Numbers and Phasors — where $j$ and the arrow picture come from.
• Capacitors and Inductors — the parts that create reactance.
• RMS and Peak Values — $V_m$ vs $V_{rms}$.
• Resonance in RLC Circuits, Power in AC Circuits — where these foundations lead next.
• Parent: Analyze simple AC circuits with reactance.