Foundations — Q factor — quality of oscillator

Before you can read the parent note, you must own every symbol it throws at you. This page builds each one from the ground up, in the order they depend on each other. Nothing here assumes you have seen an oscillator equation before.

1 · What is oscillation? (the picture behind everything)

Look at the figure. The rest point (green line) is where the object would sit forever if nothing disturbed it. When you pull it aside and let go, it overshoots the rest point, swings to the other side, comes back — over and over. That repeating trip is one oscillation or one cycle.

Everything in the Q-factor topic is about how many of these cycles you get before the motion dies. So this is symbol zero — the thing being counted.

See Simple Harmonic Motion for the idealised (frictionless) version of this motion.

2 · Position $x$ and time $t$

In the figure above, $x$ is the horizontal distance from the green rest line to the blue block. When the block is exactly at rest position, $x=0$. When it is pulled fully right, $x$ is at its biggest positive value.

We need $x$ and $t$ because the whole story is a graph of "where is it ($x$) at each moment ($t$)". Every later symbol is just a feature of that graph.

3 · Amplitude $A$ — the size of the swing

In this figure the oscillation is drawn as a wave in time. The amplitude is the height from the middle line up to a peak (red arrow). If the swing gets weaker over time, the peaks get lower — the amplitude shrinks. That shrinking is the entire drama of a damped oscillator, so $A$ is the quantity we watch decay.

The symbol $A_0$ (with a little zero) means "the amplitude at the start, at $t=0$".

4 · Period $T$ and frequency — how fast it swings

Why we need both: the parent note counts cycles ("rings for many swings"), and one cycle takes a time $T$. So to convert "how long it rang" into "how many swings", you divide by $T$ — or multiply by $f$. They are two views of the same speed.

In the wave figure, $T$ is the horizontal distance from one peak to the next peak.

5 · Angular frequency $\omega$ and the natural frequency ${\omega }_0$

Here is the first symbol that trips people up, so we build it slowly.

The figure shows a dot going around a circle at constant speed; its shadow on the vertical axis traces exactly the wave from §3. One full loop = $2\pi$ radians = one cycle. So:

Why bother? Physics formulas for oscillation come out clean in radians (no stray $2\pi$'s in the differential equation). That is why the parent note writes ${\omega }_0$ everywhere instead of $f_0$.

Read ${\omega }_0=\sqrt{k/m}$ as a tug-of-war: a stiffer spring (bigger $k$) yanks harder → faster swing → bigger ${\omega }_0$. A heavier mass (bigger $m$) is more sluggish → slower swing → smaller ${\omega }_0$. That is why $k$ is on top and $m$ underneath.

6 · The spring constant $k$ and mass $m$

These two are the ingredients of ${\omega }_0$. You need them because the parent note's example 2 gives you $m$ and $k$ and asks you to build ${\omega }_0$ yourself.

7 · Velocity $\dot{x}$, acceleration $\ddot{x}$, and the dot notation

So when the parent writes $m\ddot{x}+b\dot{x}+kx=0$, translate it out loud: "mass times acceleration, plus (a friction number) times velocity, plus (spring stiffness) times position, all balances to zero." It is just Newton's law with three forces.

8 · Damping: the friction number $b$ and the damping rate $\gamma$

Why divide by $m$? The same drag $b$ slows a light object much faster than a heavy one. Dividing by $m$ gives the drag's actual effect on the motion. That is why the parent's clean equation $\ddot{x}+\gamma \dot{x}+{\omega }_0^{2}x=0$ uses $\gamma$, not $b$.

The whole Q story is a race between two rates:

• ${\omega }_0$ = how fast it wants to swing (radians per second),
• $\gamma$ = how fast friction eats the motion (per second).

If swinging wins by a huge margin, you get many rings before it fades — high Q. If friction is comparable, it dies almost at once — low Q. Foreshadowing: $Q={\omega }_0/\gamma$ is literally "swing rate ÷ loss rate". See Damped Harmonic Motion.

9 · The exponential $e^{-\gamma t/2}$ — how a fading swing shrinks

In the figure, the wiggling wave is the actual motion; the smooth curve hugging its peaks (the envelope, dashed) is $A_0{e}^{-\gamma t/2}$. The envelope is how the amplitude shrinks. The exponent carries $\gamma /2$ (not $\gamma$) for the amplitude; energy — which goes as amplitude squared — decays with $\gamma$. See Energy in Oscillations.

10 · Energy $E$ and "$E\propto A^2$"

Why the square? A spring stretched twice as far stores four times the energy (energy in a spring is $\frac{1}{2}k{x}^2$ — the displacement appears squared). So doubling amplitude quadruples energy. This is the bridge that turns "amplitude fell to $1/e$" into "energy fell to $1/e^2$", which the parent uses constantly.

11 · Resonance, driving, and bandwidth $\Delta \omega$

The parent's third face of Q is $Q={\omega }_0/\Delta \omega$: the tall centre frequency divided by the narrow width. You now have every symbol in it. The electrical twin of this whole story lives in RLC Circuits.

Prerequisite map

Equipment checklist

Test yourself — cover the right side and answer before revealing.

Connections

• Simple Harmonic Motion — the frictionless ideal these symbols specialise from.
• Damped Harmonic Motion — home of $b$, $\gamma$ and the $e^{-\gamma t/2}$ envelope.
• Energy in Oscillations — why $E\propto A^2$.
• Resonance & Forced Oscillations — driving, the peak, and $\Delta \omega$.
• RLC Circuits — the electrical version of every symbol here.
• Q factor — quality of oscillator — the parent topic all of this feeds.