= Operating pitch circle
{title2=$r_{wi}=aN_i/(N_1+N_2)$}
= Working pitch circle
{synonym}
The <operating pitch circles> of two meshing <gears> are the imaginary <circles> that reproduce their motion by <rolling without slipping>. They are properties of the working pair and its mounting, whereas a <gear reference circle> specifies one <gear>'s nominal geometry. <Working pitch circle> and intermeshing pitch circle are alternative names.<KHK Gear Nomenclature>[“Intermeshing Pitch Circle”, definition and alternative names]{ref}
Consider an external pair of <spur gears> on fixed parallel axes, with axis separation $a$ and tooth counts $N_1,N_2$. Their operating <radii> $r_{w1},r_{w2}$ must satisfy both contact geometry and the tooth-count <gear ratio>:
$$
r_{w1}+r_{w2}=a,\qquad \frac{r_{w2}}{r_{w1}}=\frac{N_2}{N_1}.
$$
Solving gives
$$
r_{w1}=\frac{aN_1}{N_1+N_2},\qquad
r_{w2}=\frac{aN_2}{N_1+N_2},\qquad
r_{w1}\omega_1+r_{w2}\omega_2=0.
$$
Thus the operating radii scale with the actual separation, while the <gear ratio> stays fixed by tooth counts. These are kinematic equations, conditional on the teeth actually maintaining compatible contact.
For an original example, choose <gear module> $m=2\,\mathrm{mm}$ and tooth counts 20 and 40. Reference <radii> are $20\,\mathrm{mm}$ and $40\,\mathrm{mm}$; their standard separation is $60\,\mathrm{mm}$. If a compatible mounting uses $a=61\,\mathrm{mm}$ instead, operating <radii> become $20.333\ldots\,\mathrm{mm}$ and $40.666\ldots\,\mathrm{mm}$. At input <angular velocity> $30\,\mathrm{rad/s}$ the output has $-15\,\mathrm{rad/s}$, and both operating tangential speed magnitudes are $0.61\,\mathrm{m/s}$. Using the reference radii in that speed calculation would incorrectly give $0.60\,\mathrm{m/s}$.
For tooth flanks formed as <involutes>, the fixed <gear base circles> permit constant-ratio motion over a range of separations. If their radii are $r_{b1},r_{b2}$, the common-normal geometry gives
$$
a\cos\alpha_w=r_{b1}+r_{b2},
$$
where $\alpha_w$ is the <operating pressure angle>. Increasing separation therefore increases that angle, while leaving each <gear reference circle> and <gear base circle> unchanged. This geometric possibility does not guarantee adequate tooth contact, clearance or strength at an arbitrary separation: the physical tooth extents and mounting must still be checked.<KHK Gear Nomenclature>[The ABC's of Gears, Basic Guide B, §§3–4, printed pp. 20–23: involute geometry, base circles and changes of working pressure angle with axis separation]{ref}
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