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When we describe a AC signal (voltage is this case) in time domain we use an equation something like this below:

\$v(t) = V_m \mathrm{sin}(\omega 𝑑 + \phi)\$, where πœ” is the angular frequency of the voltage.

I know πœ” is the symbol for angular frequency which is equal to \$2 \pi f\$ and is measured in rad/s.

So why does the voltage equation above show the angular frequency as πœ”π‘‘ and not just πœ”? Is the \$t\$ there just to show the measurement is with respect to time?

Or is it because πœ” and πœ”π‘‘ not the same?

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No, \$\omega\$ and \$\omega t\$ and are not the same, and they don't have the same units. \$\omega t\$ has units of radians, which makes it possible to add the phase angle \$\phi\$ to it and get something sensible.

In the equation you provide, the values of \$V_m\$, \$\omega\$, and \$\phi\$ are constants. So, if we want to find the voltage as a function of time, then there must be a variable in the equation that represents the specific value of time in question. That variable is \$t\$.

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πœ”π‘‘ and πœ‘ are the same units (phase in Rads) except πœ‘ is constant and πœ”π‘‘ expresses the exact phase as a function of time, t for v(t) at freq=πœ”

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