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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 𝑡 + \phi)\$, where 𝜔 is the angular frequency of the voltage.

I know 𝜔 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 𝜔𝑡 and not just 𝜔? Is the \$t\$ there just to show the measurement is with respect to time?

Or is it because 𝜔 and 𝜔𝑡 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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𝜔𝑡 and 𝜑 are the same units (phase in Rads) except 𝜑 is constant and 𝜔𝑡 expresses the exact phase as a function of time, t for v(t) at freq=𝜔

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