IB Maths AI 1.10 Notes

Sinusoidal functions using phasors

This section develops two linked ideas.

1. First, sinusoidal functions with the same frequency can be combined into a single sinusoidal function with a new amplitude and phase shift. 
2. Second, complex numbers can be interpreted geometrically on an Argand diagram, where addition behaves like vector addition and multiplication produces a stretch and a rotation.

Suppose two sinusoidal functions have the same frequency but different phase shifts:

$f_1(t)=a\cos(\omega t+\alpha)\text{ and }f_2(t)=b\cos(\omega t+\beta)$

Their sum can always be written in the form:

$f_1(t)+f_2(t)=R\cos(\omega t+\phi)$

for some amplitude $R$ and phase shift $\phi$.

The key idea is that each sinusoidal term can be treated like a vector, or equivalently a complex number, with modulus equal to its amplitude and argument equal to its phase angle. Adding the functions is then the same as adding these vectors.

To combine expressions of the form $a\cos(\omega t+\alpha)+b\cos(\omega t+\beta)$, treat $a\operatorname{cis}\alpha$ and $b\operatorname{cis}\beta$ as complex numbers, where $\operatorname{cis}\theta=\cos\theta+i\sin\theta$. Then, add them:

$a\operatorname{cis}\alpha+b\operatorname{cis}\beta=R\operatorname{cis}\phi$

Then the required sinusoidal form is $R\cos(\omega t+\phi)$.

Two AC voltage sources are connected in a circuit, where $V_1=10\cos(40t)$ and $V_2=20\cos\left(40t+\frac{\pi}{18}\right)$. Find the total voltage in the form $V=A\cos(40t+B)$.

Represent the two voltages by the complex numbers $z_1=10\operatorname{cis}0$ and $z_2=20\operatorname{cis}\frac{\pi}{18}$.

Add them: $z=z_1+z_2=10+20\left(\cos\frac{\pi}{18}+i\sin\frac{\pi}{18}\right)$.

So $z=\left(10+20\cos\frac{\pi}{18}\right)+i\left(20\sin\frac{\pi}{18}\right)$.

Using $\cos\frac{\pi}{18}\approx0.9848$ and $\sin\frac{\pi}{18}\approx0.1736$, we get $z\approx29.696+i3.472$.

Now find the modulus: $A=|z|=\sqrt{29.696^2+3.472^2}\approx29.898$.

Now find the argument: $B=\arg(z)=\tan^{-1}\left(\frac{3.472}{29.696}\right)\approx0.116$ radians.

Hence $V\approx29.9\cos(40t+0.116)$. This phase shift is approximately $6.67^\circ$.

This method only works directly when the sinusoidal functions have the same frequency. If the frequencies are different, the sum cannot in general be written as a single sinusoidal function of the same type.