 # Phasor and sinusoid Calculus Example Problem #1

### Using Phasors, determine i(t) for the following differential equation:

$$2\frac{di}{dt} + 3i(t) = 4\cos(2t-45^{\circ})$$ By inspection we can see that the angular frequency is: $$\omega = 2\frac{rad}{s}$$ ...and from the previous page we know that the derivative of i(t) is equivalent to the following in phasor form: $$\frac{di}{dt} \iff j\omega \mathbb{I}$$ ...where: $$\mathbb{I} = phasor \; form \; of \; the \; sinusoid \; i(t)$$ This allows us to rewrite the equation as: $$2j(2)\mathbb{I} + 3\mathbb{I} = 4\cos(2t-45^{\circ}) \qquad (Expression \; 1)$$

Note that the right side of the equation is equivalent to the following: $$4\cos(2t-45^{\circ}) = 4\angle(-45^{\circ})$$ ...which means that Expression #1 becomes:

$$4j\mathbb{I} + 3\mathbb{I} = 4\angle(-45^{\circ})$$

# GOAL: isolate the phasor on left side of the equation and convert it to i(t) in sinusoid form

$$\mathbb{I} (3+j4) = 4\angle(-45^{\circ}) \qquad (Expression \; 2)$$

For the 3+j4 term: $$r = \sqrt{3^2+4^2}$$ $$r = 5$$ $$\phi_{ref} = tan^{-1} \Big( \frac{4}{3} \Big) \;\;\;,lies \; 1^{st} \; quadrant$$ $$\phi_{ref} = 53.1^{\circ} = \phi$$ Therefore: $$3+j4 = 5 \angle 53.1^{\circ}$$ ...which allows us to rewrite expression #2 as:

$$\mathbb{I} = \frac{4\angle(-45^{\circ})} {5 \angle 53.1^{\circ}} \qquad (Expression \; 3)$$

Recall the property of division for complex numbers from our review of complex numbers page $$\frac{z_1}{z_2} = \frac{r_1}{r_2} \angle (\phi_1 - \phi_2)$$ ...and use it on expression #3:

$$\mathbb{I} = \frac{4}{5} \angle (-45^{\circ} - 53.1^{\circ})$$ $$\mathbb{I} = 0.8 \angle (-98.1^{\circ})$$ which is equal to the following when expressed in exponential form: $$\mathbb{I} = 0.8e^{j(-98.1^{\circ})} \qquad (Expression \; 4)$$ We now need to multiply expression #4 by e^(jwt) and take the real part of the result in order to obtain the sinusoid i(t). Recall that omega equals 2: $$i(t) = R_e \{ 0.8e^{j(-98.1^{\circ})} e^{j2t} \}$$ $$i(t) = R_e \{ 0.8 e^{j(2t-98.1^{\circ})} \} \qquad (Expression \; 5)$$

Recall Euhler's Identity: $$e^{\pm j\phi} = \cos\phi \pm j\sin\phi$$ ...and apply it to expression #5:

$$i(t) = R_e \{ 0.8\cos(2t-98.1^{\circ}) + j0.8\sin(2t-98.1^{\circ}) \}$$ ...which we exaluate to obtain the:

$$i(t) = 0.8\cos(2t-98.1^{\circ})$$