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:

Final Answer:

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

Next we will look at an example of using expressions A and B to solve a integro-differential equation.

Continue on to using phasors to solve an integro-differential equation....