Question

the impedance z, in ohms, of a circuit element is defined as the ratio of the phasor voltage v, in volts, across the element to the phasor current i, in amperes, through the elements. that is, $z = \frac{v}{i}$. if the voltage across a circuit element is $33 + 10i$ volts and the current through the element is $4 - 5i$ amperes, determine the impedance. the impedance is $z = \square$. (type your answer in the form $a + bi$.)

Explanation:

Step1: Substitute given values

$Z = \frac{33 + 10i}{4 - 5i}$

Step2: Multiply by conjugate of denominator

Multiply numerator and denominator by $4 + 5i$:
$\frac{(33 + 10i)(4 + 5i)}{(4 - 5i)(4 + 5i)}$

Step3: Expand numerator

Use FOIL method:
$33*4 + 33*5i + 10i*4 + 10i*5i = 132 + 165i + 40i + 50i^2$
Since $i^2=-1$, this becomes:
$132 + 205i + 50(-1) = 132 - 50 + 205i = 82 + 205i$

Step4: Expand denominator

Use difference of squares:
$4^2 - (5i)^2 = 16 - 25i^2 = 16 - 25(-1) = 16 + 25 = 41$

Step5: Simplify the fraction

Divide each term by 41:
$\frac{82}{41} + \frac{205}{41}i = 2 + 5i$

Answer:

$2 + 5i$