Final answer:
The impedance of the circuit element is 5 Ω with a phase angle of 72°. The full expression for impedance is Z = 5 Ω ∠ 72° or Z = R + jX, where R and X can be determined using the phase angle.
Step-by-step explanation:
To find the impedance of the circuit element, we need to compare the voltage and current expressions. Given the voltage across the circuit element as v(t) = 150cos(8000πt+20°) V, and the current passing through the element as i(t) = 30cos(8000πt−52°), impedance can be determined.
Impedance, Z, is a complex quantity represented as Z = R + jX, where R is the resistance and X is the reactance. The magnitude of impedance is given by |Z| = Vmax/Imax, where Vmax and Imax are the maximum (or peak) voltage and current, respectively.
From the given equations:
Vmax = 150 V
Imax = 30 A
The magnitude of impedance is |Z| = Vmax/Imax = 150 V / 30 A = 5 Ω.
Now, we need to consider the phase angle difference to find the complete impedance. The voltage phase angle is 20° and the current phase angle is -52°. Therefore, the total phase angle φ = voltage phase angle - current phase angle = 20° - (-52°) = 72°.
The complete impedance is thus Z = |Z| ∠ φ = 5 Ω ∠ 72°. It can also be written in rectangular form as Z = R + jX, where R and X can be found using the magnitude and phase angle φ.