Final answer:
The complex gain of the black box at frequency ω₀ is 14.34 - j*20.48 in Cartesian form, and 25 ∠ -55 degrees in polar form, with 'j' representing the imaginary unit, and the angle given in degrees to indicate the phase lag.
Step-by-step explanation:
The gain of a black box at a specific frequency ω₀ is given as a magnitude |H(jω₀)| = 25 with the output voltage lagging the input voltage by 55 degrees. To express this gain as a complex number in Cartesian and polar forms, we can use the magnitude and phase shift information provided.
In Cartesian form, the complex gain H(jω₀) will have a real part 'a' and an imaginary part 'b' such that a + j * b is equivalent to the polar form with magnitude 25 and phase -55 degrees. The negative sign in the phase indicates lag between the output and input.
The conversion from polar to Cartesian form is given by the equations:
a = |H| * cos(ϕ) = 25 * cos(-55 degrees)
b = |H| * sin(ϕ) = 25 * sin(-55 degrees)
After calculating, we have:
a ≈ 25 * cos(-55 degrees) ≈ 14.34
b ≈ 25 * sin(-55 degrees) ≈ -20.48
Thus, in Cartesian form, the gain H(jω₀) = 14.34 - j*20.48. To express this in polar form, we can use the magnitude and phase directly as:
H(jω₀) = 25 ∠ -55 degrees.
Note that in this context, degrees are used rather than radians for the phase angle.