Final answer:
The complex power, average power, and reactive power can be calculated using the voltage and current equations, their rms values, and the phase difference. Complex power is expressed as P + jQ, and by using the power factor, cos(\phi), we can obtain the average power Pave and the reactive power Q for both scenarios presented in the question.
Step-by-step explanation:
To calculate the complex power, average power, and reactive power for the given voltage and current equations, we use the formula for complex power S = Vrms * Irms * e^{j\phi}, where \(\phi\) is the phase difference between the voltage and current. The complex power S can be expressed in rectangular form as P + jQ, where P is the average (or real) power and Q is the reactive power.
For the first question, the given voltage is v(t) = 160cos(377t)V and the current is i(t) = 4cos(377t+45^\circ)A. The rms values are Vrms = \frac{160V}{\sqrt{2}} and Irms = \frac{4A}{\sqrt{2}}, with a phase difference of 45^\circ. Using the power factor cos(\phi) = cos(45^\circ), we can calculate the average power Pave = Irms * Vrms * cos(\phi), and the reactive power is Q = Irms * Vrms * sin(\phi).
For the second question, we are given V = 80∠60∘V rms and Z = 50∠30∘Ω. The complex power for this scenario is S = V^2 / Z. We can calculate this using polar notation and then convert to rectangular form to find the average power (real part) and the reactive power (imaginary part).