Final answer:
The rms values of voltage and current for the given circuit element can be calculated by summing the rms values of their individual sinusoidal and constant components. For voltage and current waveforms that consist of multiple cosine terms, each term's rms value is obtained by first finding the square root of half the squared amplitude (peak value) of each term.
Step-by-step explanation:
To determine the rms (root mean square) values of the voltage and current waveforms for a circuit element, we must examine the given time-dependent expressions for voltage, v(t), and current, i(t).
The voltage v(t) is given by:
v(t) = 2 + 5cos(2π60t) – 3cos(4π60t + 45°) V
And the current i(t) is given by:
i(t) = 1 + 2cos(2π60t+20°) + 1.2cos(4π60t – 20°) + 0.5cos(6π60t – 30°) A
To calculate the rms value of a sinusoidal function, we use the formula:
RMS value = √(1/2)×peak value
For the voltage component, the rms value of each cosine term can be computed separately. Since the constant term (2 V) does not vary in time, its rms value is simply its magnitude. For the cosine terms, only the coefficients of the cosine functions (which represent the peak values) contribute to the rms calculations:
Voltage rms = √(1/2) × (5 V) + √(1/2) × (3 V)
This analysis is repeated for the current waveform, calculating the rms value for each cosine term individually:
Current rms = √(1/2) × (2 A) + √(1/2) × (1.2 A) + √(1/2) × (0.5 A)
After calculating these values, the total rms for v(t) and i(t) would be obtained by summing the individual components' rms values. This calculation provides the rms voltage and current for the circuit element in question.