A sinusoidal voltage Δv = 35.0 sin(100t), where Δv is in volts and t is in seconds, is applied to a series RLC circuit with L = 180 mH, C = 99.0 µF, and R = 62.0 Ω.
D) Determine the numerical value for 𝜑 (in rad) in the equation i = Imax sin(𝜔t − 𝜑).
To determine the numerical value for 𝜑 (in rad) in the equation i = Imax sin(𝜔t − 𝜑), we need to first find the current i in the RLC circuit.
Given that Δv = 35.0 sin(100t), we can find the angular frequency 𝜔 using the formula:
𝜔 = 2πf
where f is the frequency.
Since the voltage is sinusoidal, its frequency can be determined from the coefficient of t in the sine function. In this case, the coefficient is 100, so the frequency f is:
f = 100 Hz
Now we can calculate 𝜔:
𝜔 = 2π(100) = 200π rad/s
Next, we can calculate the impedance Z of the RLC circuit using the formulas:
Z = R + j(XL - XC)
where R is the resistance, XL is the inductive reactance, and XC is the capacitive reactance.
XL = 2πfL, where L is the inductance
XC = 1 / (2πfC), where C is the capacitance
Substituting the given values, we get:
XL = 2π(100)(180 × 10^(-3)) = 36π Ω
XC = 1 / (2π(100)(99 × 10^(-6))) = 1 / (198π) Ω
Now we can calculate the impedance Z:
Z = 62 + j(36π - 1 / (198π)) Ω
To find the current i, we use Ohm's Law:
i = Δv / Z
Substituting the given values, we get:
i = 35.0 sin(100t) / (62 + j(36π - 1 / (198π))) Ω
Now we can compare this equation to i = Imax sin(𝜔t − 𝜑) to determine the value of 𝜑:
We can see that the angular frequency 𝜔 in both equations is 200π rad/s.
Therefore, to find the value of 𝜑, we need to compare the arguments (angles) in the two equations.
From the equation i = 35.0 sin(100t) / (62 + j(36π - 1 / (198π))) Ω, we can determine the angle by calculating the inverse tangent of the imaginary part divided by the real part of the impedance's argument:
𝜑 = atan((36π - 1 / (198π)) / 62)
By evaluating this expression, we can find the numerical value for 𝜑 (in rad) in the equation i = Imax sin(𝜔t − 𝜑).