A sinusoidal voltage Δv = (65 V) sin (140t) is applied to a series RLC circuit with L = 100 mH, C = 175 μF, and R = 42 Ω.
(a) What is the impedance of the circuit?
R = 42 Ohms.
Xc = 1/W*C = 1/(140*175*10^-6)=40.8 Ohms
Xl = W*L = 140*100*10^-3 = 14 Ohms.
Z^2 = R^2 + (Xl-Xc)^2 = 2482.24
Z = 49.82 Ohms.
tanA = (Xl-Xc)/R = (14-40.8)/42=-0.63810
A = -32.54o = Phase angle.
Z = 49.82 Ohms @ -32.54o.
To find the impedance of the circuit, we need to calculate the total opposition to the flow of current in the circuit. The impedance (Z) of a series RLC circuit is given by the formula:
Z = √(R^2 + (XL - XC)^2)
where R is the resistance, XL is the inductive reactance, and XC is the capacitive reactance.
In this case, we have R = 42 Ω, L = 100 mH = 0.1 H, and C = 175 μF = 1.75 x 10^-4 F.
First, let's calculate the inductive reactance (XL) using the formula:
XL = 2πfL
where f is the frequency. From the given equation for Δv = (65 V) sin (140t), we can see that the angular frequency (ω) is 140 rad/s (since ω = 2πf). Therefore, we can substitute the values into the formula:
XL = 2πfL = 2π(140 rad/s)(0.1 H) = 28π Ω
Next, let's calculate the capacitive reactance (XC) using the formula:
XC = 1 / (2πfC)
Again, substituting the values:
XC = 1 / (2πfC) = 1 / (2π(140 rad/s)(1.75 x 10^-4 F)) ≈ 0.152 Ω
Now, we can substitute all the values into the formula for impedance:
Z = √(R^2 + (XL - XC)^2)
= √(42^2 + (28π - 0.152)^2) ≈ 84.5 Ω
Therefore, the impedance of the circuit is approximately 84.5 Ω.