A 80.0 Hz generator with an rms voltage of 150 V is connected in series to a 3.20 k ohm resistor and a 3.50 micro farad capacitor. Find the phase angle between the current and the voltage.
f=80 Hz, V=150V, R=3200 Ω, C=3.5•10⁻⁶ F
ω=2πf =2π•80=160π(rad/s)
ω²=1/T² =1/(2π)²LC =>
L=1/(2π ω)²C= ....
X(L) =ωL= ...
X(C) =1/ωC= ...
tan φ=(X(L) –X(C))/R = ...
Xc = 1/2pi*F*C = 1/(6.28*80*3.5*10^-6) =
569 Ohms
Z=R-jXc = 3200 - j569=3250 Ohms[-10.1o]
The negative impedance angle means the
circuit is capacitive and the current leads the applied voltage by 10.1o.
Tan A = Xc/R = -569/3200 = -0.17772
A = -10.1o
To find the phase angle between the current and the voltage, we need to use the concepts of impedance and reactance in an AC circuit.
First, let's calculate the impedance of the circuit. The impedance (Z) of a series RL 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.
Given:
Frequency (f) = 80.0 Hz
Resistance (R) = 3.20 kΩ = 3200 Ω
Capacitance (C) = 3.50 μF = 3.50 × 10^(-6) F
Inductance (L) = 1 / (4π^2f^2C)
First, let's calculate the inductive reactance (Xl):
Xl = 2πfL
Next, let's calculate the capacitive reactance (Xc):
Xc = 1 / (2πfC)
Now, we can calculate the impedance (Z):
Z = √(R^2 + (Xl - Xc)^2)
Once we have the impedance (Z), we can find the phase angle by using the formula:
Phase angle (θ) = arctan((Xl - Xc) / R)
Let's calculate the values step by step:
1. Inductive Reactance (Xl):
Xl = 2πfL
= 2 x 3.14 x 80.0 x (1 / (4 x 3.14^2 x 80.0^2 x 3.50 x 10^(-6)))
≈ 25.23 Ω
2. Capacitive Reactance (Xc):
Xc = 1 / (2πfC)
= 1 / (2 x 3.14 x 80.0 x 3.50 x 10^(-6))
≈ 567.05 Ω
3. Impedance (Z):
Z = √(R^2 + (Xl - Xc)^2)
= √((3200)^2 + (25.23 - 567.05)^2)
≈ √(10240000 + 321388.8349)
≈ √10561388.83
≈ 3251.37 Ω
4. Phase Angle (θ):
Phase angle (θ) = arctan((Xl - Xc) / R)
= arctan((25.23 - 567.05) / 3200)
≈ arctan(-19.44)
≈ -89.60° or -1.56 radians
Therefore, the phase angle between the current and the voltage is approximately -89.60° or -1.56 radians. The negative sign indicates that the current lags behind the voltage in this case.