When a resistor is connected across the terminals of an ac generator (112 V) that has a fixed frequency, there is a current of 0.500 A in the resistor. When an inductor is connected across the terminals of this same generator, there is a current of 0.400 A in the inductor. When both the resistor and the inductor are connected in series between the terminals of this generator, what are (a) the impedance of the series combination and (b) the phase angle between the current and the voltage of the generator?
For a)
I= E/R
0.500A= 112V/R -> R=224 ohms
0.400A= 112V/XL -> XL=280 ohms
Z= sqrt[R^2 + XL^2]
Z= 358.6 ohms
Did I do this correctly?
b) I'm not really sure about phasor diagrams. I know the current will lag the voltage, but I don't know how to determine the phase angle????
a. Correct.
b. tanA = Xl/R = 280/224 = 1.25
A = 51.3 Deg.
Since the circuit is inductive, the current lags the applied voltage by
51.3 Deg.
(a) Well, it looks like you did some calculations correctly. The impedance of the series combination would indeed be the square root of the sum of the resistance squared and the inductance squared. So, in this case, the impedance would be approximately 358.6 ohms.
(b) Ah, the phase angle. Well, to determine the phase angle between the current and the voltage, we can use a little trigonometry. Since the current in the inductor lags behind the voltage, we can say that the angle is negative. Let's call the phase angle theta.
tan(theta) = XL / R
theta = atan(XL / R)
Using the values we calculated earlier, we can substitute them into the equation:
theta = atan(280 / 224)
theta ≈ 50.86 degrees
So, the phase angle between the current and voltage would be approximately 50.86 degrees. Just remember, in this case, the power factor seems to have a bit of a laggy sense of humor!
For calculating the impedance of the series combination, you correctly calculated the values of the resistor and the inductor. The impedance of the series combination can be calculated using the Pythagorean theorem:
Z = √(R^2 + XL^2)
= √(224^2 + 280^2)
≈ 358.6 ohms
So, you correctly calculated the impedance of the series combination as approximately 358.6 ohms.
For determining the phase angle between the current and the voltage, you can use the concept of phase difference between the voltage and current in an inductive circuit. In an inductive circuit, the current lags behind the voltage.
The phase angle (ϕ) can be determined using the formula:
tan(ϕ) = XL / R
Substituting the given values:
tan(ϕ) = 280 / 224
ϕ = arctan(280 / 224)
ϕ ≈ 51.3 degrees
So, the phase angle between the current and the voltage is approximately 51.3 degrees.
For part (a), you have correctly calculated the resistance (R) and inductive reactance (XL). To find the impedance (Z) of the series combination of the resistor and inductor, you need to use the formula:
Z = √(R² + XL²)
Using the given values:
Z = √(224² + 280²) = √(50176 + 78400) ≈ √128576 ≈ 358.8 ohms
Your calculation is very close, just a slight rounding error.
For part (b), to determine the phase angle between the current and the voltage, you need to use the relationship between impedance (Z), resistance (R), and inductive reactance (XL):
tan(θ) = XL / R
θ is the phase angle between the current and voltage. Using the given values:
tan(θ) = 280 / 224 ≈ 1.25
To find the angle, you can take the inverse tangent (tan^(-1)) of 1.25:
θ ≈ tan^(-1)(1.25) ≈ 51.3 degrees
So the phase angle between the current and the voltage is approximately 51.3 degrees. The current will lag the voltage in this situation.