a series a.c. circuit, abcd, consists of a resistor ab, an inductor bc, of resistance r and inductance l, and a resistor cd. when a current of 6.5 a flows through the circuit, the voltage drops across various points are: vab = 65 v; vbc = 124 v; vac = 149 v. the supply voltage is 220 v at 50 hz. draw a phasor diagram to scale showing all the resistive and reactive volt drops and, from the diagram. determine the following: (a) the voltage drop vbd and the phase angle between it and the current (b) the resistance and inductance of the inductor
To draw a phasor diagram, we need to determine the voltages and phase angles for each component in the circuit.
Given information:
- Current through the circuit (I) = 6.5 A
- Voltage drop across resistor AB (Vab) = 65 V
- Voltage drop across inductor BC (Vbc) = 124 V
- Voltage drop across circuit AC (Vac) = 149 V
- Supply voltage (V) = 220 V
- Frequency (f) = 50 Hz
Let's start by calculating the voltage drop across resistor CD:
Vcd = Vac - Vab
Vcd = 149 V - 65 V
Vcd = 84 V
Now, let's calculate the impedance of the inductor, which incorporates both resistance and reactance:
Zl = Vbc / I
Zl = 124 V / 6.5 A
Zl = 19.08 Ω
Since the inductor has resistance (R) and inductance (L), we can determine the individual values. Let's split the impedance into resistance and reactance components:
Zl = R + jXl
Reactive component:
Xl = √(Zl^2 - R^2)
Xl = √((19.08 Ω)^2 - R^2)
Given that the inductor has resistance (R) = r and inductance (L) = l, the inductive reactance can be equated as:
Xl = 2πfL
Now, we can solve for resistance (R) and inductance (L) using these equations.
Phasor Diagram:
To draw the phasor diagram, we need to represent all these voltages and phase angles in a graphical representation. However, since it is not possible to draw diagrams here, I will describe the phasor diagram in words.
1. Draw a horizontal line representing the reference axis for the diagram.
2. Place a labeled vector representing the supply voltage (V) with magnitude 220 V, making an angle of 0 degrees with the reference axis.
3. Draw a vector labeled Vab with a magnitude of 65 V, making an angle with the reference axis.
4. Draw a vector labeled Vbc with a magnitude of 124 V, making an angle with the reference axis.
5. Draw a vector labeled Vcd with a magnitude of 84 V, making an angle with the reference axis.
6. Draw a vector labeled I with a magnitude of 6.5 A, making an angle with the reference axis.
7. The phase angle between Vbd and I can be determined by subtracting the angle of I from the angle of Vcd.
Please note that in a phasor diagram, angles are not drawn to scale, but the relative lengths of the vectors represent the magnitudes of the corresponding quantities.
Using the above information, you can draw the phasor diagram and calculate the resistance and inductance of the inductor.
To solve this problem, let's start by drawing the phasor diagram. We'll scale the vectors according to the given voltage drops, vab, vbc, and vac.
1. Draw a horizontal line to represent the reference axis for the current, labeled as I.
2. From the start of the line, draw a vector Vab at an angle of 0 degrees (since there is no phase difference mentioned), with a length of 65 volts. Label it as Vab.
3. From the end of Vab, draw a vector Vbc at an angle of 0 degrees, with a length of 124 volts. Label it as Vbc.
4. From the end of Vab, draw a vector Vac at an angle of 0 degrees, with a length of 149 volts. Label it as Vac.
5. Connect the end of Vbc with the start of Vac to form a triangle.
6. From the starting point of Vac, draw a line parallel to the reference axis to intersect with the line representing the current. Label this point as D.
7. Finally, draw a vector Vbd from D to B (representing the voltage drop across the inductor) and label it as Vbd.
Now that we have the phasor diagram, we can determine the requested values:
(a) The voltage drop Vbd is equal to the vector length of Vbd in the diagram. Measure the length of Vbd in the diagram and convert it to its corresponding voltage value by using the given scale. Also, measure the angle between Vbd and the current vector I. This angle represents the phase angle.
(b) To find the resistance and inductance of the inductor, we'll use the formulas:
- Resistance (R) = Vbc / I
- Inductance (L) = Vbd / (2πfI), where f is the frequency in Hz.
Using the given values for Vbc, I, and the supply frequency of 50 Hz, substitute them into the formulas to calculate the resistance and inductance.