A single- phase capacitor- start squirrel- cage induction motor take 2.5 A from a 220-V line. The current in the starting winding is 1.3 A and the current in the main winding is 1.45 A. The total power input is 550 W. What is the resistance of the main winding? Assume the auxiliary winding takes a leading current while the main winding takes a lagging current.
To find the resistance of the main winding, we need to understand the power triangle and analyze the current components.
First, let's understand the power triangle:
The power input (P) is given as 550 W.
The apparent power (S) is the product of the voltage (V) and the total current (I), which is 220 V × 2.5 A = 550 VA.
The power factor (PF) is the ratio of the real power (P) to the apparent power (S), which is P / S = 550 W / 550 VA = 1.
In our problem, the auxiliary winding takes a leading current, which means it is capacitive, and the main winding takes a lagging current, which means it is inductive.
Now, let's analyze the current components:
The current in the starting winding is given as 1.3 A (capacitive).
The current in the main winding is given as 1.45 A (inductive).
The total current is the vector sum of the starting winding current and the main winding current, which is √[(1.3 A)^2 + (1.45 A)^2] = 1.9 A.
Now we can find the real power (P) using the power factor (PF):
P = S × PF = 550 VA × 1 = 550 W
Next, we can find the reactive power (Q) using the total current (I) and the apparent power (S):
Q = √(S^2 - P^2) = √(550^2 VA^2 - 550^2 W^2) = 550 VA
Now, we can calculate the impedance (Z) using the voltage (V) and the total current (I):
Z = V / I = 220 V / 1.9 A = 115.79 Ω
Since the impedance (Z) consists of both the resistance (R) and the reactance (X), we can calculate the reactance (X) using the resistance (R) formula:
X = √(Z^2 - R^2)
Squaring both sides gives:
X^2 = Z^2 - R^2
Substituting the given values, we get:
X^2 = (115.79 Ω)^2 - R^2
To calculate the resistance (R), we rearrange the equation:
R^2 = (115.79 Ω)^2 - X^2
Now, we need to determine the reactance (X). Since the starting winding is capacitive, we can assume the reactance (X) is negative and find its value using the formula:
X = -1 / (2 * π * f * C * I)
Where:
f is the frequency of the applied voltage.
C is the capacitance of the starting winding.
I is the current in the starting winding.
Given that the starting winding current (I) is 1.3 A and assuming the frequency (f) and capacitance (C) are not provided, we cannot calculate the reactance (X) accurately.
Therefore, without the necessary information, we cannot determine the resistance of the main winding accurately.