A resistance of 400 ohms and inductance of 8H and a capacitance of 20 microfarad are connected in series. Cal. (I) the peak current (ii) the phase angle when an a.c voltage of 40v, f= 900Hz is connected across them
To calculate the peak current and phase angle in this series circuit, we need to use the formulas for impedance (Z), current (I), and phase angle (φ).
The impedance (Z) of an RLC series circuit is given by:
Z = √(R^2 + (XL - XC)^2)
Where:
- R is the resistance (400 ohms)
- XL is the inductive reactance (ωL, with ω representing the angular frequency and L being the inductance)
- XC is the capacitive reactance (1 / ωC, with C being the capacitance)
In this case, we are given that the angular frequency (ω) is 2πf, where f is the frequency in Hertz. Therefore, ω = 2π * 900 Hz.
Let's calculate the impedance (Z) first:
XL = ωL = (2π * 900 Hz) * 8H = 14400π ohms
XC = 1 / (ωC) = 1 / ((2π * 900 Hz) * 20μF) = 1 / (2π * 900 * 10^(-6) F) = 1 / (1.8π * 10^(-3)) = 1 / (1.8 * 3.14 * 10^(-3)) ≈ 9.95 / 10^(-3) ≈ 995 ohms
Z = √(R^2 + (XL - XC)^2)
= √(400^2 + (14400π - 995)^2)
To calculate the peak current (I), we can now use Ohm's law:
I = V / Z
= 40V / Z
To calculate the phase angle (φ), we can use the arctan formula:
φ = arctan((XL - XC) / R)
Let's calculate these values:
I) Peak Current:
Substitute Z into the formula to find I:
I = 40V / √(400^2 + (14400π - 995)^2)
II) Phase Angle:
Substitute XL, XC, and R into the formula to find φ:
φ = arctan((14400π - 995) / R)
Using the formulas and the given values, you can now calculate the peak current and phase angle for the provided circuit.