Final answer:
- The value of voltage δvr is 11.9 V
- The value of voltage δvl is -0.085 V
- The value of voltage δvc is 0.071 V
Step-by-step explanation:
To find the voltages δvr, δvl, and δvc, we can use the given values and formulas for resistors, inductors, and capacitors in an AC circuit.
Given:
- R = 10 Ω (resistance)
- L = 0.1 H (inductance)
- C = 10 μF (capacitance)
- f = 60 Hz (frequency)
- Vm = 12 V (maximum voltage)
1. Calculate the reactance of the inductor (XL) and the capacitor (XC):
Reactance of an inductor (XL) = 2πfL
Reactance of a capacitor (XC) = 1/(2πfC)
XL = 2π(60)(0.1) = 37.7 Ω
XC = 1/(2π(60)(10 × 10⁻⁶)) = 265.3 Ω
2. Find the total impedance (Z) of the circuit:
Total impedance (Z) = √(R² + (XL - XC)²)
Z = √(10² + (37.7 - 265.3)²) = 263.3 Ω
3. Calculate the phase angle (θ):
Phase angle (θ) = arctan((XL - XC)/R)
θ = arctan((37.7 - 265.3)/10) = -1.476 radians
4. Calculate the voltage across the resistor (δvr):
Voltage across the resistor (δvr) = Vm * cos(θ)
δvr = 12 V * cos(-1.476) = 11.9 V
5. Calculate the voltage across the inductor (δvl):
Voltage across the inductor (δvl) = Vm * cos(θ - π/2)
δvl = 12 V * cos(-1.476 - π/2) = -0.085 V
6. Calculate the voltage across the capacitor (δvc):
Voltage across the capacitor (δvc) = Vm * cos(θ + π/2)
δvc = 12 V * cos(-1.476 + π/2) = 0.071 V
In summary:
- δvr = 11.9 V
- δvl = -0.085 V
- δvc = 0.071 V
These are the voltages across the resistor, inductor, and capacitor, respectively, in the given AC circuit.
Your question is incomplete, but most probably the full question was:
Find the voltages δvr, δvl, and δvc. (Hint: R = 10 Ω L = 0.1 H C = 10 μF f = 60 Hz, Vm= 12 v)