Math Problem Statement
A series AC circuit has:
Resistor (R=10 ΩR = 10 , \Omega), Inductor (L=0.2 HL = 0.2 , \text{H}), and Capacitor (C=50 μFC = 50 , \mu\text{F}).
AC source with Vpeak=100 VV_\text{peak} = 100 , \text{V} and frequency f=50 Hzf = 50 , \text{Hz}.
Find inductive reactance (XLX_L) and capacitive reactance (XCX_C).
Calculate total impedance (ZZ).
Determine peak current (IpeakI_\text{peak}).
Find phase angle (ϕ\phi).
Is the circuit inductive, capacitive, or resistive?
Solution
Let's go step by step and solve each part of the problem systematically.
1. Inductive Reactance ()
Inductive reactance is given by the formula:
Where:
- (frequency),
- (inductance).
Substituting the given values:
2. Capacitive Reactance ()
Capacitive reactance is given by the formula:
Where:
- (frequency),
- (capacitance).
Substituting the given values:
3. Total Impedance ()
The total impedance for a series AC circuit is given by:
Where:
- (resistance),
- (inductive reactance),
- (capacitive reactance).
Now, calculate the difference between the inductive and capacitive reactance:
Now, substitute into the impedance formula:
4. Peak Current ()
The peak current is related to the peak voltage and total impedance by:
Where:
- (peak voltage),
- (total impedance).
Substitute the values:
5. Phase Angle ()
The phase angle is given by:
Substitute the values:
6. Inductive, Capacitive, or Resistive Circuit?
To determine whether the circuit is inductive, capacitive, or resistive, we look at the net reactance:
- If , the circuit is inductive.
- If , the circuit is capacitive.
- If , the circuit is purely resistive (with no net reactance).
Since (63.66 Ω > 62.83 Ω), the circuit is capacitive.
Summary of Results:
- Inductive Reactance:
- Capacitive Reactance:
- Total Impedance:
- Peak Current:
- Phase Angle:
- Circuit Type: Capacitive
Would you like further explanations or have any additional questions?
Related Questions:
- How does frequency affect the impedance of a series AC circuit?
- What happens if the resistance is increased in this circuit?
- How can we calculate the power factor from the phase angle?
- How do inductive and capacitive reactances affect the phase angle?
- What is the difference between peak and RMS values for current and voltage in AC circuits?
Tip: In AC circuits, when the impedance is nearly equal to the resistance (), the circuit behaves more resistively, and the current is almost in phase with the voltage.
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Math Problem Analysis
Mathematical Concepts
AC Circuits
Impedance
Reactance
Trigonometry
Formulas
X_L = 2πfL
X_C = 1/(2πfC)
Z = √(R^2 + (X_L - X_C)^2)
I_peak = V_peak / Z
ϕ = tan^(-1)((X_L - X_C) / R)
Theorems
Impedance in AC Circuits
Phase Angle Calculation
Suitable Grade Level
Grades 11-12
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