Math Problem Statement
No circuito da Figura abaixo, o fasor que representa a tensão da fonte é
O fasor que representa a corrente Î do circuito, dado em ampères, é:
Solution
The given circuit consists of an AC voltage source, a resistor, and an inductor. From the circuit diagram:
- The resistor has a value of ,
- The inductor is represented by a reactance ,
- The current and voltage are represented in phasor form.
To determine the phasor for the current , we can use Ohm's Law for AC circuits: where:
- is the phasor voltage from the source,
- is the total impedance of the circuit.
Step 1: Calculate the total impedance
The total impedance is the sum of the resistive and inductive reactances: where:
- (resistance),
- (inductive reactance).
Thus, the total impedance is:
Step 2: Calculate the magnitude and phase of the impedance
To find the magnitude and phase of the impedance, we use the following formulas:
and
Let me calculate these values for you.The total impedance has a magnitude of 40 ohms and a phase angle of 30°.
Step 3: Phasor current calculation
Now, the phasor current can be expressed as: If the voltage is given as , you can divide the phasor voltage by the magnitude and phase of the impedance to find the phasor current .
Would you like me to proceed with calculating assuming a certain voltage, or do you have the value of available for the source? Let me know if you have any questions!
Here are some questions you can think about:
- How do you determine the phase angle of the current relative to the voltage?
- How does the inductive reactance affect the total impedance?
- What would happen if the circuit had a capacitive reactance instead of an inductive one?
- How would the current change if the resistor's value increased?
- What are the real and reactive power components in this AC circuit?
Tip: In AC circuits, the impedance angle reflects the phase difference between voltage and current due to reactive components.
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Math Problem Analysis
Mathematical Concepts
AC Circuits
Impedance
Phasors
Ohm's Law
Formulas
Ohm's Law for AC Circuits: \( \hat{I} = \frac{\hat{V}}{Z} \)
Theorems
-
Suitable Grade Level
Advanced College
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