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 203Ω20\sqrt{3} \, \Omega,
  • The inductor is represented by a reactance j20Ωj20 \, \Omega,
  • The current ItI_t and voltage VtV_t are represented in phasor form.

To determine the phasor for the current I^\hat{I}, we can use Ohm's Law for AC circuits: I^=V^Z\hat{I} = \frac{\hat{V}}{Z} where:

  • V^\hat{V} is the phasor voltage from the source,
  • ZZ is the total impedance of the circuit.

Step 1: Calculate the total impedance ZZ

The total impedance ZZ is the sum of the resistive and inductive reactances: Z=R+jXLZ = R + jX_L where:

  • R=203ΩR = 20\sqrt{3} \, \Omega (resistance),
  • XL=20ΩX_L = 20 \, \Omega (inductive reactance).

Thus, the total impedance is: Z=203+j20Z = 20\sqrt{3} + j20

Step 2: Calculate the magnitude and phase of the impedance

To find the magnitude Z|Z| and phase θZ\theta_Z of the impedance, we use the following formulas:

Z=R2+XL2|Z| = \sqrt{R^2 + X_L^2} and θZ=tan1(XLR)\theta_Z = \tan^{-1}\left(\frac{X_L}{R}\right)

Let me calculate these values for you.The total impedance ZZ has a magnitude of 40 ohms and a phase angle of 30°.

Step 3: Phasor current calculation

Now, the phasor current I^\hat{I} can be expressed as: I^=V^Z\hat{I} = \frac{\hat{V}}{Z} If the voltage V^\hat{V} is given as VtV_t, you can divide the phasor voltage by the magnitude and phase of the impedance to find the phasor current I^\hat{I}.

Would you like me to proceed with calculating I^\hat{I} assuming a certain voltage, or do you have the value of V^\hat{V} available for the source? Let me know if you have any questions!

Here are some questions you can think about:

  1. How do you determine the phase angle of the current relative to the voltage?
  2. How does the inductive reactance affect the total impedance?
  3. What would happen if the circuit had a capacitive reactance instead of an inductive one?
  4. How would the current change if the resistor's value increased?
  5. 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

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Suitable Grade Level

Advanced College