Vs(t) And Is(t) In Phase: Find The Frequency!

by ADMIN 46 views

Alright, let's dive into figuring out when the voltage Vs(t) and current is(t) are in phase in a circuit where Vs(t) is given by 20sin(ωt - 40°). This is a classic AC circuit problem, and to solve it, we need to understand the concepts of impedance, phase angles, and how inductors and capacitors behave in AC circuits. So, buckle up, guys, it's gonna be an electrifying ride!

Understanding the Basics

Before we jump into the specifics, let's make sure we're all on the same page with some fundamental concepts:

  • AC Circuits: In alternating current (AC) circuits, the voltage and current vary sinusoidally with time. This is different from direct current (DC) circuits, where the voltage and current are constant.
  • Phasors: Phasors are a way to represent sinusoidal voltages and currents as rotating vectors. This simplifies the analysis of AC circuits because we can use vector algebra instead of dealing with trigonometric functions directly.
  • Impedance (Z): Impedance is the AC equivalent of resistance in DC circuits. It's the total opposition to current flow in an AC circuit and includes both resistance (R) and reactance (X). Reactance is the opposition to current flow due to inductors and capacitors.
  • Inductive Reactance (XL): Inductors oppose changes in current. The inductive reactance XL is given by XL = ωL, where ω is the angular frequency (ω = 2πf) and L is the inductance.
  • Capacitive Reactance (XC): Capacitors oppose changes in voltage. The capacitive reactance XC is given by XC = 1/(ωC), where ω is the angular frequency and C is the capacitance.
  • Phase Angle (Φ): The phase angle is the difference in phase between the voltage and current in an AC circuit. When the voltage and current are in phase, the phase angle is zero.

Analyzing the Circuit

To find the frequency at which Vs(t) and is(t) are in phase, we need to determine the condition where the impedance of the circuit is purely resistive. This means that the net reactance (the difference between inductive and capacitive reactance) must be zero. In other words, XL - XC = 0.

Let's break down the circuit analysis step by step:

  1. Identify Components: First, identify all the inductors (L) and capacitors (C) in the circuit, as well as their respective values. Also, identify any resistors (R) and their values. This is crucial because the values of these components determine the impedance of the circuit.
  2. Calculate Reactances: Calculate the inductive reactance XL and capacitive reactance XC using the formulas XL = ωL and XC = 1/(ωC). Remember that ω = 2πf, so XL = 2πfL and XC = 1/(2πfC).
  3. Set Net Reactance to Zero: To find the frequency at which the voltage and current are in phase, set the net reactance equal to zero: XL - XC = 0. This gives us the equation 2πfL - 1/(2πfC) = 0.
  4. Solve for Frequency: Solve the equation 2πfL - 1/(2πfC) = 0 for the frequency f. Here's how we do it:
    • 2πfL = 1/(2πfC)
    • (2πfL)(2πfC) = 1
    • (2πf)²LC = 1
    • (2πf)² = 1/(LC)
    • 2πf = √(1/(LC))
    • f = 1/(2π√(LC))

This frequency is known as the resonant frequency of the circuit. At the resonant frequency, the impedance of the circuit is at its minimum, and the voltage and current are in phase.

Applying the Formula

Now that we have the formula for the resonant frequency, f = 1/(2π√(LC)), we can plug in the values of the inductance L and capacitance C to find the frequency at which Vs(t) and is(t) are in phase.

Example:

Let's say the circuit has an inductance of L = 1 mH (1 × 10⁻³ H) and a capacitance of C = 10 μF (10 × 10⁻⁶ F). Plugging these values into the formula, we get:

f = 1/(2π√((1 × 10⁻³ H)(10 × 10⁻⁶ F)))

f = 1/(2π√(1 × 10⁻⁸))

f = 1/(2π × 1 × 10⁻⁴)

f = 1/(2π × 0.0001)

f ≈ 1591.55 Hz

So, in this example, the voltage and current are in phase at approximately 1591.55 Hz.

Key Considerations

  • Circuit Complexity: This analysis assumes a simple series or parallel RLC circuit. If the circuit is more complex (e.g., with multiple inductors and capacitors in different configurations), the analysis may require more advanced techniques, such as mesh analysis or nodal analysis.
  • Ideal Components: We've assumed ideal inductors and capacitors, meaning they have no internal resistance. In reality, inductors have some resistance due to the wire used to make the coil, and capacitors have some equivalent series resistance (ESR). These non-ideal characteristics can affect the phase relationship between voltage and current, especially at high frequencies.
  • Source Impedance: The impedance of the voltage source Vs(t) can also affect the circuit's behavior. If the source impedance is significant, it should be included in the circuit analysis.

Practical Implications

Understanding the frequency at which voltage and current are in phase is crucial in many applications, including:

  • Power Factor Correction: In electrical power systems, it's desirable to have the voltage and current as close to being in phase as possible. This minimizes reactive power and improves the power factor, which leads to more efficient energy transfer.
  • Resonant Circuits: Resonant circuits are used in many electronic devices, such as radio receivers and transmitters. By tuning the circuit to the resonant frequency, we can selectively amplify or filter signals at that frequency.
  • Impedance Matching: In high-frequency circuits, impedance matching is essential to maximize power transfer and minimize signal reflections. This often involves adjusting the circuit components to ensure that the impedance of the source and load are matched.

Conclusion

In conclusion, determining the frequency at which Vs(t) and is(t) are in phase involves finding the resonant frequency of the circuit. This occurs when the inductive reactance equals the capacitive reactance, resulting in a purely resistive impedance. By using the formula f = 1/(2π√(LC)), we can calculate the resonant frequency and ensure that our AC circuits operate efficiently and effectively. So keep those circuits humming and those electrons flowing, guys! Remember to always double-check your component values and circuit configurations to get the most accurate results. Happy circuit analyzing!