To understand the concept of reactance (of an inductor) and its frequency dependence. When an inductor is connected to a voltage

Question

To understand the concept of reactance (of an inductor) and its frequency dependence.

When an inductor is connected to a voltage source that varies sinusoidally, a sinusoidal current will flow through the inductor, its magnitude depending on the frequency. This is the essence of AC (alternating current) circuits used in radio, TV, and stereos. Circuit elements like inductors, capacitors, and resistors are linear devices, so the amplitude I0 of the current will be proportional to the amplitude V0 of the voltage. However, the current and voltage may not be in phase with each other. This new relationship between voltage and current is summarized by the reactance, the ratio of voltage and current amplitudes, V0, and I0: XL=V0/I0, where the subscript L indicates that this formula applies to an inductor.

A.To find the reactance XL of an inductor, imagine that a current I(t)=I0sin(?t), is flowing through the inductor. What is the voltage V(t) across this inductor?

Express your answer in terms of I0, ?, and the inductance L.

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Mộc Miên 4 years 2021-08-06T22:06:37+00:00 2 Answers 104 views 0

Answers ( )

  1. thienhuong
    0
    2021-08-06T22:07:54+00:00
    Reply

    Answer:

    V(t) = ωLI₀sin(π/2 – ωt)

    Explanation:

    We know from the Maxwell’s equation that

    V(t) = dФ/dt

    Where Ф is the magnetic flux and is given by

    Ф = LI(t)

    Where L is the inductance of the inductor

    V(t) = dLI(t)/dt

    In case of DC circuits, the rate of change of current dI/dt is zero therefore, the voltage is zero. But for AC circuits, the current is time varying therefore, the voltage is not zero.

    If the inductance is constant then voltage becomes

    V(t) = LdI(t)/dt

    When the current is given in the form of

    I(t) = I₀sin(ωt)

    Where ωt is the frequency in rad/s

    Then the voltage across inductor is

    V(t) = Ld(I₀sin(ωt))/dt

    V(t) = LI₀ωcos(ωt)

    We know that cos(ωt) = sin(π/2 – ωt)

    V(t) = ωLI₀sin(π/2 – ωt)

    Which means that the voltage and current are out of phase.

    The phase difference is π/2 or 90°

    And the inductive reactance XL is

    XL = ωL

    or

    XL = 2πfL

    Where f is the frequency in Hertz

    Therefore, the voltage across inductor is

    V(t) = ωLI₀sin(π/2 – ωt)

  2. ngochoa
    0
    2021-08-06T22:08:10+00:00
    Reply

    Answer:

    The voltage across the inductor I’m terms of I0, ω and L is expressed as V(t) = I0ωLsinωt

    Explanation:

    In an R-L AC circuit, the voltage across the inductor leads the current by 90°.

    Voltage V(t) across the inductor can be expressed as

    V(t) = V0sinωt … (1) where;

    ω is the angular velocity which is a function of its frequency.

    Since the inductive reactance

    XL = V0/I0

    V0 = I0XL

    also if XL = ωL

    V0 = I0(ωL)…(2)

    Substituting equation 2 into 1 to get V(t),

    V(t) = I0ωLsin(ωt)

    V(t) = I0ωLsinωt

    The voltage across the inductor in terms of I0, ω and L is expressed as V(t) = I0ωLsinωt

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