RLC Circuit Differential Equation | Lecture 25 | Differential Equations for Engineers

00:11:17
https://www.youtube.com/watch?v=RF9EyZCGGx0

Zusammenfassung

TLDRThe video explores how second-order differential equations apply to RLC circuits in electrical engineering. An RLC circuit includes a resistor, inductor, and capacitor, connected to an AC current. By using Kirchhoff's Law and the constitutive relationships for voltage drops across these components, a second-order linear differential equation is established. The equation is parameterized by constants L, R, C, E0, and omega. To simplify, the equation is non-dimensionalized, introducing parameters alpha and beta, leading to a form without dimensions. Understanding the natural frequency in the circuit and manipulating time and charge units offer a clearer representation of the circuit behavior. The video hints at applying similar methods to other physical problems in future discussions.

Mitbringsel

  • 🔌 Introduction of RLC circuits in electrical engineering.
  • 💡 Application of second-order differential equations to circuit analysis.
  • ⚡ Explanation of Kirchhoff's Law in circuit differential equations.
  • 🔄 Non-dimensionalization to simplify circuit equations.
  • 📉 Understanding of natural frequency in RLC circuits.
  • 🔗 Connection between current, charge, and voltage in circuits.
  • 🎛️ Parameters in the non-dimensional equation: alpha and beta.
  • 🧩 Similar equations can apply to different physical problems.
  • 🧮 Relationship of equation terms to circuit elements.
  • 🔍 Overview of simplifying the number of parameters in equations.

Zeitleiste

  • 00:00:00 - 00:05:00

    The video discusses second-order differential equations in the context of an RLC circuit in electrical engineering. It explains the voltage drop across each component: resistors, capacitors, and inductors, all leading to Kirchhoff's Law. An AC current with EMF E(t) is considered, and the differential equation for the circuit is derived, involving parameters like L, R, C, E_0, and omega. This second-order linear differential equation with constant coefficients is what the audience has been studying.

  • 00:05:00 - 00:11:17

    The equation is then simplified by non-dimensionalizing using the natural frequency of the circuit, omega_0, which is 1/√(LC), and non-dimensional variables like tau for time and Q for charge. The resulting dimensionless equation has two parameters, alpha and beta, which simplifies the RLC circuit's analysis. By minimizing parameters, the physical understanding of the circuit dynamics improves. The video concludes, promising to explore two other applications of similar equations in different contexts, highlighting the universality of these mathematical concepts.

Mind Map

Mind Map

Häufig gestellte Fragen

  • What is an RLC circuit?

    An RLC circuit consists of a resistor (R), inductor (L), and capacitor (C) connected to an AC current.

  • What are the constitutive relations in a circuit?

    Constitutive relations describe the voltage drop across each circuit element: resistor, capacitor, and inductor.

  • How is the differential equation for an RLC circuit derived?

    The equation is derived using Kirchhoff's Law and the constitutive relations for each component.

  • What is Kirchhoff's Law in this context?

    Kirchhoff's Law states that the supplied voltage is equal to the sum of the voltage drops across circuit elements.

  • Why is non-dimensionalization used in the equation?

    To reduce the parameters in the equation, making it unit-less and easier to analyze.

  • What is the natural frequency in an RLC circuit?

    The natural frequency, without resistance, is defined as omega_0 = 1/(sqrt(LC)).

  • What parameters are used in the non-dimensional equation?

    Alpha and beta are the two non-dimensional parameters used in the simplified equation.

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Automatisches Blättern:
  • 00:00:10
    In the next three videos,
  • 00:00:13
    I want to show you some nice applications of  these second-order differential equations.
  • 00:00:22
    The first one is from electrical engineering,
  • 00:00:25
    is the RLC circuit: resistor, capacitor, inductor,
  • 00:00:32
    connected to an AC current with an EMF,
  • 00:00:37
    E of t. To write down the  differential equation here,
  • 00:00:42
    we need the constitutive relations  for these circuit elements.
  • 00:00:48
    So, I will just write them down.
  • 00:00:52
    So, the voltage drop across a resistor,
  • 00:00:58
    is equal to the current in the  circuit times the resistance.
  • 00:01:05
    The voltage drop across a capacitor,
  • 00:01:09
    is equal to the charge on the  capacitor divided by the capacitance C,
  • 00:01:16
    and the voltage drop across the inductor,
  • 00:01:23
    is equal to the derivative of  the current with respect to time,
  • 00:01:32
    times L. So, these define  essentially the circuit elements,
  • 00:01:40
    and then there's a charge on the  capacitor and a current in the circuit.
  • 00:01:44
    Those are related by the current is equal  to the time derivative of the charge.
  • 00:01:56
    How do we get a differential equation?
  • 00:01:59
    The differential equation  comes from Kirchhoff's Law,
  • 00:02:11
    which says that, the voltage  supplied by the battery,
  • 00:02:20
    or here it's the AC voltage,  epsilon of t, or E of t,
  • 00:02:27
    is equal to the voltage drop across each
  • 00:02:31
    of the circuit elements when they are in series.
  • 00:02:36
    So, here let me write that as V_L
  • 00:02:41
    plus V_R plus V_C.
  • 00:02:47
    And because these circuit  elements have derivatives in them,
  • 00:02:53
    this is a differential equation.
  • 00:02:58
    The AC current can be modeled as E of
  • 00:03:05
    t equals some amplitude E_0 times
  • 00:03:12
    cosine omega t.
  • 00:03:16
    So, that's a sinusoidal AC current.
  • 00:03:21
    So, putting this together,
  • 00:03:22
    what is the differential equation?
  • 00:03:25
    So, V_L is, L times di/dt,
  • 00:03:30
    but i is dq/dt.
  • 00:03:32
    So, we can write a differential  equation in terms of q.
  • 00:03:37
    So, V_L becomes L,
  • 00:03:40
    d squared q dt squared.
  • 00:03:46
    V_R is i times R, i is dq/dt.
  • 00:03:52
    So, plus V_R becomes plus R dq/dt,
  • 00:04:01
    and then VC is just q over C,
  • 00:04:05
    plus q over C. That's equal  to the AC current which
  • 00:04:11
    is E_0 times cosine omega t.
  • 00:04:20
    And that's our differential equation.
  • 00:04:23
    You should recognize this L, R,
  • 00:04:27
    C, E_0 are parameters that are constant.
  • 00:04:33
    Omega is a constant.
  • 00:04:35
    So, this is a second-order  linear differential equation
  • 00:04:41
    inhomogeneous with constant coefficients.
  • 00:04:46
    So, that's exactly what we've been studying.
  • 00:04:50
    This equation has a lot of parameters in it,
  • 00:04:53
    L, R, C, E_0, omega.
  • 00:04:57
    It pays to reduce the number of parameters.
  • 00:05:03
    If you divide through by L,
  • 00:05:06
    you see this term here has a q over LC.
  • 00:05:10
    If you remember our oscillator equation x
  • 00:05:14
    double dot plus omega_0  squared x equals some force,
  • 00:05:20
    then the term omega_0 squared is the one.
  • 00:05:24
    When this becomes d squared q dt squared,
  • 00:05:27
    the term omega_0 squared  is the term multiplying q,
  • 00:05:32
    and here it's equal to one over LC.
  • 00:05:35
    So, we can define an omega_0 equal  to one over the square root of LC,
  • 00:05:44
    and that's the natural frequency of this  RLC circuit when you don't have a resistor,
  • 00:05:51
    when there is no damping.
  • 00:05:54
    This has units of one over time.
  • 00:05:57
    So, we can non-dimensionalize this equation  to give us an equation with fewer parameters.
  • 00:06:06
    We can use this omega_0,
  • 00:06:08
    which has units of one over  time, to non-dimensionalize time.
  • 00:06:13
    So, we can define the non-dimensional time
  • 00:06:18
    tau equal to omega_0 times t. Then,
  • 00:06:23
    tau is unitless, t has units  of whatever units of time,
  • 00:06:30
    but now we've non-dimensionalized it.
  • 00:06:33
    So, when tau goes from zero to one,
  • 00:06:38
    t is going from zero to one over omega_0.
  • 00:06:42
    So, in some sense we've chosen one  over omega_0 as our unit of time.
  • 00:06:48
    Okay. Then, we can also non-dimensionalize q.
  • 00:06:52
    You can play with this equation and try to  determine how do you define the charge q,
  • 00:06:59
    so that in the nondimensional  equation this term will clear.
  • 00:07:06
    I won't do that here,
  • 00:07:07
    but I can write it down.
  • 00:07:09
    We define the dimensionless charge q by capital Q,
  • 00:07:14
    and that will be omega_0 squared L,
  • 00:07:19
    divided by E_0 times the charge q.
  • 00:07:24
    This has units of one over charge times charge,
  • 00:07:28
    and then we'll get a dimensionless charge.
  • 00:07:32
    You take these definitions,
  • 00:07:36
    and substitute it into the differential equation,
  • 00:07:39
    and you end up with a dimensionless equation,
  • 00:07:42
    which looks like d squared Q over d tau squared,
  • 00:07:51
    plus alpha dq/d tau, plus
  • 00:07:58
    Q equals cosine beta tau.
  • 00:08:07
    Where you have two nondimensional  parameters in this equation,
  • 00:08:12
    the alpha and the beta.
  • 00:08:15
    alpha is given by R over L omega_0,
  • 00:08:24
    and beta is given by omega divided by omega_0,
  • 00:08:32
    and these are dimensionless.
  • 00:08:35
    So, we went from a full dimensional  equation for the RLC circuit.
  • 00:08:41
    By redefining dimensionless variables,
  • 00:08:45
    we end up with an equation  that no longer has dimensions.
  • 00:08:50
    Each term is unit-less,
  • 00:08:53
    and we end up with two parameters, alpha and beta.
  • 00:08:58
    Okay. Let me review here.
  • 00:09:03
    We have the RLC circuit which is
  • 00:09:08
    a simple circuit from electrical  engineering with an AC current.
  • 00:09:13
    If we want to write down the  differential equation for this circuit,
  • 00:09:17
    we need the constitutive relations  for the circuit elements.
  • 00:09:22
    This defines what it means to be a resistor,
  • 00:09:25
    a capacitor, and an inductor.
  • 00:09:29
    There is a relationship between current  and charge through the derivative.
  • 00:09:34
    Then, we write down Kirchhoff's Law,
  • 00:09:36
    which is E of t is equal to the voltage drop
  • 00:09:40
    across the resistor plus the  voltage drop across the capacitor,
  • 00:09:44
    plus the voltage drop across the inductor.
  • 00:09:47
    And that gives us our second-order  differential equation.
  • 00:09:52
    We want to define something,
  • 00:09:54
    a parameter that has units of one over time.
  • 00:09:58
    We can do that by dividing through by L,
  • 00:10:01
    and picking up the one over LC times the q term.
  • 00:10:07
    One over LC is our omega_0 squared.
  • 00:10:11
    So, we can define the natural frequency  of the circuit without a resistor,
  • 00:10:17
    is one over root LC,
  • 00:10:20
    make a nondimensional time,  a nondimensional charge,
  • 00:10:24
    and we end up with this second-order  differential equation with two parameters.
  • 00:10:31
    It's interesting, but we're  going to see that we're going
  • 00:10:34
    to discuss two more applications,
  • 00:10:38
    completely different physical problems,
  • 00:10:41
    but we are going to end up with  exactly the same equation in the end.
  • 00:10:46
    So, that will be interesting.
  • 00:10:48
    The only thing different will be the  definitions of these dimensionless parameters.
  • 00:10:54
    So, let's continue with the next two videos,
  • 00:10:57
    and see two more applications.
  • 00:11:00
    I'm Jeff Chasnov, thanks for watching,
  • 00:11:02
    and I'll see you in the next video.
Tags
  • RLC circuit
  • second-order differential equations
  • Kirchhoff's Law
  • non-dimensionalization
  • natural frequency
  • electrical engineering
  • voltage drop
  • AC current
  • constitutive relations