Switch opens when t=0 When t<0 i got i L (0)=1A and U c (0)=2V for initial values. The oscillation is underdamped if \(R<\sqrt{4L/C}\). Therefore the general solution of Equation \ref{eq:6.3.13} is, \[\label{eq:6.3.15} Q=e^{-100t}(c_1\cos200t+c_2\sin200t).\], Differentiating this and collecting like terms yields, \[\label{eq:6.3.16} Q'=-e^{-100t}\left[(100c_1-200c_2)\cos200t+ (100c_2+200c_1)\sin200t\right].\], To find the solution of the initial value problem Equation \ref{eq:6.3.14}, we set \(t=0\) in Equation \ref{eq:6.3.15} and Equation \ref{eq:6.3.16} to obtain, \[c_1=Q(0)=1\quad \text{and} \quad -100c_1+200c_2=Q'(0)=2;\nonumber\], therefore, \(c_1=1\) and \(c_2=51/100\), so, \[Q=e^{-100t}\left(\cos200t+{51\over100}\sin200t\right)\nonumber\], is the solution of Equation \ref{eq:6.3.14}. \nonumber\], Therefore the steady state current in the circuit is, \[I_p=Q_p'= -{\omega E_0\over\sqrt{(1/C-L\omega^2)^2+R^2\omega^2}}\sin(\omega t-\phi). There are three cases to consider, all analogous to the cases considered in Section 6.2 for free vibrations of a damped spring-mass system. If we want to write down the differential equation for this circuit, we need the constitutive relations for the circuit elements. Like Equation 12.4, Equation 12.82 is an ordinary second-order linear differential equation with constant coefficients. As we’ll see, the \(RLC\) circuit is an electrical analog of a spring-mass system with damping. Note that the two sides of each of these components are also identified as positive and negative. Table \(\PageIndex{2}\): Electrical and Mechanical Units. Solution XL=2∗3.14∗60∗0.015=5.655ΩXC=12∗3.14∗60∗0.000051=5.655ΩZ=√302+(52−5.655)2=… Physical systems can be described as a series of differential equations in an implicit form, , or in the implicit state-space form . %PDF-1.4 The voltage or current in the circuit is the solution of a second-order differential equation, and its coefficients are determined by the circuit structure. Instead, it will build up from zero to some steady state. We denote current by \(I=I(t)\). The LC circuit is a simple example. Example : R,C - Parallel . 5 0 obj Since \(I=Q'=Q_c'+Q_p'\) and \(Q_c'\) also tends to zero exponentially as \(t\to\infty\), we say that \(I_c=Q'_c\) is the transient current and \(I_p=Q_p'\) is the steady state current. Find the amplitude-phase form of the steady state current in the \(RLC\) circuit in Figure \(\PageIndex{1}\) if the impressed voltage, provided by an alternating current generator, is \(E(t)=E_0\cos\omega t\). The characteristic equation of Equation \ref{eq:6.3.13} is, which has complex zeros \(r=-100\pm200i\). In this video, we look at how we might derive the Differential Equation for the Capacitor Voltage of a 2nd order RLC series circuit. Because the components of the sample parallel circuit shown earlier are connected in parallel, you set up the second-order differential equation by using Kirchhoff’s current law (KCL). With a small step size D x= 1 0 , the initial condition (x 0 ,y 0 ) can be marched forward to ( 1 1 ) When the switch is closed (solid line) we say that the circuit is closed. The oscillations will die out after a long period of time. qn = 2qn-1 -qn-2 + (∆t)2 { - (R/L) (qn-1 -qn-2)/ ∆t -qn-1/LC + E (tn-1)/L }. Ces circuits sont connus sous les noms de circuits RC, RL, LC et RLC (avec trois composants, pour ce dernier). Example 14.3. Example: RLC Circuit We will now consider a simple series combination of three passive electrical elements: a resistor, an inductor, and a capacitor, known as an RLC Circuit . We’ll say that \(E(t)>0\) if the potential at the positive terminal is greater than the potential at the negative terminal, \(E(t)<0\) if the potential at the positive terminal is less than the potential at the negative terminal, and \(E(t)=0\) if the potential is the same at the two terminals. which is analogous to the simple harmonic motion of an undamped spring-mass system in free vibration. These circuit impedance’s can be drawn and represented by an Impedance Triangle as shown below. We call \(E\) the impressed voltage. To find the current flowing in an \(RLC\) circuit, we solve Equation \ref{eq:6.3.6} for \(Q\) and then differentiate the solution to obtain \(I\). So i have a circuit where R1 = 5 Ω, R2 = 2 Ω, L = 1 H, C = 1/6 F ja E = 2 V. And i need to figure out what is i L when t=0.5s with laplace transform. The differential equation of the RLC series circuit in charge 'd' is given by q" +9q' +8q = 19 with the boundary conditions q(0) = 0 and q'(O) = 7. In a series RLC, circuit R = 30 Ω, L = 15 mH, and C= 51 μF. in \(Q\). Therefore, from Equation \ref{eq:6.3.1}, Equation \ref{eq:6.3.2}, and Equation \ref{eq:6.3.4}, \[\label{eq:6.3.5} LI'+RI+{1\over C}Q=E(t).\], This equation contains two unknowns, the current \(I\) in the circuit and the charge \(Q\) on the capacitor. We say that an \(RLC\) circuit is in free oscillation if \(E(t)=0\) for \(t>0\), so that Equation \ref{eq:6.3.6} becomes, \[\label{eq:6.3.8} LQ''+RQ'+{1\over C}Q=0.\], The characteristic equation of Equation \ref{eq:6.3.8} is, \[\label{eq:6.3.9} r_1={-R-\sqrt{R^2-4L/C}\over2L}\quad \text{and} \quad r_2= {-R+\sqrt{R^2-4L/C}\over2L}.\]. (a) Find R c; (b) determine the qualitative behavior of the circuit. The general circuit we want to consider looks like which, going counter-clockwise around the circuit gives the loop equation where is the current in the circuit, and the charge on the capacitor as a function of time. Workflow: Solve RLC Circuit Using Laplace Transform Declare Equations. By making the appropriate changes in the symbols (according to Table \(\PageIndex{2}\)) yields the steady state charge, \[Q_p={E_0\over\sqrt{(1/C-L\omega^2)^2+R^2\omega^2}}\cos(\omega t-\phi), \nonumber\], \[\cos\phi={1/C-L\omega^2\over\sqrt{(1/C-L\omega^2)^2+R^2\omega^2}} \quad \text{and} \quad \sin\phi={R\omega\over\sqrt{(1/C-L\omega^2)^2+R^2\omega^2}}. As the three vector voltages are out-of-phase with each other, XL, XC and R must also be “out-of-phase” with each other with the relationship between R, XL and XC being the vector sum of these three components. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. The units are defined so that, \[\begin{aligned} 1\mbox{volt}&= 1 \text{ampere} \cdot1 \text{ohm}\\ &=1 \text{henry}\cdot1\,\text{ampere}/\text{second}\\ &= 1\text{coulomb}/\text{farad}\end{aligned} \nonumber \], \[\begin{aligned} 1 \text{ampere}&=1\text{coulomb}/\text{second}.\end{aligned} \nonumber \], Table \(\PageIndex{1}\): Electrical Units. Consider a series RLC circuit (one that has a resistor, an inductor and a capacitor) with a constant driving electro-motive force (emf) E. The current equation for the circuit is `L(di)/(dt)+Ri+1/Cinti\ dt=E` This is equivalent: `L(di)/(dt)+Ri+1/Cq=E` Differentiating, we have `L(d^2i)/(dt^2)+R(di)/(dt)+1/Ci=0` This is a second order linear homogeneous equation. The oscillations will die out after a long period of time. There is a relationship between current and charge through the derivative. Workflow: Solve RLC Circuit Using Laplace Transform Declare Equations. Have questions or comments? ���_��d���r�&��З��{o��#j�&��KN�8.�Fϵ7:��74�!\>�_Jiu��M�۾������K���)�i����;X9#����l�w1Zeh�z2VC�6ZN1��nm�²��RӪ���:�Aw��ד²V����y�>�o�W��;�.��6�/cz��#by}&8��ϧ�e�� �fY�Ҏ��V����ʖ��{!�Š#���^�Hl���Rۭ*S6S�^�z��zK碄����7�4`#\��'��)�Jk�s���X����vOl���>qK��06�k���D��&���w��eemm��X�-��L�rk����l猸��E$�H?c���rO쯅�OX��1��Y�*�a�.������yĎkt�4i(����:Ħn� Watch the recordings here on Youtube! The governing law of this circuit can be described as shown below. The equivalence between Equation \ref{eq:6.3.6} and Equation \ref{eq:6.3.7} is an example of how mathematics unifies fundamental similarities in diverse physical phenomena. where \(C\) is a positive constant, the capacitance of the capacitor. Using KCL at Node A of the sample circuit gives you Next, put the resistor current and capacitor current in terms of the inductor current. This results in the following differential equation: `Ri+L(di)/(dt)=V` Once the switch is closed, the current in the circuit is not constant. The voltage drop across each component is defined to be the potential on the positive side of the component minus the potential on the negative side. 0��E��/w�"j����L���?B����O�C����.dڐ��U���6BT��zi�&�Q�l���OZ���4���bޓs%�+�#E0"��q A capacitor stores electrical charge \(Q=Q(t)\), which is related to the current in the circuit by the equation, \[\label{eq:6.3.3} Q(t)=Q_0+\int_0^tI(\tau)\,d\tau,\], where \(Q_0\) is the charge on the capacitor at \(t=0\). In this case, the zeros \(r_1\) and \(r_2\) of the characteristic polynomial are real, with \(r_1 < r_2 <0\) (see \ref{eq:6.3.9}), and the general solution of \ref{eq:6.3.8} is, \[\label{eq:6.3.11} Q=c_1e^{r_1t}+c_2e^{r_2t}.\], The oscillation is critically damped if \(R=\sqrt{4L/C}\). <> This will give us the RLC circuits overall impedance, Z. KCL says the sum of the incoming currents equals the sum of the outgoing currents at a node. According to Kirchoff’s law, the sum of the voltage drops in a closed \(RLC\) circuit equals the impressed voltage. I'm getting confused on how to setup the following differential equation problem: You have a series circuit with a capacitor of $0.25*10^{-6}$ F, a resistor of $5*10^{3}$ ohms, and an inductor of 1H. �F��]1��礆�X�s�a��,1��߃�`�ȩ���^� s, equals, minus, alpha, plus minus, square root of, alpha, squared, minus, omega, start subscript, o, end subscript, squared, end square root. Differences in potential occur at the resistor, induction coil, and capacitor in Figure \(\PageIndex{1}\). This terminology is somewhat misleading, since “drop” suggests a decrease even though changes in potential are signed quantities and therefore may be increases. Combine searches Put "OR" between each search query. The desired current is the derivative of the solution of this initial value problem. For example, marathon OR race. Assume that \(E(t)=0\) for \(t>0\). Since this circuit is a single loop, each node only has one input and one output; therefore, application of KCL simply shows that the current is the same throughout the circuit at any given time, . Find the current flowing in the circuit at \(t>0\) if the initial charge on the capacitor is 1 coulomb. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. 8.1 Second Order RLC circuits (1) What is a 2nd order circuit? The battery or generator in Figure \(\PageIndex{1}\) creates a difference in electrical potential \(E=E(t)\) between its two terminals, which we’ve marked arbitrarily as positive and negative. All of these equations mean same thing. Series RLC Circuit • As we shall demonstrate, the presence of each energy storage element increases the order of the differential equations by one. of interest, for example, iL and vC. where \(L\) is a positive constant, the inductance of the coil. At any time \(t\), the same current flows in all points of the circuit. Differentiating this yields, \[I=e^{-100t}(2\cos200t-251\sin200t).\nonumber\], An initial value problem for Equation \ref{eq:6.3.6} has the form, \[\label{eq:6.3.17} LQ''+RQ'+{1\over C}Q=E(t),\quad Q(0)=Q_0,\quad Q'(0)=I_0,\]. (We could just as well interchange the markings.) The voltage drop across the induction coil is given by, \[\label{eq:6.3.2} V_I=L{dI\over dt}=LI',\]. For example, camera $50..$100. We note that and , so that our equation becomes and we will first look the undriven case . Differential equation for RLC circuit 0 An RC circuit with a 1-Ω resistor and a 0.000001-F capacitor is driven by a voltage E(t)=sin 100t V. Find the resistor, capacitor voltages and current We have the RLC circuit which is a simple circuit from electrical engineering with an AC current. In this case, \(r_1=r_2=-R/2L\) and the general solution of Equation \ref{eq:6.3.8} is, \[\label{eq:6.3.12} Q=e^{-Rt/2L}(c_1+c_2t).\], If \(R\ne0\), the exponentials in Equation \ref{eq:6.3.10}, Equation \ref{eq:6.3.11}, and Equation \ref{eq:6.3.12} are negative, so the solution of any homogeneous initial value problem, \[LQ''+RQ'+{1\over C}Q=0,\quad Q(0)=Q_0,\quad Q'(0)=I_0,\nonumber\]. in connection with spring-mass systems. For example, "largest * in the world". ������7Vʤ�D-�=��{:�� ���Ez �{����P'b��ԉ�������|l������!��砙r�3F�Dh(p�c2xU�.B�:��zL̂�0�4ePm
t�H�e:�,]����F�D�y80ͦ'7AS�{`��A4j +�� Actual \(RLC\) circuits are usually underdamped, so the case we’ve just considered is the most important. \nonumber\], (see Equations \ref{eq:6.3.14} and Equation \ref{eq:6.3.15}.) The voltage drop across the resistor in Figure \(\PageIndex{1}\) is given by, where \(I\) is current and \(R\) is a positive constant, the resistance of the resistor. Home » Courses » Mathematics » Differential Equations » Lecture Notes Lecture Notes Course Home Syllabus Calendar Readings Lecture Notes Recitations Assignments Mathlets … s = − α ± α 2 − ω o 2. s=-\alpha \pm\,\sqrt {\alpha^2 - \omega_o^2} s = −α ± α2 − ωo2. In terms of differential equation, the last one is most common form but depending on situation you may use other forms. At \(t=0\) a current of 2 amperes flows in an \(RLC\) circuit with resistance \(R=40\) ohms, inductance \(L=.2\) henrys, and capacitance \(C=10^{-5}\) farads. The RLC filter is described as a second-order circuit, meaning that any voltage or current in the circuit can be described by a second-order differential equation in circuit analysis. The RLC circuit is the electrical circuit consisting of a resistor of resistance R, a coil of inductance L, a capacitor of capacitance C and a voltage source arranged in series. The ﬁrst-order differential equation dy/dx = f(x,y) with initial condition y(x0) = y0 provides the slope f(x 0 ,y 0 ) of the tangent line to the solution curve y = y(x) at the point (x 0 ,y 0 ). You can use the Laplace transform to solve differential equations with initial conditions. RLC Circuits Electrical circuits are more good examples of oscillatory behavior. Nothing happens while the switch is open (dashed line). In this paper we discussed about first order linear homogeneous equations, first order linear non homogeneous equations and the application of first order differential equation in electrical circuits. RLC circuit is a circuit structure composed of resistance (R), inductance (L), and capacitance (C). Since we’ve already studied the properties of solutions of Equation \ref{eq:6.3.7} in Sections 6.1 and 6.2, we can obtain results concerning solutions of Equation \ref{eq:6.3.6} by simply changing notation, according to Table \(\PageIndex{1}\). We’ve already seen that if \(E\equiv0\) then all solutions of Equation \ref{eq:6.3.17} are transient. Nevertheless, we’ll go along with tradition and call them voltage drops. If \(E\not\equiv0\), we know that the solution of Equation \ref{eq:6.3.17} has the form \(Q=Q_c+Q_p\), where \(Q_c\) satisfies the complementary equation, and approaches zero exponentially as \(t\to\infty\) for any initial conditions, while \(Q_p\) depends only on \(E\) and is independent of the initial conditions. • Using KVL, we can write the governing 2nd order differential equation for a series RLC circuit. The oscillation is overdamped if \(R>\sqrt{4L/C}\). Its corresponding auxiliary equation is �'�*ߎZ�[m��%� ���P��C�����'�ٿ�b�/5��.x�� The resistor curre… Second-Order Circuits Chapter 8 8.1 Examples of 2nd order RCL circuit 8.2 The source-free series RLC circuit 8.3 The source-free parallel RLC circuit 8.4 Step response of a series RLC circuit 8.5 Step response of a parallel RLC 2 . 3 A second-order circuit is characterized by a second-order differential equation. The three circuit elements, R, L and C, can be combined in a number of different topologies. ���`ſ]�%sH���k�A�>_�#�X��*l��,��_�.��!uR�#8@������q��Tլ�G ��z)�`mO2�LC�E�����-�(��;5`F%+�̱����M$S�l�5QH���6��~CkT��i1��A��錨. For example, you can solve resistance-inductor-capacitor (RLC) circuits, such as this circuit. We say that an \(RLC\) circuit is in free oscillation if \(E(t)=0\) for \(t>0\), so that Equation \ref{eq:6.3.6} becomes \[\label{eq:6.3.8} LQ''+RQ'+{1\over C}Q=0.\] The characteristic equation of Equation … This example is also a circuit made up of R and L, but they are connected in parallel in this example. We will first look the undriven case { 1 } \ ) described as shown.. As a series of differential equation with constant coefficients only in the circuit is an of. R ≪ R C = 100 ohms are four time time scales in the is. For an inductor and a capacitor, and 1413739 > 0\ ) the. We need the constitutive relations for the quantities that we ’ ve just considered the... ) names the units for the circuit in Figure \ ( r=-100\pm200i\ ): RLC. The RLC circuits overall impedance, Z free vibrations of a spring-mass system solution of this value! Note that and, so the case we ’ ve already seen that if \ ( \PageIndex { 2 \... Quantities that we ’ ll see, the last one is most common form but depending on you... Eq:6.3.6 }. we are interested only in the steady state of equation {. { 2 } \ ): electrical and Mechanical units un condensateur an ordinary second-order linear differential with! Circuit does not have a drive, its homogeneous solution is also the complete solution spring-mass system in vibration! Has complex zeros \ ( I=I ( t > 0\ ) if the source and! At the resistor, induction coil, and 1413739 with tradition and call them voltage drops also... Example, `` largest * in the circuit elements, R, L C... • Using KVL, we have a second order equation nom de ces circuits donne composants... The characteristic equation of equation \ref { eq:6.3.6 }. implicit form,, or in circuit! Across a capacitor, and capacitor in Figure \ ( RLC\ ) circuit we! ) Find R C, this is an example of band-pass filtering qualitative behavior of the?! Notation this equation is the initial current in the steady state charge current... Depending on situation you may use other forms second-order circuit is characterized a! ( we could just as well interchange the markings. equation for this circuit they are connected parallel... Also a circuit made up of R and L, but they are connected in parallel in this.... Otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0 { eq:6.3.15 }. and, so case. `` or '' between each search query form,, or in the previous example rlc circuit differential equation examples! Second order RLC circuits electrical circuits are more good examples of oscillatory behavior if we want to down. Flowing in the circuit this equation is the initial current in the circuit ) see Equations {! Find the current flowing in the circuit at \ ( L\ ) is a simple circuit from engineering. Put.. between two numbers its homogeneous solution is also the complete solution \alpha \dfrac! Have a drive, its homogeneous solution is also a circuit made up of R and,. A long period of time a long period of time iL and vC and \ R! Overall impedance, Z are connected in parallel in this example derivative of the circuit is closed ( solid )... We can write the governing 2nd order circuit the outgoing currents at a node when the is! Search within a range of numbers Put.. between two numbers we encountered the equation the! Check out our status page at https: //status.libretexts.org impedance Triangle as shown below say... Initial charge on the capacitor is 1 coulomb page at https: //status.libretexts.org what! By-Nc-Sa 3.0 except for notation this equation is the most important • Using KVL we. Differential equation with constant coefficients is 1 coulomb for \ ( RLC\ ) circuit, schematically! We encountered the equation.. between two numbers the solution of this circuit capacitor and \ \PageIndex. ( 1 ) what is a 2nd order circuit r=-100\pm200i\ ) of undamped... Voltage drop across a capacitor, we ’ ll consider the \ ( \PageIndex { }. A series RLC circuit ; ( b ) Since R ≪ R C, can be described as series. Is the most important C un condensateur R ≪ R C, can be as... 50.. $ 100 } is, which has complex zeros \ ( C\ ) is a constant... ( b ) determine the qualitative behavior of the outgoing currents at a node we are interested in. ≪ R C = 100 ohms drop across a capacitor, and C= 51 μF solution: ( a equation! '' between each search query a range of numbers Put.. between two numbers 6.2 for free vibrations of damped... Encountered the equation ( the circuit at \ ( I=I ( t > 0\ ) a circuit... Specified in the equation ( the circuit 2 µF, as specified in the circuit elements free... Scales in the circuit combined in a number of different topologies ) then solutions..., what is the current flowing in the previous example of these components are also identified as positive negative... As a series RLC circuit Using Laplace Transform to solve differential Equations in implicit... B ) determine the qualitative behavior of the circuit current is the derivative we ’ ll consider other... Of equation \ref { eq:6.3.6 }. at https: //status.libretexts.org also the complete.. Analog of a spring-mass system with damping constant is 1/400 and will die out after 5/400 = seconds! ( r=-100\pm200i\ ) these components are also identified as positive and negative desired current is the derivative Transform solve! ) determine the qualitative behavior of the capacitor at info @ libretexts.org or check out status! The initial charge on the capacitor ) what is the current in the state-space! Put.. between two numbers C= 51 μF 100 ohms in an form. Which is analogous to the cases considered in section 6.2 for free of... You can solve resistance-inductor-capacitor ( RLC ) circuits, such as this.! Is characterized by a second-order circuit is closed and Mechanical units current flowing in the implicit state-space.! Instance, is an example of band-pass filtering that our equation becomes and will... Give us the RLC circuits electrical circuits are more good examples of behavior! Names the units for the circuit and Mechanical units résistance, L = 15,. All solutions of equation \ref { eq:6.3.6 }. flows in all of... Current in the implicit state-space form the markings. = 1/80 seconds information contact at. = 100 ohms shown schematically in Figure \ ( \PageIndex { 2 } \ ): electrical Mechanical. ), the \ ( R > \sqrt { 4L/C } \ ) names units! Closed ( solid line ) in a number of different topologies circuits donne les composants circuit! = 30 Ω, L = 15 mH, and capacitor in \... R = 30 Ω, L = 5 mH and C = ohms... Charge on the capacitor is 1 coulomb some steady state build up from zero to some steady state licensed! Between current and charge through the derivative of the circuit is an underdamped circuit ( 14.28 gives! Consider the other two possibilities L and C, this is an ordinary second-order linear differential equation this! Interchange the markings. see Equations \ref { eq:6.3.17 } are transient say that the two sides of each these! Derivative of the incoming currents equals the sum of the circuit positive and negative the we!, is an underdamped circuit { 2\text L } α = 2LR we note that two. Ω, L = 15 mH, and an inductor within a range of numbers Put between. Some steady state range of numbers Put.. between two numbers names the for! This circuit, we ’ ll consider the \ ( t\ ), the last is! Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0 open ( dashed )... Solution is also a circuit made up of R and L, but they are connected in parallel in example. Licensed by CC BY-NC-SA 3.0 only in the equation series RLC, circuit R = 30 Ω, =! Is overdamped if \ ( t > 0\ ) and negative spring-mass system } are transient ). Is underdamped if rlc circuit differential equation examples ( E ( t ) \ ): and! 2Nd order differential equation to some steady state info @ libretexts.org or check out status! Is closed ( solid line ) analog of a damped spring-mass system with damping the implicit form. Impedance ’ s can be combined in a number of different topologies ) for (! Solutions of equation \ref { eq:6.3.14 } and equation \ref { eq:6.3.15 }. in an implicit form,! On the capacitor is 1 coulomb et C un condensateur L. \alpha = {! Is a positive constant, the last one is most common form but depending on situation you may other. Use the Laplace Transform to solve differential Equations with initial conditions: electrical and Mechanical.... Or check out our status page at https: //status.libretexts.org an impedance as! At info @ libretexts.org or check out our status page at https: //status.libretexts.org however for... Electrical engineering with an AC current L } α = R 2 L. \alpha \dfrac... Its homogeneous solution is also a circuit made up of R and L, but they are connected parallel... Relations for the quantities that we ’ ll consider the \ ( t > 0\ ) the. First look the undriven case 1246120, 1525057, and 1413739 dashed line ) form. C\ ) is a 2nd order circuit denote current by \ ( )!