Application of Second-Order Differential Equations

Study Case Application of Second-Order Differential Equations in RLC Electrical Circuit Analysis Michael Kurniawan (2018123456) University of Cincinnati, 2600 Clifton Ave, Cincinnati, OH 45221, United States kurniaml@mail.uc.edu Abstract Second-order ordinary differential equations (ODEs) are widely used to model dynamic phenomena in engineering, including RLC (Resistor, Inductor, Capacitor) electrical circuits. This paper discusses the application of second-order ODEs in analyzing the relationship between charge Q and current I in an RLC circuit with an external voltage source. The methods used include characteristic equations for homogeneous solutions, which represent natural damped oscillations, and indeterminate coefficient methods for nonhomogeneous solutions, which describe steady-state responses. The initial conditions Q(0)=0 and I(0)=0 are used to determine the integration constants, thereby obtaining specific solutions for charge and current as functions of time. The results show that the current and charge undergo damped oscillations toward steady-state values, proving the effectiveness of Second-Order PDB in solving real-world problems in electrical circuit analysis. Keywords: Ordinary Differential Equations, RLC Circuit, Damped Oscillations, SteadyState Response, Electrical Engineering. INTRODUCTION Background Second-order Ordinary Differential Equations (ODEs) are mathematical equations that relate a function to its first and second derivatives with respect to an independent variable, which is typically time or position. The general form of a second-order ODE is: 𝑦 ′′ + 𝑝(𝑥)𝑦 ′ + 𝑞(𝑥)𝑦 = 𝑟(𝑥) MICHAEL KURNIAWAN 1 Study Case Objective This paper aims to explain the working mechanism and application of second-order ordinary differential equations in the field of Electronics, specifically in RLC (Resistor, Inductor, and Capacitor) Electrical Circuits. DISCUSSION Direct Current Circuit (RLC) Consider a resistor (R ohms), inductor (L henries), and capacitor (C farads) arranged in series with a voltage source E(t) volts at time t. Example Problem Given: Determine the charge Q and current I as functions of time t for the RLC circuit with parameters: , 𝐿 = 0.02 H, 𝐶 = 2 × 10−4 F, 𝐸 = 12 V 𝑅 = 16 Ω assuming that the initial current and charge are zero when the switch is closed. Solution The applicable differential equation: 𝑑2𝑄 𝑑𝑄 1 𝐿 2 +𝑅 + 𝑄 = 𝐸(𝑡) 𝑑𝑡 𝑑𝑡 𝐶 With substitution of the values of R, L, and C: 0.02 𝑑2 𝑄 𝑑𝑄 + 16 + 5000𝑄 = 12 2 𝑑𝑡 𝑑𝑡 Multiply by 50 for convenience: 𝑄 ′′ + 800𝑄 ′ + 250,000𝑄 = 600 Homogeneous Solution Characteristic equation: MICHAEL KURNIAWAN 2 Study Case 𝑟 2 + 800𝑟 + 250,000 = 0 Roots of the equation: 𝑟= −800 ± √8002 − 4(1)(250,000) = −400 ± 300𝑖 2 Therefore, the homogeneous solution is: 𝑄ℎ (𝑡) = 𝑒 −400𝑡 (𝑐1 cos⁡ 300𝑡 + 𝑐2 sin⁡ 300𝑡) Nonhomogeneous Solution Using the method of undetermined coefficients, we obtain: 𝑄𝑝 = 2.4 × 10−3 Therefore, the general solution is: 𝑄(𝑡) = 𝑄ℎ (𝑡) + 𝑄𝑝 = 𝑒 −400𝑡 (𝑐1 cos⁡ 300𝑡 + 𝑐2 sin⁡ 300𝑡) + 2.4 × 10−3 With initial conditions Q(0)=0 and I(0)=0, the values of constants c₁ and c₂ are obtained. Determination of Constants (Special Solution) Initial conditions: 𝑄(0) = 𝑒 0 (𝑐1 ⋅ 1 + 𝑐2 ⋅ 0) + 2.4 × 10−3 = 0 ⇒ 𝑐1 = −2.4 × 10−3 𝐼(0) = 𝑄 ′ (0) = 0 Using the product rule: 𝑄 ′ (𝑡) = 𝑢′ 𝑣 + 𝑢𝑣 ′ where 𝑢 = 𝑒 −400𝑡 ⁡and v = c₁ cos 300t + c₂ sin 300t 𝑄 ′ (0) = −400𝑐1 + 300𝑐2 = 0 ⇒ 𝑐2 = −3.2 × 10−3 So the particular solution is: 𝑄(𝑡) = 𝑒 −400𝑡 (−2.4 × 10−3 cos⁡ 300𝑡 − 3.2 × 10−3 sin⁡ 300𝑡) + 2.4 × 10−3 Current Calculation The current I(t) is obtained by differentiating Q(t) with respect to time: 𝐼(𝑡) = 𝑄 ′ (𝑡) = 2𝑒 −400𝑡 sin⁡ 300𝑡 Thus, the current at time t is: 𝑰(𝒕) = 𝟐𝒆−𝟒𝟎𝟎𝒕 𝐬𝐢𝐧⁡ 𝟑𝟎𝟎𝒕 MICHAEL KURNIAWAN 3 Study Case CONCLUSION This paper discusses the application of Second-Order Ordinary Differential Equations (ODEs) in Electrical Engineering, specifically in RLC (Resistor, Inductor, and Capacitor) electrical circuits. Second-order ODEs are used to model the relationship between charge Q and current I in RLC circuits, taking into account the effects of resistance, inductance, capacitance, and external voltage sources. For example, the following parameters are used: ⁡𝐿 = 0.02 H, 𝐶 = 2 × 10−4 F, 𝐸 = 12 V ⁡𝑅 = 16 Ω The solution is obtained through the characteristic equation for homogeneous components and the method of undetermined coefficients for non-homogeneous components. • The homogeneous solution describes the natural response of the circuit, in the form of damped oscillations with a negative exponential factor e^(-400t) and an angular frequency of 300“ rad/s” . • The nonhomogeneous solution provides a steady-state response, in the form of a constant 2.4×10^(-3), which indicates the final charge of the capacitor. By applying the initial conditions Q(0)=0 and I(0)=0, the integration constants c_1 and c_2 are determined, so that the particular solutions for charge and current as functions of time are obtained: Charge:⁡𝑄(𝑡) = 𝑒 −400𝑡 (−2.4 × 10−3 cos⁡ 300𝑡 − 3.2 × 10−3 sin⁡ 300𝑡) + 2.4 × 10−3 Current ∶ 𝐼(𝑡) = 2𝑒 −400𝑡 sin⁡ 300𝑡 These results show that the current and charge undergo damped oscillations towards steady-state values. Thus, this paper confirms how the mathematical concept of Second Order PDB can be effectively applied to solve real problems in the field of Electrical Engineering, particularly in the analysis of RLC electrical circuits. MICHAEL KURNIAWAN 4 Study Case REFERENCE MICHAEL KURNIAWAN 5