Analysis of Results and Discussion on LC and RLC

Study Case Analysis of Results and Discussion on LC and RLC Series Circuits Michael Kurniawan (2018123456) University of Cincinnati, 2600 Clifton Ave, Cincinnati, OH 45221, United States kurniaml@mail.uc.edu Abstract Physical phenomena in electrical systems can be represented through mathematical models that enable analytical solutions to be determined. Second-order electrical circuits are electrical systems that can be modeled using second-order differential equations. Examples include LC series circuits and RLC series circuits. This study discusses the process of analyzing and solving mathematical models for both types of circuits using a second-order differential equation approach, both for homogeneous and non-homogeneous cases. Keywords: mathematical model, second-order differential equation, LC circuit, RLC circuit. INTRODUCTION Mathematics is a discipline that plays an important role in representing real phenomena in mathematical models. Such modeling is often used to explain various physical phenomena, including phenomena in electrical systems. In principle, electrical circuits can be modeled using second-order differential equations. In general, second-order differential equation models can be classified into the following four types: 1. Type 1: 𝑑2 𝑦 = 𝑓(𝑥) 𝑑𝑥 2 MICHAEL KURNIAWAN 1 Study Case 2. Type 2: 𝑑2 𝑦 𝑑𝑦 = 𝑓(𝑥) 2 𝑑𝑥 𝑑𝑥 3. Type 3: 𝑎 4. 𝑑2𝑦 𝑑𝑦 +𝑏 +𝑐𝑦 = 0 2 𝑑𝑥 𝑑𝑥 Type 4: 𝑎 𝑑2𝑦 𝑑𝑦 +𝑏 + 𝑐 𝑦 = 𝑓(𝑥) 2 𝑑𝑥 𝑑𝑥 LC Circuit An LC circuit is a simple electronic circuit consisting of a DC voltage source (Vd), an inductor (L), and a capacitor (C), arranged in series. Since there are no resistors in this configuration, the LC system is called an undamped circuit, i.e., a circuit without damping. This can be seen from the form of the second-order differential equation, where the damping ratio is zero. Figure 1. Undamped Series-Resonant Circuit (LC Circuit) An undamped LC circuit (Undamped Series-Resonant Circuit) is shown in Figure 1, with input voltage V_d at time t_0. The initial conditions of the system are determined by the state variables: the current in the inductor i_L and the voltage across the capacitor v_C, each with values I_L0 and V_C0 at time t_0. RLC Circuit A series RLC circuit in an alternating current (AC) system consists of a resistor (R), an inductor (L), and a capacitor (C) connected in series with an AC voltage source. The resistance caused by the resistor is called resistance, while the resistance caused by the inductor is called inductive reactance (X_L), and that caused by the capacitor is called capacitive reactance (X_C). These three types of resistance, when combined, form the total impedance (Z) of the system. MICHAEL KURNIAWAN 2 Study Case Figure 2. RLC Series Circuit Kirchhoff's Laws Most simple electrical circuits (such as in Figure 1) cannot be analyzed simply by replacing the series-parallel combination of resistors, especially in systems with many elements. Therefore, Kirchhoff's Laws are used as the basis for circuit analysis. Figure 3. Example of a Simple Electrical Circuit 1. Kirchhoff's First Law (Current Law) Kirchhoff's first law states that the sum of the electric currents entering a junction point is equal to the sum of the currents leaving that point. ∑Iin =∑Iout Figure 4. Illustration of Incoming and Outgoing Currents at a Junction Point 2. Kirchhoff's Second Law (Voltage Law) Kirchhoff's second law states that in any closed circuit, the algebraic sum of all potential differences (voltages) must be zero. This law is also known as the loop rule, and is a direct application of the law of conservation of energy. If a charge Q is at a point with potential V, then its energy is QV. When the charge moves around the loop, it can gain or lose energy when passing through resistors, MICHAEL KURNIAWAN 3 Study Case voltage sources, or other elements. However, after returning to the starting point, its total energy returns to QV. Mathematically, this law is expressed as: Figure 5. Circuit with Two Batteries and Three External Resistors METHOD The method used in this study is a literature study, reviewing textbooks and online scientific journals related to second-order differential equations to obtain homogeneous and non-homogeneous solutions in series LC and series RLC circuits. The research steps include: 1. Determining a mathematical model in the form of a second-order differential equation. 2. Deriving formulas to obtain homogeneous and non-homogeneous solutions. 3. Applying the model in the analysis of second-order electrical circuits. 4. Presenting the general form of mathematical models for LC and RLC circuits. 5. Analyzing the results of solving both types of solutions. RESULTS AND DISCUSSION Series LC Circuit A series LC circuit with a constant voltage source of E volts can be illustrated as shown in Figure 6. Based on Kirchhoff's Voltage Law, the mathematical model can be written as follows: 𝑉𝐿 + 𝑉𝐶 = 𝐸 MICHAEL KURNIAWAN 4 Study Case where: 𝑑𝐼 • V_L= voltage across inductor L, i.e., 𝑉𝐿 = 𝐿 𝑑𝑡 • V_C = voltage across capacitor C, i.e., 𝑉𝐶 = 𝐶 ∫ 𝐼 𝑑𝑡 Since current 𝐼 = 1 𝑑𝑄 𝑑𝑡 (where Q is charge in coulombs), the equation can be written as: 𝐿 𝑑𝐼 1 + ∫ 𝐼 𝑑𝑡 = 𝐸 𝑑𝑡 𝐶 To eliminate the integral sign, both sides of the equation are differentiated with respect to time t: 𝑑2 𝐼 1 𝑑𝐸 𝐿 2+ 𝐼= 𝑑𝑡 𝐶 𝑑𝑡 or in terms of charge Q(t): 𝑑2𝑄 1 𝐿 2 + 𝑄=𝐸 𝑑𝑡 𝐶 Case A — No Voltage Source (E=0) If E = 0, then the equation becomes: 𝐿 𝑑2𝐼 1 + 𝐼=0 𝑑𝑡 2 𝐶 𝑑2𝐼 1 + 𝐼=0 2 𝑑𝑡 𝐿𝐶 The characteristic equation is: 𝑟2 + 1 =0 𝐿𝐶 Thus, we obtain the root: 𝑟1,2 = ±𝑖√ 1 𝐿𝐶 Therefore, the general solution for current I(t) is: 𝐼(𝑡) = 𝐴cos⁡(√ 1 1 𝑡) + 𝐵sin⁡(√ 𝑡) 𝐿𝐶 𝐿𝐶 This solution describes simple harmonic oscillation without damping (undamped oscillation). MICHAEL KURNIAWAN 5 Study Case Case B — Constant Voltage Source (E = “constant”) The model equation becomes: 𝐿 𝑑2𝑄 1 + 𝑄=𝐸 𝑑𝑡 2 𝐶 or 𝑑2 𝑄 1 𝐸 + 𝑄= 2 𝑑𝑡 𝐿𝐶 𝐿 General solution: 𝑄(𝑡) = 𝐴cos⁡(√ 1 1 𝑡) + 𝐵sin⁡(√ 𝑡) + 𝐸𝐶 𝐿𝐶 𝐿𝐶 Case C — Sinusoidal Voltage Source (E = E₀ cos ωt) Model equation: 𝐿 𝑑2 𝑄 1 + 𝑄 = 𝐸0 cos⁡ 𝜔𝑡 𝑑𝑡 2 𝐶 𝑑2 𝑄 1 𝐸0 + 𝑄 = cos⁡ 𝜔𝑡 2 𝑑𝑡 𝐿𝐶 𝐿 The solution consists of homogeneous and inhomogeneous parts: 𝑄ℎ (𝑡) = 𝐶1 cos⁡(𝜔0 𝑡 − 𝜃), 𝑄𝑝 (𝑡) = 𝐸0 cos⁡ 𝜔𝑡 𝐿(𝜔02 − 𝜔 2 ) 1 with 𝜔02 = 𝐿𝐶. 1 Case D — Resonance (𝜔 = 𝜔0 = √𝐿𝐶) When the source frequency is equal to the natural frequency of the circuit, resonance occurs. The model equation is: 𝑑2 𝑄 1 𝐿 2 + 𝑄 = 𝐸0 cos⁡ 𝜔𝑡 𝑑𝑡 𝐶 𝑑2 𝑄 𝐸0 2 + 𝜔 𝑄 = cos⁡ 𝜔𝑡 𝑑𝑡 2 𝐿 Homogeneous solution: 𝑄ℎ (𝑡) = 𝐴cos⁡ 𝜔𝑡 + 𝐵sin⁡ 𝜔𝑡 Particular solution: 𝑄𝑝 (𝑡) = 𝑡(𝐾cos⁡ 𝜔𝑡 + 𝑀sin⁡ 𝜔𝑡) Then the complete solution: 𝑄(𝑡) = 𝐶cos⁡(𝜔𝑡 − 𝜃) + 2𝐸0 𝜔0 𝐿 𝑡sin⁡ 𝜔𝑡 MICHAEL KURNIAWAN 6 Study Case Table 1. Parameters and Solutions for Series LC Circuits Series RLC Circuit For a series RLC circuit with voltage source E(t), Kirchhoff's law gives: 𝑉𝑅 + 𝑉𝐿 + 𝑉𝐶 = 𝐸 where: • • 𝑉𝑅 = 𝑅𝐼 𝑑𝐼 𝑉𝐿 = 𝐿 𝑑𝑡 • 𝑉𝐶 = 𝐶 ∫ 𝐼 𝑑𝑡 Since 𝐼 = 1 𝑑𝑄 𝑑𝑡 , then: 𝑅 𝑑𝑄 𝑑2 𝑄 1 +𝐿 2 + 𝑄 = 𝐸 𝑑𝑡 𝑑𝑡 𝐶 Case A — Without Source (E=0) 𝐿 𝑑2 𝐼 𝑑𝐼 1 +𝑅 + 𝐼 =0 2 𝑑𝑡 𝑑𝑡 𝐶 Characteristic equation: 𝐿𝑟 2 + 𝑅𝑟 + MICHAEL KURNIAWAN 7 1 =0 𝐶 Study Case Roots: 𝑟1,2 = −𝑅 ± √𝑅 2 − 4𝐿/𝐶 2𝐿 Three possible conditions: 1. Overdamped (𝑅 2 > 4𝐿/𝐶) 𝐼(𝑡) = 𝐶1 𝑒 𝑟1𝑡 + 𝐶2 𝑒 𝑟2𝑡 2. Critically damped (𝑅 2 = 4𝐿/𝐶) 𝐼(𝑡) = (𝐶1 + 𝐶2 𝑡)𝑒 𝑟𝑡 3. Underdamped (𝑅 2 < 4𝐿/𝐶) 𝐼(𝑡) = 𝑒 𝛼𝑡 [𝐴cos⁡(𝛽𝑡) + 𝐵sin⁡(𝛽𝑡)] 𝑅 1 𝑅2 with 𝛼 = − 2𝐿, 𝛽 = √𝐿𝐶 − 4𝐿2 Case B — Sinusoidal Source (E = E₀ cos(ωt)) 𝑑2𝐼 𝑑𝐼 1 𝐿 2 + 𝑅 + 𝐼 = 𝐸0 cos⁡ 𝜔𝑡 𝑑𝑡 𝑑𝑡 𝐶 Particular solution (steady-state): 𝐼𝑝 (𝑡) = 𝐸0 √𝑅 2 + (𝑋𝐿 − 𝑋𝐶 )2 cos⁡(𝜔𝑡 − 𝜙) where: 𝑋𝐿 = 𝜔𝐿, 𝑋𝐶 = 1 𝑋𝐿 − 𝑋𝐶 , tan⁡ 𝜙 = 𝜔𝐶 𝑅 Example 1 Given R=16“ Ω, L=0.02” H, C=2×10^(-4) F, and E=12" V. At t=0, I=0 and Q=0. 𝐿 𝑑2 𝑄 𝑑𝑄 1 +𝑅 + 𝑄=𝐸 2 𝑑𝑡 𝑑𝑡 𝐶 Substituting the above values yields: 𝑄ℎ (𝑡) = 𝑒 −400𝑡 (𝐶1 cos⁡ 300𝑡 + 𝐶2 sin⁡ 300𝑡) 𝑄(𝑡) = 2.4 × 10−3 + 𝑒 −400𝑡 (𝐶1 cos⁡ 300𝑡 + 𝐶2 sin⁡ 300𝑡) Current: 𝐼(𝑡) = 𝑑𝑄 = 𝑒 −400𝑡 [−400(𝐶1 cos⁡ 300𝑡 + 𝐶2 sin⁡ 300𝑡) − 300𝐶1 sin⁡ 300𝑡 𝑑𝑡 + 300𝐶2 cos⁡ 300𝑡] MICHAEL KURNIAWAN 8 Study Case Example 2 A 2 H inductor, 16 Ω resistor, and 0.02 F capacitor are connected in series with a voltage source ε=100sin 3t. At t=0, the charge and current are zero. 2 𝑑2𝑄 𝑑𝑄 + 16 + 25𝑄 = 20sin⁡ 3𝑡 2 𝑑𝑡 𝑑𝑡 Solution: 25 25 −4𝑡 [2sin⁡ 3𝑡 − 3cos⁡ 3𝑡] + 𝑒 (3cos⁡ 3𝑡 + 2sin⁡ 3𝑡) 52 52 𝑑𝑄 75 25 −4𝑡 𝐼(𝑡) = = [2cos⁡ 3𝑡 + 3sin⁡ 3𝑡] − 𝑒 (17sin⁡ 3𝑡 + 6cos⁡ 3𝑡) 𝑑𝑡 52 52 𝑄(𝑡) = The first term describes the steady-state current, while the second term is the transient current that decreases over time. CONCLUSION 1. The mathematical model for series LC and series RLC circuits using second-order differential equations produces homogeneous and non-homogeneous solutions. 2. In series LC circuits, constant inductor and capacitor conditions produce identical homogeneous solutions, while non-homogeneous solutions depend on the form of the voltage source. 3. In series RLC circuits, there are two current components: transient current (which fades over time) and steady-state current (which remains constant). 4. For further research, it is recommended to analyze parallel LC and RLC circuits in order to expand the understanding of resonance and damping system characteristics. MICHAEL KURNIAWAN 9 Study Case REFERENCE MICHAEL KURNIAWAN 10