{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Praktikum Komputasi 2: Simulasi Transien Rangkaian RLC Seri (Julia)\n", "\n", "Selamat datang di modul simulasi komputasi untuk **Persamaan Diferensial Orde 2** pada aplikasi rangkaian RLC seri.\n", "\n", "## 📌 Pemodelan Rangkaian\n", "Berdasarkan Hukum Tegangan Kirchhoff (KVL) pada rangkaian RLC seri tanpa sumber tegangan luar (respons alami / *zero-input response*):\n", "$$ L \\frac{d^2i}{dt^2} + R \\frac{di}{dt} + \\frac{1}{C} i = 0 $$\n", "\n", "Dengan mendefinisikan koefisien redaman $\\alpha$ dan frekuensi sudut natural $\\omega_0$:\n", "$$ \\alpha = \\frac{R}{2L}, \\quad \\omega_0 = \\frac{1}{\\sqrt{LC}} $$\n", "\n", "Klasifikasi respons:\n", "1. **Underdamped (Kurang Redam):** $\\alpha < \\omega_0$ (Arus berosilasi sambil meluruh secara eksponensial)\n", "2. **Critically Damped (Redam Kritis):** $\\alpha = \\omega_0$ (Arus kembali ke nol paling cepat tanpa berosilasi)\n", "3. **Overdamped (Lebih Redam):** $\\alpha > \\omega_0$ (Arus meluruh lambat menuju nol tanpa osilasi)\n", "\n", "Di sini kita akan memecah PDB orde 2 menjadi sistem 2 PDB orde 1 dan menyelesaikannya secara numerik menggunakan Julia." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "using Plots" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 1. Reduksi Orde PDB ke Sistem Orde 1\n", "Misalkan variabel keadaan (state variables):\n", "$$ x_1 = i(t), \\quad x_2 = \\frac{di}{dt} $$\n", "\n", "Sistem persamaan diferensial orde 1:\n", "$$ \\frac{dx_1}{dt} = x_2 $$\n", "$$ \\frac{dx_2}{dt} = -\\frac{R}{L}x_2 - \\frac{1}{LC}x_1 $$" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# Parameter Rangkaian\n", "L = 1.0 # Induktansi (Henry)\n", "C = 0.01 # Kapasitansi (Farad)\n", "omega_0 = 1.0 / sqrt(L * C) # Frekuensi natural = 10 rad/s\n", "\n", "# Nilai R untuk 3 skenario redaman\n", "R_under = 4.0 # alpha = 2 < 10 (Underdamped)\n", "R_crit = 20.0 # alpha = 10 = 10 (Critically Damped)\n", "R_over = 50.0 # alpha = 25 > 10 (Overdamped)\n", "\n", "println(\"Frekuensi sudut natural ω₀ = $omega_0 rad/s\")\n", "println(\"R Kritis = $(2 * L * omega_0) Ω\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 2. Implementasi Numerik (Metode Runge-Kutta Orde 4 / RK4)\n", "Metode RK4 memberikan akurasi yang jauh lebih tinggi daripada metode Euler untuk sistem osilasi orde 2." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "function solve_rlc_rk4(R, L, C, i0, didt0, t_max, dt)\n", " t_vec = 0.0:dt:t_max\n", " N = length(t_vec)\n", " \n", " # State: x[1] = i, x[2] = di/dt\n", " x = zeros(2, N)\n", " x[:, 1] = [i0, didt0]\n", " \n", " # Turunan: dx/dt = f(x)\n", " function f(state)\n", " i_val, didt_val = state[1], state[2]\n", " return [didt_val, -(R/L)*didt_val - (1.0/(L*C))*i_val]\n", " end\n", " \n", " for k in 1:(N-1)\n", " k1 = f(x[:, k])\n", " k2 = f(x[:, k] + 0.5 * dt * k1)\n", " k3 = f(x[:, k] + 0.5 * dt * k2)\n", " k4 = f(x[:, k] + dt * k3)\n", " \n", " x[:, k+1] = x[:, k] + (dt / 6.0) * (k1 + 2*k2 + 2*k3 + k4)\n", " end\n", " \n", " return t_vec, x[1, :]\n", "end\n", "\n", "# Simulasi 3 kondisi\n", "t_max = 1.5\n", "dt = 0.001\n", "i0 = 0.0 # Arus awal 0 A\n", "didt0 = 10.0 # Laju perubahan awal 10 A/s (akibat tegangan awal kapasitor)\n", "\n", "t_arr, i_under = solve_rlc_rk4(R_under, L, C, i0, didt0, t_max, dt)\n", "_, i_crit = solve_rlc_rk4(R_crit, L, C, i0, didt0, t_max, dt)\n", "_, i_over = solve_rlc_rk4(R_over, L, C, i0, didt0, t_max, dt)\n", "\n", "# Plot perbandingan ketiga respon\n", "plot(t_arr, i_under, label=\"Underdamped (R = 4 Ω)\", lw=2.5, color=:blue, xlabel=\"Waktu (detik)\", ylabel=\"Arus i(t) [Ampere]\")\n", "plot!(t_arr, i_crit, label=\"Critically Damped (R = 20 Ω)\", lw=2.5, color=:green)\n", "plot!(t_arr, i_over, label=\"Overdamped (R = 50 Ω)\", lw=2.5, color=:red)\n", "hline!([0], color=:black, ls=:dash, label=\"\")\n", "title!(\"Perbandingan Respons Transien RLC Seri (Julia RK4)\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 3. Penyelesaian Menggunakan SciML (DifferentialEquations.jl)\n", "Mari kita selesaikan kasus yang sama menggunakan pustaka standar industri `DifferentialEquations.jl`." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "using DifferentialEquations\n", "\n", "# Definisikan PDB dalam format SciML: f(du, u, p, t)\n", "# u[1] = i, u[2] = di/dt\n", "# p = [R, L, C]\n", "function rlc_ode!(du, u, p, t)\n", " R, L, C = p\n", " du[1] = u[2]\n", " du[2] = -(R/L)*u[2] - (1.0/(L*C))*u[1]\n", "end\n", "\n", "u0 = [0.0, 10.0]\n", "tspan = (0.0, 1.5)\n", "params = [R_under, L, C]\n", "\n", "prob = ODEProblem(rlc_ode!, u0, tspan, params)\n", "sol = solve(prob, Tsit5(), reltol=1e-6, abstol=1e-6)\n", "\n", "# Plot arus i(t)\n", "plot(sol, vars=(0, 1), label=\"Arus i(t) - SciML Tsit5\", lw=2, color=:purple, xlabel=\"Waktu (detik)\", ylabel=\"Arus (A)\")\n", "title!(\"Solusi Adaptif RLC Seri dengan Tsit5()\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 4. Tugas Eksplorasi Mahasiswa\n", "1. **Analisis Frekuensi:** Pada kasus *underdamped*, hitung secara analitik frekuensi osilasi teredam $\\omega_d = \\sqrt{\\omega_0^2 - \\alpha^2}$. Cocokkan periodenya ($T = 2\\pi / \\omega_d$) dengan jarak puncak ke puncak pada grafik!\n", "2. **PDB Non-Homogen:** Ubahlah fungsi diferensial dengan menambahkan tegangan generator sinusoidal $V(t) = 10 \\sin(10t)$. Amati fenomena resonansi yang terjadi ketika frekuensi sumber mendekati frekuensi natural $\\omega_0$!" ] } ], "metadata": { "kernelspec": { "display_name": "Julia 1.10", "language": "julia", "name": "julia-1.10" }, "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "1.10.4" } }, "nbformat": 4, "nbformat_minor": 4 }