ME2016 Mini Project 2

ME_2016/mini_project_2/.ipynb_checkpoints/direct-checkpoint.ipynb

@@ -0,0 +1,154 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "bb0a3f4e-6f1a-459c-b7af-df7d82b82a97",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "import sympy as sm\n",
+    "from scipy.optimize import fsolve\n",
+    "from scipy.integrate import quad\n",
+    "import matplotlib.pyplot as plt"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 21,
+   "id": "8504b0c4-befd-4b9e-8c32-dfb65a82a831",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "f_pprime, f_prime, f = sm.symbols('f\\'\\', f\\', f')\n",
+    "v_prime, v = sm.symbols('v\\', v')\n",
+    "u_prime, u = sm.symbols('u\\', u')\n",
+    "epsilon, lambduh, alpha_1, mu, alpha_2, g = sm.symbols('epsilon, lambda, alpha_1, mu, alpha_2, g')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 22,
+   "id": "9301e042-edff-4579-b00b-53d2a6162e6f",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\epsilon^{2} f' \\lambda + f + f'' = \\epsilon v'$"
+      ],
+      "text/plain": [
+       "Eq(epsilon**2*f'*lambda + f + f'', epsilon*v')"
+      ]
+     },
+     "execution_count": 22,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "nine_a = sm.Eq(f_pprime+epsilon**2 *lambduh*f_prime + f, epsilon*v_prime)\n",
+    "nine_a"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 23,
+   "id": "3710753d-59ca-42c3-8791-15b02f317001",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\alpha_{1} v - u' + v' = - \\epsilon f$"
+      ],
+      "text/plain": [
+       "Eq(alpha_1*v - u' + v', -epsilon*f)"
+      ]
+     },
+     "execution_count": 23,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "nine_b = sm.Eq((v_prime-u_prime) + alpha_1*v, -epsilon*f)\n",
+    "nine_b"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 24,
+   "id": "a6796232-a2c0-40e5-8190-5a2c64bdb50d",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\alpha_{2} u - \\mu \\left(- u' + v'\\right) + u' = g$"
+      ],
+      "text/plain": [
+       "Eq(alpha_2*u - mu*(-u' + v') + u', g)"
+      ]
+     },
+     "execution_count": 24,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "nine_c = sm.Eq(-mu*(v_prime-u_prime) + u_prime + alpha_2*u, g)\n",
+    "nine_c"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 32,
+   "id": "a100efcd-77d8-4d9b-a1a3-277c318549b0",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "{f'': -alpha_1*epsilon*mu*v - alpha_1*epsilon*v - alpha_2*epsilon*u - epsilon**2*f*mu - epsilon**2*f - epsilon**2*f'*lambda + epsilon*g - f,\n",
+       " u': -alpha_1*mu*v - alpha_2*u - epsilon*f*mu + g,\n",
+       " v': -alpha_1*mu*v - alpha_1*v - alpha_2*u - epsilon*f*mu - epsilon*f + g}"
+      ]
+     },
+     "execution_count": 32,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "# Solve the system of equations\n",
+    "solutions = sm.solve([nine_a, nine_b, nine_c], (v_prime, u_prime, f_pprime))\n",
+    "\n",
+    "# Display the solutions\n",
+    "solutions"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.12.3"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}

ME_2016/mini_project_2/.ipynb_checkpoints/main-checkpoint.py

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+import numpy as np
+from scipy.optimize import fsolve
+from scipy.integrate import quad
+import matplotlib.pyplot as plt
+
+# Persistent storage for fsolve initial conditions
+nsolve_ics_storage = {'nsolve_ics': None}
+
+# Helper functions for g1, g1timescos, and g1timessin
+def g1(phi0, x, params):
+    """
+    Purpose of the function:
+    Computes the transistor's nondimensionalized function `g1`, which represents 
+    the piecewise relationship between input and output voltages.
+
+    Inputs:
+    1. phi0 (float or array-like): Phase angle(s) at which the function is evaluated.
+    2. x (array-like): Fourier coefficients [U, G, H, X, Y].
+    3. params (dict): Dictionary containing nondimensional parameters:
+        - ep, gamma, u0.
+
+    Outputs:
+    1. g1out (array-like): The value of the `g1` function at the input phase angles.
+    
+    Notes:
+    - Implements piecewise behavior for the `g1` function based on the conditional 
+      `ep * v1mu1 >= u0 - 1`.
+    - This is based on equations 28 and 34 in the referenced model.
+    """
+
+    U, V, G, H, X, Y = x[0], 0, x[1], x[2], x[3], x[4]
+    ep = params['ep']
+    gamma = params['gamma']
+    u0 = params['u0']
+    
+    # Equation 34
+    v1mu1 = (V - U) + (G - X) * np.cos(phi0) + (H - Y) * np.sin(phi0)
+    
+    # Equation 28
+    cond = ep * v1mu1 >= u0 - 1
+    g1out = np.where(
+        cond,
+        gamma * v1mu1 * (2 * (1 - u0) + ep * v1mu1),
+        -gamma / ep * (1 - u0)**2
+    )
+    return g1out
+
+def g1timescos(phi0, x, params):
+    """Compute g1(phi0) * cos(phi0)."""
+    return g1(phi0, x, params) * np.cos(phi0)
+
+def g1timessin(phi0, x, params):
+    """Compute g1(phi0) * sin(phi0)."""
+    return g1(phi0, x, params) * np.sin(phi0)
+
+# Harmonic balance system (hbsystem_case1)
+def hbsystem_case1(x, params):
+    """
+    Purpose of the function:
+    Computes the residuals for the harmonic balance system (Eq. 33a-f). 
+    These residuals are minimized by `fsolve` to find the Fourier coefficients.
+
+    Inputs:
+    1. x (array-like): A vector of Fourier coefficients [U, G, H, X, Y].
+    2. params (dict): Dictionary containing nondimensional parameters:
+        - mu, a1, a2, ep, gamma, a0 (current state), and u0 (steady-state solution).
+
+    Outputs:
+    1. F (array-like): A vector of residuals corresponding to equations 33a-f.
+
+    Notes:
+    - The residuals involve integrals over `phi0` (0 to 2π), which are computed using 
+      numerical quadrature (`quad`).
+    - The function incorporates `g1`, `g1timescos`, and `g1timessin`.
+    """ 
+    mu = params['mu']
+    a1 = params['a1']
+    a2 = params['a2']
+    ep = params['ep']
+    gamma = params['gamma']
+    a0 = params['a0cur']
+    u0 = params['u0']
+    
+    # Fourier coefficients
+    U, V, G, H, X, Y = x[0], 0, x[1], x[2], x[3], x[4]
+    
+    # Equations 33a-f
+    F = np.zeros(5)
+    F[0] = -a2 * U + (1 / (2 * np.pi)) * quad(lambda phi: g1(phi, x, params), 0, 2 * np.pi)[0]
+    F[1] = H - Y + a1 * G
+    F[2] = -mu * H + (mu + 1) * Y + a2 * X - (1 / np.pi) * quad(lambda phi: g1timescos(phi, x, params), 0, 2 * np.pi)[0]
+    F[3] = -G + X + a1 * H + a0
+    F[4] = mu * G - (mu + 1) * X + a2 * Y - (1 / np.pi) * quad(lambda phi: g1timessin(phi, x, params), 0, 2 * np.pi)[0]
+    #print(f'F: {F}')
+    return F
+
+# ODE function for A0
+def odefun_A0(t, A0, params):
+    """
+    Purpose of the function:
+    Defines the ordinary differential equation (ODE) for `A0` as part of the 
+    larger Colpitts oscillator system. Computes the time derivative of `A0`.
+
+    Inputs:
+    1. t (float): Current time (unused in this specific ODE but required by solver).
+    2. A0 (float): Current value of `A0`.
+    3. params (dict): Dictionary containing nondimensional parameters:
+        - lambda, a0cur (current A0 state), and others used in `hbsystem_case1`.
+
+    Outputs:
+    1. dA0 (float): The time derivative of `A0`.
+
+    Notes:
+    - Uses `fsolve` to solve the harmonic balance equations (via `hbsystem_case1`).
+    - Returns `dA0` using the computed `H` value (harmonic balance coefficient).
+    """
+    global nsolve_ics_storage
+    
+    # Initialize fsolve initial conditions if not set
+    if nsolve_ics_storage['nsolve_ics'] is None:
+        nsolve_ics_storage['nsolve_ics'] = np.array([100, 100, 100, 100, 100])
+    
+    # Update current A0 in parameters
+    params['a0cur'] = A0
+    lambda_ = params['lambda']
+    
+    # Solve harmonic balance system
+    nsolve_ics = nsolve_ics_storage['nsolve_ics']
+    x = fsolve(lambda x: hbsystem_case1(x, params), nsolve_ics, xtol=1e-4)
+    
+    # Update fsolve initial conditions for next iteration
+    nsolve_ics_storage['nsolve_ics'] = x
+    
+    # Extract H from the solution
+    H = x[2]
+    
+    # Compute dA0/d_eta_2
+    dA0 = (-1 / 2) * (lambda_ * A0 - H)
+    #print(f'dA0 {dA0}')
+    print(f't {t}')
+    return dA0
+
+# Main function to solve the system and plot results
+def main():
+    # Dimensional parameters
+    Cm = 2.201e-15
+    Cs = 1.62e-11
+    Cg = 2.32e-11
+    Lm = 4.496e-2
+    Rm = 5.62e1
+    Rs = 7.444e3
+    Rg = 5.168e4
+    Vth = -0.95
+    beta = 0.12
+    B = 0.8
+    
+    # Compute nondimensional parameters
+    mu = Cg / Cs
+    a1 = np.sqrt(Lm * Cm) / (Cg * Rg)
+    a2 = np.sqrt(Lm * Cm) / (Cs * Rs)
+    ep = np.sqrt(Cm / Cg)
+    lmbda = Rm * Cg / np.sqrt(Lm * Cm)
+    gamma = beta * abs(Vth) * np.sqrt(Lm * Cm) / Cs
+    u0 = (1 + a2 / (2 * gamma)) - np.sqrt((1 + a2 / (2 * gamma))**2 - 1)
+    
+    # Parameters
+    params = {
+        'mu': mu,
+        'a1': a1,
+        'a2': a2,
+        'ep': ep,
+        'lambda': lmbda,
+        'gamma': gamma,
+        'u0': u0,
+        'a0cur': 0, 
+        'v0': 0,
+    }
+    
+    # Initial conditions and time span
+    a0ic = 0
+    tspan = [0, 200]
+    
+    # Solve the ODE
+    from scipy.integrate import solve_ivp
+    sol = solve_ivp(
+        fun=lambda t, A0: odefun_A0(t, A0, params),
+        t_span=tspan,
+        y0=[a0ic],
+        method='RK45',
+        max_step=1e-0
+    )
+    
+    # Plot the results
+    plt.plot(sol.t / params['ep']**2, np.abs(sol.y[0]))
+    plt.xlabel('Scaled Time (t / ep^2)')
+    plt.ylabel('A0 (Amplitude)')
+    plt.title('Dynamics of A0 Over Time')
+    plt.grid()
+    plt.savefig('test.png')
+
+# Run the main function
+if __name__ == "__main__":
+    main()

ME_2016/mini_project_2/README.md

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+Okay, full disclosure. This folder and the organization of this code is MESSY. 
+Here's the layout of things to understand what's what.
+
+# "direct.ipynb"
+This is the symbolic manipulation done to create the state space equations for the direct numerical part.
+
+# "main.py"
+Is the averaging solution and the direct numerical solution actually being done.
+
+**The top half is the averaging solution, adapted from code in MATLAB given by Nikhil Bajaj.** Most of this code is translated into Python with the help of ChatGPT 4.0, and then checked for congruency by hand. 
+
+The second half lazily redefines all the parameters and then implements a simple ODE solver on the state-space equations for the direct simulation.
+
+Finally, everything is plotted and the figure is saved.
+
+If you have any questions, contact me at dane.sabo@pitt.edu

ME_2016/mini_project_2/main.py

@@ -178,23 +178,65 @@ def main():
     
     # Initial conditions and time span
     a0ic = 0
-    tspan = [0, 200]
+    tspan = [0, 100]
     
     # Solve the ODE
+    print('AVERAGING SOLUTION')
     from scipy.integrate import solve_ivp
     sol = solve_ivp(
         fun=lambda t, A0: odefun_A0(t, A0, params),
         t_span=tspan,
         y0=[a0ic],
         method='RK45',
-        max_step=1e-0
+        atol = 1e-3
+    )
+    #### DIRECT INTEGRATION
+        # Compute nondimensional parameters
+    mu = Cg / Cs
+    alpha_1 = np.sqrt(Lm * Cm) / (Cg * Rg)
+    alpha_2 = np.sqrt(Lm * Cm) / (Cs * Rs)
+    epsilon = np.sqrt(Cm / Cg)
+    lambduh = Rm * Cg / np.sqrt(Lm * Cm)
+    gamma = beta * abs(Vth) * np.sqrt(Lm * Cm) / Cs
+    tspan_2 = [0, 1e6]
+
+    def g(value):
+        if value >= 0:
+            output = gamma*value**2
+        else:
+            output = 0
+        return output
+            
+
+    def colpitts(t, X):
+        x_1, x_2, x_3, x_4 = X
+        dx1_dt = x_2
+        dx2_dt = -alpha_1*epsilon*mu*x_4 - alpha_1*epsilon*x_4 - \
+                    alpha_2*epsilon*x_3 - epsilon**2*x_1*mu - epsilon**2*x_1 - \
+                    epsilon**2*x_2*lambduh + epsilon*g(x_4-x_3+1) - x_1
+        dx3_dt = -alpha_1*mu*x_4 - alpha_2*x_3 - epsilon*x_1*mu + g(x_4-x_3+1)
+        dx4_dt = -alpha_1*mu*x_4 - alpha_1*x_4 - alpha_2*x_3 - epsilon*x_1*mu - epsilon*x_1 + g(x_4-x_3+1)
+        print(f't: {t*epsilon**2}')
+        return [dx1_dt, dx2_dt, dx3_dt, dx4_dt]
+
+        
+# Solve the ODE
+    print('DIRECT SOLUTION')
+    sol2 = solve_ivp(
+        fun=colpitts,
+        t_span=tspan_2,
+        y0=[0,0,0,0],
+        method='RK45',
+        atol = 1e-13,
+        rtol = 1e-6
     )
 
     # Plot the results
-    plt.plot(sol.t / params['ep']**2, np.abs(sol.y[0]))
+    plt.plot(sol.t / params['ep']**2, np.abs(sol.y[0]), sol2.t, np.abs(sol2.y[0]))
     plt.xlabel('Scaled Time (t / ep^2)')
-    plt.ylabel('A0 (Amplitude)')
+    plt.ylabel('Amplitude')
-    plt.title('Dynamics of A0 Over Time')
+    plt.title('Averaging solution compared to direct numerical solution')
+    plt.legend(['A0','||f||'])
     plt.grid()
     plt.savefig('test.png')