🦎 The Lizard: Maxon DC Motor Modeling, Parameter Identification & Multi-Domain Simulation

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1. Executive Summary

The Lizard is a first-principles electro-mechanical model, analytical transfer function derivation, parameter identification, and multi-domain simulation suite for the Maxon DC-max 16 S (24V) graphite-brushed DC motor.

The motor dynamics are implemented and cross-validated across three independent simulation paradigms in MATLAB & Simulink:

1. Analytical Transfer Function Model (SYS1, SYS2, SYS3 via Model Linearizer): Evaluated in the frequency and Laplace $s$-domains.
2. Differential Equation Block Diagram Model: Direct state-variable integration of Kirchhoff’s Voltage Law (KVL) and Newton’s second rotational law.
3. Simscape Multi-Domain Physical Network: A physics-based causal network modeling real-world electrical (current/voltage sensors, reference grounds) and mechanical (rotor inertia, damping, torque loading) domains.

All three implementations are benchmarked against official manufacturer specifications (Cataloge-Page-EN-122.pdf) across No-Load, Nominal Load, and Stall regimes, achieving $<1%$ steady-state tracking error.

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2. Simulation Architectures

Simulink Differential Equation Block Diagram

Figure 1: Closed-loop electro-mechanical block diagram with armature back-EMF feedback, torque constant gain $k_\phi$, and mechanical damping.

Simscape Physical Multi-Domain Network

Figure 2: Simscape physical schematic showing coupled electrical domain (blue) and rotational mechanical domain (green) with physical-signal sensors.

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3. Mathematical Formulation & Derivations

Governing Time-Domain Differential Equations

The dynamic behavior of a permanent magnet DC motor is governed by coupled electrical and rotational mechanical equations:

1. Electrical Subsystem (Kirchhoff's Voltage Law): $$V_a(t) = R_a i_a(t) + L_a \frac{di_a(t)}{dt} + e_b(t)$$ where the counter-electromotive force (back-EMF) is proportional to rotational velocity: $$e_b(t) = k_\phi \omega_m(t)$$

2. Mechanical Subsystem (Newton's Second Law for Rotation): $$J_m \frac{d\omega_m(t)}{dt} = T_m(t) - B_m \omega_m(t) - T_L(t)$$ where the generated electromagnetic torque is proportional to armature current: $$T_m(t) = k_\phi i_a(t)$$

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Laplace S-Domain Transfer Functions

Assuming zero initial conditions and no external load torque ($T_L = 0$):

$$\begin{aligned} V_a(s) &= (L_a s + R_a) I_a(s) + k_\phi \Omega_m(s) \\ k_\phi I_a(s) &= (J_m s + B_m) \Omega_m(s) \end{aligned}$$

1. Angular Velocity Transfer Function $\frac{\Omega_m(s)}{V_a(s)}$:

$$\frac{\Omega_m(s)}{V_a(s)} = \frac{k_\phi}{J_m L_a s^2 + (J_m R_a + B_m L_a) s + (R_a B_m + k_\phi^2)}$$

2. Armature Current Transfer Function $\frac{I_a(s)}{V_a(s)}$:

$$\frac{I_a(s)}{V_a(s)} = \frac{J_m s + B_m}{J_m L_a s^2 + (J_m R_a + B_m L_a) s + (R_a B_m + k_\phi^2)}$$

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4. Parameter Identification (Maxon DC-max 16 S)

Extracted from the official Maxon catalog datasheet:

Parameter	Symbol	Catalog Value	Units	Description / Derivation
Nominal Voltage	$V_{an}$	$24.0$	$\text{V}$	Rated input armature voltage
Terminal Resistance	$R_a$	$37.5$	$\Omega$	Phase winding resistance
Terminal Inductance	$L_a$	$1.37 \times 10^{-3}$	$\text{H}$	Armature winding inductance
Torque / Speed Constant	$k_\phi$	$0.0225$	$\text{N}\cdot\text{m/A}$ ($\text{V}\cdot\text{s/rad}$)	Flux linkage constant
Rotor Inertia	$J_m$	$1.16 \times 10^{-7}$	$\text{kg}\cdot\text{m}^2$	Mass moment of inertia of rotor
No-Load Current	$I_{anl}$	$0.0168$	$\text{A}$	Idling current draw ($16.8\text{ mA}$)
Nominal Torque	$T_n$	$4.76 \times 10^{-3}$	$\text{N}\cdot\text{m}$	Continuous operating load ($4.76\text{ mN}\cdot\text{m}$)
Stall Torque	$T_s$	$14.4 \times 10^{-3}$	$\text{N}\cdot\text{m}$	Peak stall torque ($14.4\text{ mN}\cdot\text{m}$)
Viscous Friction Coeff.	$B_m$	$3.64 \times 10^{-7}$	$\text{N}\cdot\text{m}\cdot\text{s/rad}$	Identified via steady-state no-load equation: $B_m = \frac{k_\phi^2 I_{anl}}{V_{an} - R_a I_{anl}}$

Time Constant Evaluation

• Electrical Time Constant ($\tau_e$): $$\tau_e = \frac{L_a}{R_a} = \frac{1.37 \times 10^{-3}\text{ H}}{37.5\ \Omega} = 36.5\ \mu\text{s}$$
• Mechanical Time Constant ($\tau_m$): $$\tau_m = \frac{R_a J_m}{k_\phi^2} = \frac{(37.5)(1.16 \times 10^{-7})}{(0.0225)^2} = 8.59\text{ ms}$$ Because $\tau_m \gg \tau_e$ (ratio $> 235:1$), mechanical inertia dominates the system bandwidth.

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5. Verification Benchmark: Datasheet vs. Simulation

Performance Metric	Datasheet Value	Simulink Block Diagram	Simscape Physical Model	Steady-State Error
No-Load Speed (ω_nl)	9,920 RPM	9,919 RPM	9,909 RPM	< 0.11%
No-Load Current (I_nl)	16.8 mA	16.4 mA	16.77 mA	< 0.20%
Nominal Speed (ω_n)	6,580 RPM	6,640 RPM	6,637 RPM	< 0.91%
Nominal Current (I_n)	0.227 A	0.2227 A	0.2226 A	< 1.90%
Stall Current (I_s)	0.640 A	0.6398 A	0.6394 A	< 0.09%

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6. Dynamic Transient Response Gallery

Nominal Load Regime (Continuous Rated Operation)

Angular Velocity Step Response (6,640 RPM)	Armature Current Transient Response (0.22 A)

No-Load Regime (Unloaded Idling)

No-Load Velocity Steady-State (9,919 RPM)	No-Load Current Idling (16.4 mA)

Stall Condition Comparison (Peak Torque & Current)

Differential Equation Block Diagram Stall Current	Simscape Physical Network Stall Current

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7. Thermal Analysis & Safe Ambient Limits

Under continuous nominal load operation ($T_n = 4.76\text{ mN}\cdot\text{m}$, $I_n = 0.227\text{ A}$):

1. Winding Joule Losses: $$P_{\text{loss}} = I_n^2 R_a = (0.227\text{ A})^2 \times 37.5\ \Omega = 1.93\text{ W}$$
2. Thermal Temperature Rise: Given the thermal resistance winding-to-housing $R_{th1} = 9.1\text{ K/W}$ and housing-to-ambient $R_{th2} = 35.0\text{ K/W}$ (total $R_{th} = 44.1\text{ K/W}$): $$\Delta T = P_{\text{loss}} \times R_{th} = 1.93\text{ W} \times 44.1\text{ K/W} \approx 85.1^\circ\text{C}$$
3. Maximum Safe Ambient Operating Temperature: For a maximum rated winding temperature $T_{\text{max}} = 125^\circ\text{C}$: $$T_{\text{amb, max}} = T_{\text{max}} - \Delta T = 125^\circ\text{C} - 85.1^\circ\text{C} \approx 39.9^\circ\text{C}$$

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8. How to Run the Simulation

Prerequisites

• MATLAB (R2022b or later recommended)
• Simulink
• Simscape / Simscape Driveline / Simscape Electrical
• Control System Toolbox

Execution Steps

1. Clone this repository:

   git clone https://github.com/sbbxiii/The-Lizard.git
   cd The-Lizard

2. Open MATLAB and navigate to the simulation/ directory.
3. Run simulation/mainEMD.m:

   run('mainEMD.m')

   • Computes motor parameters and stores variables in the base workspace.
   • Derives closed-loop transfer functions SYS1, SYS2, and SYS3.
   • Solves partial fraction residue decomposition for the $24\text{V}$ unit step input.
4. Open the Simulink model simulation/mainEMDmode.slx:

   open_system('mainEMDmode.slx')

5. Click Run to execute the multi-domain simulation and observe the transient curves on the Scopes.

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9. Repository Structure

The-Lizard/
├── simulation/
│   ├── mainEMD.m                       # Core MATLAB parameter script & TF derivation
│   ├── mainEMDmode.slx                 # Simulink & Simscape multi-paradigm model
│   ├── emdlin.mat                      # Linearized state-space model object
│   └── matlab.mat                      # Saved workspace variable state
├── docs/
│   ├── DC_Motor_Simulation_Report.pdf  # Comprehensive academic technical report
│   ├── Maxon_DC_max_16S_Catalog.pdf    # Official Maxon manufacturer datasheet
│   └── images/                         # Simulation diagrams & transient response plots
│       ├── simulink_block_diagram.png
│       ├── simscape_physical_model.png
│       ├── nominal_speed_response.png
│       ├── nominal_current_response.png
│       ├── no_load_speed_response.png
│       ├── no_load_current_response.png
│       ├── stall_current_block_diagram.png
│       └── stall_current_simscape.png
├── .gitignore
├── LICENSE
└── README.md

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10. Author & License

• Engineered by: Osakwe Nmesoma Chukwukadibia (@sbbxiii)
• License: MIT License