DC & BLDC Motor — Physics & Math Summary

Symbol	Meaning	Unit
$V$	Applied voltage (bus)	V
$I$	Armature current	A
$R$	Winding resistance	$\Omega$
$L$	Winding inductance (optional)	H
$\omega$	Angular speed	rad/s (=$2\pi\,\text{rpm}/60$)
$\tau$	Torque (load or shaft)	N·m
$K_e$	Back‑EMF constant	V·s/rad
$K_t$	Torque constant	N·m/A

Symbol	Meaning	Unit
$J$	Total inertia (ref. to motor)	kg·m$^2$
$b$	Viscous friction coeff.	N·m·s/rad
$\tau_c$	Coulomb friction (breakaway)	N·m
$N$	Gear ratio (motor:output)	–
$\eta$	Efficiency (= $P_\mathrm{mech}/P_\mathrm{el}$)	%
$P_\mathrm{el}$	Electrical power	W
$P_\mathrm{mech}$	Mechanical power	W

Electrical constants

$$R = \dfrac{V}{I_{\mathrm{stall}}},\quad K_e = \dfrac{V - I_{\mathrm{nl}} R}{\omega_{\mathrm{nl}}}$$

Torque constant

Using friction‑as‑current offset (simple, common): $$K_t = \dfrac{\tau_{\mathrm{stall}}}{I_{\mathrm{stall}}-I_{\mathrm{nl}}}$$ Using explicit friction (more physical): set $K_t=\dfrac{\tau_{\mathrm{stall}}}{I_{\mathrm{stall}}}$ and identify $\tau_c$ and $b$ from off‑rating points.

Speed–Torque line

Combine $I=(V-K_e\omega)/R$ and $\tau_e=K_t I$: $$\tau_e = \tfrac{K_t}{R}(V - K_e\,\omega) \;\Rightarrow\; \omega = \underbrace{\tfrac{V}{K_e}}_{\omega_{\mathrm{nl}}} - \underbrace{\tfrac{R}{K_e K_t}}_{\omega/\tau\;\text{slope}}\,\tau_e$$ With load torque $\tau$ and friction: $\tau_e=\tau + b\,\omega + \tau_c\,\mathrm{sign}(\omega)$.

Power & efficiency

$$P_{\mathrm{el}} = V I = \tfrac{V}{R}(V-K_e\omega),\qquad P_{\mathrm{mech}} = \tau\,\omega$$ Ignoring $I_{\mathrm{nl}}$ and $\tau_c$, peak mechanical power occurs near $\omega\approx\tfrac12\,\omega_{\mathrm{nl}}$ (i.e., $\tau\approx\tfrac12\,\tau_{\mathrm{stall}}$). With nonzero $I_{\mathrm{nl}}$ and $b$, the peak shifts slightly toward lower torque.

No inductance (algebraic current)

Substitute $I=(V-K_e\omega)/R$ into the mechanical equation: $$J\,\dot{\omega} = \tfrac{K_t}{R}(V-K_e\omega) - b\,\omega - \tau_c\,\mathrm{sign}(\omega) - \tau_{\mathrm{load}}$$ For small signals about a positive speed (ignore $\tau_c$): $$\dot{\omega} + \underbrace{\tfrac{b + K_tK_e/R}{J}}_{\color{#007acc}{1/\tau_{\mathrm{mech}}}}\omega = \tfrac{K_t}{JR}V - \tfrac{1}{J}\tau_{\mathrm{load}}$$ Time constant: $$\boxed{\tau_{\mathrm{mech}} = \dfrac{J}{\;b + K_tK_e/R\;}}$$

Including inductance (2nd order)

$$\begin{cases} L\,\dot I + R I + K_e\omega = V\\ J\,\dot{\omega} + b\,\omega = K_t I - \tau_{\mathrm{load}} \end{cases}$$ Natural electrical time constant $\tau_e=L/R$. If PWM frequency $\gg 1/\tau_e$, the averaged (no‑$L$) model is typically adequate.

DC‑Equivalent Constants (from Kv & stall)

Use a single‑input single‑output model identical to brushed DC by mapping BLDC specs to effective constants:

• Back‑EMF constant
  $K_e\approx \frac{60}{2\pi \cdot K_v}$ (Kv in rpm/V, typical RC spec is line‑to‑line no‑load)
• Torque constant
  $K_t\approx K_e$ (in SI units N·m/A)
• Effective resistance
  $R_{\text{eff}}=\frac{V_{\text{bus}}}{I_{\text{stall}}}$ (calibrate from measured stall current at your bus voltage)
• Sanity checks:
  ${\omega }_{\text{nl}}\approx \frac{V_{\text{bus}}}{K_e}$ , $I_{\text{stall}}\approx \frac{V_{\text{bus}}}{R_{\text{eff}}}$

ESC Averaging & Commands

PWM mode: $V_{\text{cmd}}=d\cdot V_{\text{bus}}$ with duty $d$ in [−1,1].

Speed mode (simple like the sim): feed‑forward plus proportional correction, clipped to the DC bus: use a small $K_{\text{speed}}$ (≈ V/ω_nl).

$V_{\text{cmd}}=\text{clip}(K_e\cdot {\omega }_{\mathrm{ref}}+I_{\mathrm{nl}}\cdot R_{\text{eff}}+K_{\text{speed}}\cdot ({\omega }_{\mathrm{ref}}-\omega ),[-V_{\text{bus}},V_{\text{bus}}\left]\right)$

Use the Same DC Equations

$I=\frac{V_{\text{cmd}}-K_e\cdot \omega }{R_{\text{eff}}},\tau =K_t\cdot (I-I_{\mathrm{nl}}\cdot \mathrm{sgn}(\omega \left)\right)$

Then integrate the same mechanics already on the page: $J\cdot \dot{\omega }=\tau -b\cdot \omega -{\tau }_{\text{load}}$.

Effective time constant (small‑signal about forward speed): ${\tau }_{\text{mech}}=\frac{J}{b+\frac{K_t\cdot K_e}{R_{\text{eff}}}}$

Worked Example (Kv→Kt→τ_max)

Kv = 200 rpm/V, V_bus = 24 V, I_max = 30 A.

$$K_e\approx \frac{60}{2\pi \cdot 200}=0.0955\text{V·s/rad}$$ $$K_t\approx 0.0955\text{N·m/A}$$

Torque limit (ESC/inverter): ${\tau }_{\text{max}}\approx K_t\cdot I_{\text{max}}=2.86\text{N·m}$.

If you know measured I_stall, set R_eff = V_bus / I_stall and you can reuse the DC stall/no‑load checks.

Kinematics

$$\omega_m = N\,\omega_{out},\qquad \tau_{out} = \eta_g\,N\,\tau_m$$

Reflected inertia

Load inertia reflected to motor: $$J_{\mathrm{eq}} = J_m + N^2 J_{\mathrm{load}}$$ (Friction reflected similarly; $\eta_g$ mainly affects torque/power, not the $N^2$ inertia scaling.)