\section{Model Formulation}\label{sec:model}

We develop a mechanism-driven continuous-time model for smartphone battery drain that couples
(i) component-level power mapping from user/device inputs,
(ii) an equivalent-circuit battery model (ECM) with polarization memory,
(iii) a constant-power-load (CPL) algebraic closure for the discharge current,
(iv) lumped thermal dynamics, and
(v) slow health degradation (SOH).
All symbols are used consistently throughout.

\subsection{Total Power Decomposition $P_{\rm tot}$ (Screen/CPU/Network)}\label{sec:ptot}
Let the state vector be
\begin{equation}
\mathbf{x}(t)=\big[z(t),\,v_p(t),\,T_b(t),\,S(t),\,w(t)\big]^\top,
\end{equation}
where $z$ is the state-of-charge (SOC), $v_p$ is the polarization voltage, $T_b$ is the battery temperature,
$S$ is the state-of-health (SOH, capacity fraction), and $w$ is a continuous radio-tail state.
The exogenous input vector is
\begin{equation}
\mathbf{u}(t)=\big[L(t),\,C(t),\,N(t),\,\Psi(t),\,T_a(t)\big]^\top,
\end{equation}
where $L$ is screen brightness, $C$ is CPU load, $N$ is network activity intensity,
$\Psi$ is signal quality (larger is better), and $T_a$ is ambient temperature.

We model the instantaneous total power demand as an additive decomposition
\begin{equation}\label{eq:ptot_def}
P_{\mathrm{tot}}(t)=P_{\mathrm{bg}}+P_{\mathrm{scr}}(L(t))+P_{\mathrm{cpu}}(C(t))+P_{\mathrm{net}}(N(t),\Psi(t),w(t)),
\end{equation}
where $P_{\mathrm{bg}}$ is background/baseline power. The component mappings are chosen to be explicit and
mechanism-consistent:
\begin{align}
P_{\mathrm{scr}}(L)&=P_{\mathrm{scr},0}+k_L L^\gamma,\qquad \gamma>1,\label{eq:pscr}\\
P_{\mathrm{cpu}}(C)&=P_{\mathrm{cpu},0}+k_C C^\eta,\qquad \eta>1,\label{eq:pcpu}\\
P_{\mathrm{net}}(N,\Psi,w)&=P_{\mathrm{net},0}+k_N\frac{N}{(\Psi+\varepsilon)^\kappa}+k_{\mathrm{tail}}w,
\qquad \kappa>0,\ \varepsilon>0.\label{eq:pnet}
\end{align}
Here $(\Psi+\varepsilon)^{-\kappa}$ captures the increased radio power required under poor signal quality,
and $k_{\mathrm{tail}}w$ represents residual ``tail'' consumption after network bursts.

\subsection{Continuous Radio-Tail Dynamics $w(t)$}\label{sec:tail}
Instead of a discrete finite-state-machine tail model, we introduce a continuous tail state $w(t)\in[0,1]$:
\begin{equation}\label{eq:w_dyn}
\dot w(t)=\frac{\sigma(N(t))-w(t)}{\tau(N(t))},
\end{equation}
where
\begin{equation}\label{eq:sigma_tau}
\sigma(N)=\min(1,N),\qquad
\tau(N)=
\begin{cases}
\tau_\uparrow, & \sigma(N)\ge w,\\
\tau_\downarrow,& \sigma(N)< w,
\end{cases}
\qquad \tau_\uparrow\ll\tau_\downarrow.
\end{equation}
This formulation yields fast engagement of the tail state during activity increases and slow decay after activity
drops, while maintaining continuity and numerical robustness.

\subsection{ECM Terminal Voltage Equation}\label{sec:ecm}
We adopt a first-order Thevenin ECM with an ohmic resistance and one polarization branch:
\begin{equation}\label{eq:vterm}
V_{\mathrm{term}}(t)=V_{\mathrm{oc}}(z(t)) - v_p(t) - I(t)\,R_0(T_b(t),S(t)),
\end{equation}
where $V_{\mathrm{oc}}(z)$ is the open-circuit voltage (OCV) as a function of SOC, and
$R_0(T_b,S)$ is the temperature- and SOH-dependent ohmic resistance.

\subsection{CPL Closure: Quadratic Current and Discriminant $\Delta$}\label{sec:cpl}
Smartphone loads are well-approximated as constant-power over short time scales.
We therefore impose a CPL constraint:
\begin{equation}\label{eq:cpl}
P_{\mathrm{tot}}(t)=V_{\mathrm{term}}(t)\,I(t)
=\big(V_{\mathrm{oc}}(z)-v_p-I R_0(T_b,S)\big)I.
\end{equation}
This yields a quadratic equation in $I$ with discriminant
\begin{equation}\label{eq:delta}
\Delta(t)=\big(V_{\mathrm{oc}}(z)-v_p\big)^2-4R_0(T_b,S)P_{\mathrm{tot}}(t).
\end{equation}
Feasibility requires $\Delta(t)\ge 0$. When feasible, the physically consistent branch is
\begin{equation}\label{eq:I_cpl}
I_{\mathrm{CPL}}(t)=\frac{V_{\mathrm{oc}}(z)-v_p-\sqrt{\Delta(t)}}{2R_0(T_b,S)}.
\end{equation}
If $\Delta(t)<0$, the demanded power is not deliverable under the CPL assumption, indicating voltage-collapse risk.

\subsection{Coupled ODEs: SOC--Polarization--Thermal--SOH--Tail}\label{sec:odes}
Given $I(t)$, the coupled state dynamics are
\begin{align}
\dot z(t)&=-\frac{I(t)}{3600\,Q_{\mathrm{eff}}(T_b(t),S(t))},\label{eq:dz}\\
\dot v_p(t)&=\frac{I(t)}{C_1}-\frac{v_p(t)}{R_1C_1},\label{eq:dvp}\\
\dot T_b(t)&=\frac{1}{C_{\mathrm{th}}}\Big(I(t)^2R_0(T_b,S)+\frac{v_p(t)^2}{R_1}-hA\big(T_b(t)-T_a(t)\big)\Big),\label{eq:dTb}\\
\dot S(t)&=-\lambda_{\mathrm{sei}}|I(t)|^{m}\exp\!\left(-\frac{E_{\mathrm{sei}}}{R_gT_b(t)}\right),\qquad 0\le m\le 1,\label{eq:dS}\\
\dot w(t)&=\frac{\sigma(N(t))-w(t)}{\tau(N(t))}.\label{eq:dw}
\end{align}
Equation \eqref{eq:dTb} uses a nonnegative polarization dissipation term $v_p^2/R_1$ for energetic consistency.

\subsection{Constitutive Relations: OCV, $R_0(T_b,S)$, and $Q_{\rm eff}(T_b,S)$}\label{sec:constitutive}
\paragraph{OCV (modified Shepherd).}
We use a modified Shepherd form:
\begin{equation}\label{eq:voc_raw}
V_{\mathrm{oc}}(z)=E_0-K\Big(\frac{1}{z}-1\Big)+A e^{-B(1-z)}.
\end{equation}

\paragraph{Ohmic resistance with Arrhenius temperature dependence and SOH correction.}
\begin{equation}\label{eq:R0}
R_0(T_b,S)=R_{\mathrm{ref}}\exp\!\Big[\frac{E_a}{R_g}\Big(\frac{1}{T_b}-\frac{1}{T_{\mathrm{ref}}}\Big)\Big]\big(1+\eta_R(1-S)\big).
\end{equation}

\paragraph{Effective capacity.}
\begin{equation}\label{eq:Qeff}
Q_{\mathrm{eff}}(T_b,S)=Q_{\mathrm{nom}}\,S\Big[1-\alpha_Q(T_{\mathrm{ref}}-T_b)\Big]_+,
\qquad [x]_+=\max(x,0).
\end{equation}

\subsection{Incorporating Three Lightweight Refinements}\label{sec:refinements}
To improve robustness while preserving the mechanistic structure, we incorporate three ``micro-refinements.''

\paragraph{(i) Low-SOC singularity protection in $V_{\mathrm{oc}}$.}
The term $1/z$ in \eqref{eq:voc_raw} is numerically singular as $z\to 0$.
We introduce an effective SOC
\begin{equation}\label{eq:zeff}
z_{\mathrm{eff}}(t)=\max\{z(t),z_{\min}\},
\end{equation}
with a small reserve threshold $z_{\min}\in(0,1)$ (e.g., $z_{\min}=0.02$) reflecting a practical BMS ``unavailable''
low-SOC region. We then evaluate OCV using $z_{\mathrm{eff}}$:
\begin{equation}\label{eq:voc}
V_{\mathrm{oc}}(z)=E_0-K\Big(\frac{1}{z_{\mathrm{eff}}}-1\Big)+A e^{-B(1-z_{\mathrm{eff}})}.
\end{equation}

\paragraph{(ii) Nonnegative polarization heating.}
Thermal generation is written as $I^2R_0+v_p^2/R_1$, which is always nonnegative and aligns with resistive dissipation
in the polarization branch. This choice avoids sign ambiguities that can arise with alternative $Iv_p$ forms.

\paragraph{(iii) Lightweight current saturation (throttling/PMIC limiting).}
Real devices may throttle performance or limit current under low voltage or high temperature.
We model this with a temperature-dependent current cap:
\begin{equation}\label{eq:I_sat}
I(t)=\min\big(I_{\mathrm{CPL}}(t),\,I_{\max}(T_b(t))\big),
\end{equation}
where a simple continuous form is
\begin{equation}\label{eq:Imax}
I_{\max}(T_b)=I_{\max,0}\Big[1-\rho_T\,(T_b-T_{\mathrm{ref}})\Big]_+,\qquad \rho_T\ge 0.
\end{equation}
When $I_{\mathrm{CPL}}>I_{\max}$, the device operates in a degraded regime with delivered power
$P_{\mathrm{del}}(t)=V_{\mathrm{term}}(t)I(t)\le P_{\mathrm{tot}}(t)$, corresponding to throttling.

\subsection{Initial Conditions and Termination Definitions (TTE and optional $t_\Delta$)}\label{sec:ic_tte}
We use
\begin{equation}\label{eq:ic}
z(0)=z_0,\qquad v_p(0)=0,\qquad T_b(0)=T_a(0),\qquad S(0)=S_0,\qquad w(0)=0.
\end{equation}
We define the time-to-end (time-to-empty / time-to-shutdown) as
\begin{equation}\label{eq:TTE}
\mathrm{TTE}=\inf\Big\{t>0:\ V_{\mathrm{term}}(t)\le V_{\mathrm{cut}}\ \ \text{or}\ \ z(t)\le 0\Big\}.
\end{equation}
Optionally, to quantify CPL infeasibility as a voltage-collapse risk indicator, we define
\begin{equation}\label{eq:tDelta}
t_{\Delta}=\inf\Big\{t>0:\ \Delta(t)\le 0\Big\}.
\end{equation}
With throttling \eqref{eq:I_sat}, $t_\Delta$ is interpreted as the onset time at which pure CPL operation becomes
infeasible, even if the system may continue operating in a degraded mode.

\subsection{Closed-Loop Structure Summary}\label{sec:summary_loop}
The model forms a closed-loop chain:
\begin{equation}\label{eq:loop}
\mathbf{u}(t)\ \Rightarrow\ P_{\mathrm{tot}}(t)\ \Rightarrow\
\big(V_{\mathrm{oc}}(z_{\mathrm{eff}}),R_0(T_b,S),\Delta(t)\big)\ \Rightarrow\
I(t)\ \Rightarrow\ \dot{\mathbf{x}}(t)\ \Rightarrow\ \big(V_{\mathrm{term}}(t),z(t),\mathrm{TTE}\big).
\end{equation}
Nonlinear feedback arises because $P_{\mathrm{tot}}$ is enforced via CPL, while $R_0$ and $Q_{\mathrm{eff}}$
depend on $(T_b,S)$, which in turn evolve under the dissipated power.

\subsection{(Optional) Scaling and Time-Scale Discussion}\label{sec:scaling}
Although not required for computation, a brief scale analysis clarifies stiffness and numerical choices.
Let $\tau_p=R_1C_1$ denote the polarization time constant, and $\tau_{\mathrm{th}}=C_{\mathrm{th}}/(hA)$ the thermal
time constant. Typically $\tau_p\ll \tau_{\mathrm{th}}$, implying fast electrical transients and slower thermal drift.
Moreover, the tail dynamics introduce $\tau_\uparrow\ll \tau_\downarrow$.
These separated time scales motivate a time step that resolves $\tau_p$ and $\tau_\uparrow$ in explicit integration,
as enforced later in the numerical method.