Capacity-based inputs evaluate the power fraction directly from the airflow fraction. Power-based inputs invert the same curve, returning zero airflow and zero power when the requested power is below the no-load power.

\begin{equation}\label{eq:vfd-piecewise-power} f_P = \begin{cases} 1, & \text{if } f_Q > 1 \\ \frac{f_{P,TD} - f_{P,NL}}{f_{Q,TD}} f_Q + f_{P,NL}, & \text{if } f_Q < f_{Q,TD} \\ a f_Q^2 + b f_Q + c, & \text{if } f_{Q,TD} \le f_Q \le 1 \end{cases} \end{equation}

\begin{equation}\label{eq:vfd-piecewise-airflow} f_Q = \begin{cases} 1, & \text{if } f_P = 1 \\ 0, & \text{if } f_P < f_{P,NL} \\ \frac{f_{Q,TD}}{f_{P,TD} - f_{P,NL}}(f_P - f_{P,NL}), & \text{if } f_{P,NL} \le f_P < f_{P,TD} \\ \frac{-b + \sqrt{b^2 - 4a(c - f_P)}}{2a}, & \text{if } f_P \ge f_{P,TD} \end{cases} \end{equation}

Symbols

\(f_P\)	Fraction of full-load power \([\unit{ \unitless}]\)
\(f_Q\)	Fraction of full-load airflow \([\unit{ \unitless}]\)
\(f_{P,TD}\)	Turndown power fraction \([\unit{ \unitless}]\)
\(f_{P,NL}\)	No-load power fraction \([\unit{ \unitless}]\)
\(f_{Q,TD}\)	Turndown airflow fraction \([\unit{ \unitless}]\)
\(a,b,c\)	Second-order curve-fit coefficients \([\unit{ \unitless}]\)