Sensitivity functions
As described in the RST and voltage loop synthesis (see RST and ILC design for I/B-Loop
and Controller design for V-Loop), one of the output arguments from the optimization
algorithm is df_sens
; for current and field control, this variable is a list where each index of the list contains
multiple sensitivity functions (in a dataframe format). The gain [dB] and phase [deg] are given for each
sensitivity function. The example below shows the sensitivity function columns names for current and field
control.
>>> df_sens[0].columns MultiIndex([( 'Sensitivity_r_y', 'gain'), ( 'Sensitivity_r_y', 'phase'), ( 'Sensitivity_dy_y', 'gain'), ( 'Sensitivity_dy_y', 'phase'), ( 'Sensitivity_dv_y', 'gain'), ( 'Sensitivity_dv_y', 'phase'), ( 'Sensitivity_n_y', 'gain'), ( 'Sensitivity_n_y', 'phase'), ( 'Sensitivity_r_u', 'gain'), ( 'Sensitivity_r_u', 'phase'), ('Sensitivity_r_y_DESIRED', 'gain'), ('Sensitivity_r_y_DESIRED', 'phase')], )
These sensitivity functions correspond to the functions used in Current/Field Control.
\[\begin{split}\begin{aligned}
\mathcal{S}_{ry} & \Rightarrow \texttt{Sensitivity_r_y} \\
\mathcal{S}_{ry}^d & \Rightarrow \texttt{Sensitivity_r_y_DESIRED} \\
\mathcal{S}_{rv_r} & \Rightarrow \texttt{Sensitivity_r_u} \\
\mathcal{S}_{d_yy} & \Rightarrow \texttt{Sensitivity_dy_y} \\
\mathcal{S}_{d_vy} & \Rightarrow \texttt{Sensitivity_dv_y} \\
\mathcal{S}_{n_yy} & \Rightarrow \texttt{Sensitivity_n_y}
\end{aligned}\end{split}\]
For voltage control, the output variable df_sens
is simply a dataframe.
>>> df_sens.columns MultiIndex([( 'Sensitivity_r_y_damping', 'gain'), ( 'Sensitivity_r_y_damping', 'phase'), ( 'Sensitivity_r_y_voltage', 'gain'), ( 'Sensitivity_r_y_voltage', 'phase'), ( 'Sensitivity_dy_y_damp', 'gain'), ( 'Sensitivity_dy_y_damp', 'phase'), ( 'Sensitivity_dy_y_volt', 'gain'), ( 'Sensitivity_dy_y_volt', 'phase'), ('Sensitivity_r_y_damping_DESIRED', 'gain'), ('Sensitivity_r_y_damping_DESIRED', 'phase'), ('Sensitivity_r_y_voltage_DESIRED', 'gain'), ('Sensitivity_r_y_voltage_DESIRED', 'phase')], )
These sensitivity functions correspond to the functions used in Voltage Control.
\[\begin{split}\begin{aligned}
\mathcal{S}_{v_dv_0} & \Rightarrow \texttt{Sensitivity_r_y_damping} \\
\mathcal{S}_{v_dv_0}^d & \Rightarrow \texttt{Sensitivity_r_y_damping_DESIRED} \\
\mathcal{S}_{v_rv_0} & \Rightarrow \texttt{Sensitivity_r_y_voltage} \\
\mathcal{S}_{v_rv_0}^d & \Rightarrow \texttt{Sensitivity_r_y_voltage_DESIRED} \\
\mathcal{S}_{d_{v_0}v_0} \text{(with voltage loop open)} & \Rightarrow \texttt{Sensitivity_dy_y_damp} \\
\mathcal{S}_{d_{v_0}v_0} \text{(with voltage loop closed)} & \Rightarrow \texttt{Sensitivity_dy_y_volt}
\end{aligned}\end{split}\]