The equivalent circuit used for the steady-state analysis of the proposed connection for the TWSPSEIG is shown in Fig. 2. The steady-state equivalent circuit is based on the double revolving field theory under the assumptions that neglecting core loss and all the generator parameters are constants and unaffected by saturation except the magnetizing reactances.

### No-load model

At no load, the loop equations for the main and auxiliary currents can be written as:

$$ Z_{0} \cdot I_{{{\text{S}}0}} = 0 , $$

(1)

$$ I_{{{\text{S}}0}} = \left[ {\begin{array}{*{20}c} {I_{\text{M}} } \\ {I_{\text{A}} } \\ \end{array} } \right], $$

(2)

$$ Z_{0} = \left[ {\begin{array}{*{20}c} {Z_{\text{M}} - j\frac{{\left( {X_{\text{CM}} + X_{\text{CA}} } \right)}}{{a^{2} }}} & {\left( {j\frac{{Z_{\text{FBA}} }}{t} + j\frac{{X_{\text{CA}} }}{{a^{2} }}} \right)} \\ {\left( {j\frac{{X_{\text{CA}} }}{{a^{2} }} - jZ_{\text{FBM}} \cdot t} \right)} & {(Z_{\text{A}} - j\frac{{X_{\text{CA}} }}{{a^{2} }})} \\ \end{array} } \right]. $$

(3)

Since for steady state \( I_{{{\text{S}}0}} \ne 0 \), then \( \left| {Z_{\text{o}} } \right|\, = \,0 \) (i.e., *Z*_{o} must be singular matrix). This means that the real and imaginary parts of the determinant of “*Z*_{o}” must be separately zero; this can be simplified to the following equations, where the imaginary part is as in (4) and the real part is as in (5):

$$ Z_{{1{\text{M}}}} = \frac{{R_{{1{\text{M}}}} }}{a} + jX_{{1{\text{M}}}} , \;\; Z_{{1{\text{A}}}} = \frac{{R_{{1{\text{A}}}} }}{a} + jX_{{1{\text{A}}}} , $$

$$ Z_{\text{FM}} = \frac{{\left[ {\frac{{R_{2} }}{{\left[ {2\left( {a - b} \right)} \right]}} + j\frac{{X_{2} }}{2}} \right] \cdot \left( {j\frac{{X_{\text{mag}} }}{2}} \right)}}{{\frac{{R_{2} }}{{\left[ {2\left( {a - b} \right)} \right]}} + j\frac{{X_{2} }}{2} + j\frac{{X_{\text{mag}} }}{2}}},\;\;\;\;Z_{\text{BM}} = \frac{{\left[ {\frac{{R_{2} }}{{\left[ {2\left( {a + b} \right)} \right]}} + j\frac{{X_{2} }}{2}} \right] \cdot \left( {j\frac{{X_{\text{mag}} }}{2}} \right)}}{{\frac{{R_{2} }}{{\left[ {2\left( {a + b} \right)} \right]}} + j\frac{{X_{2} }}{2} + j\frac{{X_{\text{mag}} }}{2}}}, $$

$$ Z_{\text{FA}} = \frac{{\left[ {\frac{{R_{{2{\text{A}}}} }}{{\left[ {2\left( {a - b} \right)} \right]}} + j\frac{{X_{{2{\text{A}}}} }}{2}} \right] \cdot \left( {j\frac{{X_{\text{magA}} }}{2}} \right)}}{{\frac{{R_{{2{\text{A}}}} }}{{\left[ {2\left( {a - b} \right)} \right]}} + j\frac{{X_{{2{\text{A}}}} }}{2} + j\frac{{X_{\text{magA}} }}{2}}},\;\;\;\; Z_{\text{BA}} = \frac{{\left[ {\frac{{R_{{2{\text{A}}}} }}{{\left[ {2\left( {a + b} \right)} \right]}} + j\frac{{X_{{2{\text{A}}}} }}{2}} \right] \cdot \left( {j\frac{{X_{\text{magA}} }}{2}} \right)}}{{\frac{{R_{{2{\text{A}}}} }}{{\left[ {2\left( {a + b} \right)} \right]}} + j\frac{{X_{{2{\text{A}}}} }}{2} + j\frac{{X_{\text{magA}} }}{2}}}, $$

$$ Z_{\text{M}} = Z_{{1{\text{M}}}} + Z_{\text{FM}} + Z_{\text{BM}} ,\;\;\;\;Z_{\text{A}} = Z_{{1{\text{A}}}} + Z_{\text{FA}} + Z_{\text{BA}} , $$

$$ Z_{\text{FBM}} = Z_{\text{FM}} - Z_{\text{BM}} , \;\;\;Z_{\text{FBA}} = Z_{\text{FA}} - Z_{\text{BA}} , $$

$$ X_{\text{CM}} = \frac{{a^{2} }}{{R_{\text{A}} }}*\left( {A + B*X_{\text{CA}} } \right), $$

(4)

$$ {\text{AA}}*X_{\text{CA}}^{2} + {\text{BB}}*X_{\text{CA}} + {\text{CC}} = 0, $$

(5)

$$ A = X_{\text{MA}} - X_{\text{FBMA}} ,\;\;\;B = \frac{1}{{a^{2} }}*\left( { \frac{{X_{\text{FBA}} }}{t} - t*X_{\text{FBM}} - R_{\text{M}} - R_{\text{A}} } \right), $$

$$ C = R_{\text{MA}} - R_{\text{FBMA}} ,\;\;\;\;D = \frac{1}{{a^{2} }}*\left( { \frac{{R_{\text{FBA}} }}{t} - t*R_{\text{FBM}} + X_{\text{M}} + X_{\text{A}} } \right) $$

$$ {\text{AA}} = \frac{B}{{a^{2} *R_{\text{A}} }},\;\;\;\;{\text{BB}} = \frac{A}{{a^{2} *R_{\text{A}} }} - \frac{{B*X_{\text{A}} }}{{R_{\text{A}} }} - D,\;\;\;\;{\text{CC}} = - \frac{{A*X_{\text{A}} }}{{R_{\text{A}} }} - C, $$

$$ Z_{\text{FBMA}} = Z_{\text{FBM}} *Z_{\text{FBA}} = R_{\text{FBMA}} + jX_{\text{FBMA}} ,\;\;\;\;Z_{\text{MA}} = Z_{\text{M}} *Z_{\text{A}} = R_{\text{MA}} + jX_{\text{MA}} . $$

The saturation portions of the magnetizing reactance of the main and the auxiliary against the air gap voltage can be piecewise linearized and expressed arithmetically in the form:

$$ X_{\text{mag}} = K_{1} - K_{2} *\left( {\frac{{V_{\text{g}} }}{a}} \right), $$

(6)

$$ X_{\text{magA}} = K_{3} - K_{4} *\left( {\frac{{V_{\text{gA}} }}{a}} \right), $$

(7)

where *K*_{1}, *K*_{2}, *K*_{3} and *K*_{4} are constants. Based on the analytical technique explained above, the necessary set values of the p.u. speed “*b*”, the auxiliary capacitor “*C*_{A}” and the main capacitor “*C*_{M}”, respectively, to insure self-excitation at the desired values of the no-load terminal voltage “*V*_{T}” and the p.u. frequency “*a*”, could be computed as shown in the flowchart of Fig. 3.

### Inductive load model

In this section, a direct and simple technique, to calculate the necessary values of the main and auxiliary capacitors for inductive load conditions is developed to attain the desired values of terminal voltage and frequency. The loop equations for the currents (*I*_{M}, *I*_{A} and *I*_{L}) are given as:

$$ Z \cdot I_{\text{S}} = 0, $$

(8)

$$ I_{\text{s}} = \left[ {\begin{array}{*{20}c} {I_{\text{M}} } \\ {I_{\text{A}} } \\ {I_{\text{L}} } \\ \end{array} } \right], $$

(9)

$$ Z = \left[ {\begin{array}{*{20}c} {Z_{\text{M}} - j\frac{{\left( {X_{\text{CM}} + X_{\text{CA}} } \right)}}{{a^{2} }}} & {\left( {j\frac{{Z_{\text{FBA}} }}{t} + j\frac{{X_{\text{CA}} }}{{a^{2} }}} \right)} & {j\frac{{X_{\text{CM}} }}{{a^{2} }}} \\ {\left( {j\frac{{X_{\text{CA}} }}{{a^{2} }} - jZ_{\text{FBM}} \cdot t} \right)} & {\left( {Z_{\text{A}} - j\frac{{X_{\text{CA}} }}{{a^{2} }}} \right)} & 0 \\ {j\frac{{X_{\text{CM}} }}{{a^{2} }}} & 0 & {\left( {Z_{\text{L}} - j\frac{{X_{\text{CM}} }}{{a^{2} }}} \right)} \\ \end{array} } \right], $$

(10)

$$ Z_{\text{L}} = \frac{{R_{\text{L}} }}{a} + jX_{\text{L}} . $$

From (9) since under steady state, \( I_{\text{s}} \ne 0 \), then \( \left| Z \right|\, = \,0 \) (i.e., *Z*_{o} must be singular matrix). This means that the real and imaginary parts of the determinant of “*Z*” must be separately zero; this could be rearranged to the following two nonlinear equations where the imaginary part is as in (11) and the real part is as in (12):

$$ X_{\text{CA}} = \frac{{X_{{{\text{H}}1}} + R_{{{\text{H}}2}} \cdot \frac{{X_{\text{CM}} }}{{a^{2} }}}}{{A_{2} \cdot \left( {\frac{{X_{\text{CM}} }}{{a^{4} }} } \right) - A_{1} }}, $$

(11)

$$ {\text{AA}}1*X_{\text{CM}}^{2} + {\text{BB}}1*X_{\text{CM}} + {\text{CC}}1 = 0, $$

(12)

where:

$$ {\text{AA}}1 = \frac{{A_{4} *R_{{{\text{H}}2}} - A_{2} *X_{{{\text{H}}2}} }}{{a^{6} }}, \;\;\;\;{\text{BB}}1 = \frac{{A_{2} *R_{{{\text{H}}1}} }}{{a^{4} }} + \frac{{A_{1} *X_{{{\text{H}}2}} }}{{a^{2} }} + \frac{{A_{3} *R_{{{\text{H}}2}} }}{{a^{4} }} + \frac{{A_{4} *X_{{{\text{H}}1}} }}{{a^{4} }}, $$

$$ {\text{CC}}1 = \frac{{A_{3} *X_{{{\text{H}}1}} }}{{a^{2} }} - A_{1} *R_{{{\text{H}}1}} , $$

$$ A_{1} = X_{{{\text{H}}3}} - R_{{{\text{H}}4}} ,\;\;\;A_{2} = X_{{{\text{H}}5}} + R_{{{\text{H}}6}} , $$

$$ A_{3} = R_{{{\text{H}}3}} + X_{{{\text{H}}4}} ,\;\;\;A_{4} = X_{{{\text{H}}6}} - R_{{{\text{H}}5}} , $$

$$ Z_{{{\text{H}}1}} = Z_{\text{MAL}} - Z_{\text{FBMAL}} = R_{{{\text{H}}1}} + jX_{{{\text{H}}1}} , \;\;\;\;Z_{{{\text{H}}2}} = Z_{\text{FBMA}} - Z_{\text{MA}} - Z_{\text{AL}} = R_{{{\text{H}}2}} + jX_{{{\text{H}}2}} , $$

$$ Z_{{{\text{H}}3}} = \frac{{Z_{\text{FBAL}} }}{t} - Z_{\text{FBML}} *t = R_{{{\text{H}}3}} + jX_{{{\text{H}}3}} , \;\;\;Z_{{{\text{H}}4}} = Z_{\text{ML}} + Z_{\text{AL}} = R_{{{\text{H}}4}} + jX_{H4} , $$

$$ Z_{{{\text{H}}5}} = Z_{\text{M}} + Z_{\text{A}} + Z_{\text{L}} = R_{{{\text{H}}5}} + jX_{{{\text{H}}5}} ,\;\;\;\;\;Z_{{{\text{H}}6}} = \frac{{Z_{\text{FBA}} }}{t} - Z_{\text{FBM}} *t = R_{{{\text{H}}6}} + jX_{{{\text{H}}6}} , $$

$$ Z_{\text{FBMAL}} = Z_{\text{FBM}} *Z_{\text{FBA}} *Z_{\text{L}} = R_{\text{FBMAL}} + jX_{\text{FBMAL}} ,\;\;\;\;Z_{\text{FBML}} = Z_{\text{FBM}} *Z_{\text{L}} = R_{\text{FBML}} + jX_{\text{FBML}} , $$

$$ Z_{\text{FBAL}} = Z_{\text{FBA}} *Z_{\text{L}} = R_{\text{FBAL}} + jX_{\text{FBAL}} , \;\;\;\;Z_{\text{MAL}} = Z_{\text{M}} *Z_{\text{A}} *Z_{\text{L}} = R_{\text{MAL}} + jX_{\text{MAL}} , $$

$$ Z_{\text{ML}} = Z_{\text{M}} *Z_{\text{L}} = R_{\text{ML}} + jX_{\text{ML}} ,\;\;\;\; Z_{\text{AL}} = Z_{\text{A}} *Z_{\text{L}} = R_{\text{AL}} + jX_{\text{AL}} . $$

Based on the analytical technique explained above, the necessary set values of the p.u. speed “*b*”, the auxiliary capacitor “*C*_{A}” and the main capacitor “*C*_{M}” respectively, to attain the desired values of the load voltage “*V*_{T}” and the operating p.u. frequency (*a*), at generally load impedance (*Z*_{L}) could be computed as shown in the flowchart of Fig. 4.