# Analysis of a proposed connection for the two-winding single-phase self-excited induction generator operating at constant voltage and frequency

## The steady-state model of TWSPSEIG

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.

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., Zo must be singular matrix). This means that the real and imaginary parts of the determinant of “Zo” 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 K1, K2, K3 and K4 are constants. Based on the analytical technique explained above, the necessary set values of the p.u. speed “b”, the auxiliary capacitor “CA” and the main capacitor “CM”, respectively, to insure self-excitation at the desired values of the no-load terminal voltage “VT” and the p.u. frequency “a”, could be computed as shown in the flowchart of Fig. 3.

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 (IM, IA and IL) 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., Zo 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 “CA” and the main capacitor “CM” respectively, to attain the desired values of the load voltage “VT” and the operating p.u. frequency (a), at generally load impedance (ZL) could be computed as shown in the flowchart of Fig. 4.

## Dynamic model of TWSPSEIG

Generally, the induction machine models are developed using simplified model in q-d reference frame attached to stator, rotor or synchronous rotating reference. In this work, a q-d reference frame attached to the stator as shown in Fig. 1 is used to develop the dynamic model of the proposed connection of the TWSPSEIG. The voltage differential equations in the q-d reference frame can be represented as in (13).

$$\left[ {\begin{array}{*{20}c} {V_{\text{qs}} } \\ {V_{\text{ds}} } \\ {V_{\text{qr}} } \\ {V_{\text{dr}} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {R_{{1{\text{M}}}} + pL_{{1{\text{M}}}} } & 0 & {pL_{\text{mag}} } & 0 \\ 0 & {R_{{1{\text{A}}}} + pL_{{1{\text{A}}}} } & 0 & {pL_{\text{magA}} } \\ {pL_{\text{mag}} } & {\frac{{ - \omega_{\text{r}} L_{\text{magA}} }}{t}} & {R_{2} + pL_{2} } & {\frac{{ - \omega_{\text{r}} L_{{2{\text{A}}}} }}{t}} \\ {\omega_{\text{r}} L_{\text{mag}} \cdot t} & {pL_{\text{magA}} } & {\omega_{\text{r}} L_{2} \cdot t} & {R_{{2{\text{A}}}} + pL_{{2{\text{A}}}} } \\ \end{array} } \right] \cdot \left[ {\begin{array}{*{20}c} {I_{\text{qs}} } \\ {I_{\text{ds}} } \\ {I_{\text{qr}} } \\ {I_{\text{dr}} } \\ \end{array} } \right].$$
(13)

At no-load condition, the main and the auxiliary winding currents and the excitation capacitor current are given in (14), the magnetization current components in the quadrature and direct axes, and the magnitude of the magnetizing current are given in (15).

$$I_{\text{qs}} = I_{\text{M}} = I_{\text{CM}} , \;\;\;\;I_{\text{ds}} = I_{\text{A}} ,\;\;\;\;I_{\text{CA}} = I_{\text{qs}} - I_{\text{ds}} ,$$
(14)
$$I_{\text{magq}} = I_{\text{qs}} + I_{\text{qr}} ,\;\;\; I_{\text{magd}} = I_{\text{ds}} + I_{\text{dr}} , \;\;\;\;I_{\text{mag}} = \frac{{\sqrt {I_{{{\text{magq}} + }}^{2} I_{{{\text{magd}} }}^{2} } }}{\sqrt 2 }.$$
(15)

The generator terminal voltages as a function of the generator currents at no-load condition are given by:

$$pV_{\text{T}} = \frac{{I_{\text{qs}} }}{{C_{\text{M}} }},\;\;\;pV_{\text{ds}} = \frac{{(I_{\text{qs}} - I_{\text{ds}} )}}{{C_{\text{A}} }},\;\;\;\;V_{\text{qs}} = - V_{\text{ds}} - V_{\text{T}} .$$
(16)

At load condition, the TWSPSEIG is loaded with a load connected to the generator terminals across the main winding (compensation) capacitor “CM”. To develop the generator dynamic model considering the load, the q-d components of the stator currents have to be computed as in (17).

$$I_{\text{qs}} = I_{\text{CM}} + I_{\text{Lq}} ,\;\;\;\;I_{\text{ds}} = I_{\text{A}} ,\;\;\;\;I_{\text{CA}} = I_{\text{qs}} - I_{\text{ds}} .$$
(17)

The generator terminal voltages (Vqs, Vds) as a function of the generator currents (Iqs, Ids) and the load current (ILq) at load condition are given by:

\begin{aligned} pV_{\text{ds}} = \frac{{(I_{\text{qs}} - I_{\text{ds}} )}}{{C_{\text{A}} }},\;\;\;\;\;\;\;\;pV_{\text{T}} = \frac{{(I_{\text{qs}} - I_{\text{Lq}} )}}{{C_{\text{M}} }}, \hfill \\ pI_{\text{Lq}} = \frac{{(V_{\text{T}} - R_{\text{L}} \cdot I_{\text{Lq}} )}}{{L_{\text{L}} }},\;\;\; V_{\text{qs}} = - V_{\text{T}} - V_{\text{ds}} . \hfill \\ \end{aligned}
(18)

## Results and discussion

The performance characteristics of 0.75 kW, 230 V, 6 A, 4-poles and 50 Hz single-phase induction generator are carried out via the proposed technique. Table 1 gives the parameters of the generator. The magnetization characteristics of the TWSPSEIG are shown in Fig. 5. Results for steady-state and dynamic simulations are provided in this section. In these results, the terminal voltage of the generator is maintained constant at 380 V. This value of the terminal voltage is one of the merits of the proposed connection, since it is higher than the rated value of the used machine. In steady state, MATLAB program was used to estimate the values of the main and the auxiliary winding capacitors, and the relation between the speed and the values of capacitors under the variation of the load and its power factor will be presented.

Figure 6a shows the needed capacitances for self-excitation versus the P.U. speed at constant terminal voltage and rated frequency at no-load conditions. It is noticeable that the needed main capacitance increases with the increase of the speed, while the auxiliary capacitance decreases. Figure 6b shows the variation of the P.U. speed limits versus the terminal voltage to maintain the operating frequency, under no load, constant at the rated value. It may be noted that the minimum speed is almost constant with the increase of the terminal voltage.

If the generator has a variable speed prime mover, it is needed to continuously vary the capacitors to deal with the change of the prime mover speed. Under such condition, it is required to examine the speed ranges within which the generator could generate constant load voltage and frequency for different loads. Figure 7a shows the variation of the P.U. speed limits for the resistive loads, while Fig. 7b shows the variation of the P.U. speed limits for inductive loads with a power factor of 0.97. To maintain constant load voltage and frequency at any speed within the speed ranges, the main and auxiliary capacitors will be determined using the suggested technique. Figure 8 gives the calculated values of the main and auxiliary capacitances versus the operating speed at different loads.

### Dynamic results

For dynamic simulation, MATLAB program was used to show the dynamic response of different voltages and currents building up processes of the studied TWSPSEIG under no-load and load conditions. Figure 9 shows the dynamic response of voltages building up process and the frequency of the terminal voltage of the proposed connection during self-excitation under no-load condition. The generator initially runs at a p.u. speed of 1.0038 (1506 rpm) with the main capacitance of 13.414 µF and an auxiliary capacitance of 11.5 µF. The corresponding main and the auxiliary currents for building up wave forms closely resemble the voltages waveforms. Figure 10 shows the dynamic response of the studied no-load proposed TWSPSEIG connection subjected to a sudden connection of a resistive load of 120 Ω at time 6 s., with a P.U. speed of 1.063, 28.227 µF as the main capacitance and 10.5 µF as the auxiliary capacitance, and then sudden changing of the P.U. speed to 1.065 at time 7 s., with the main capacitance of 30.712 µF and the auxiliary capacitance of 7.05 µF.

Figure 11 shows the dynamic response of the studied no-load proposed TWSPSEIG connection subjected to a sudden connection of an inductive load of 205 Ω with 0.97 power factor at time 6 s. with a p.u. speed of 1.037, 24.266 µF as the main capacitance and 9.7 µF as the auxiliary capacitance, and then sudden changing of the p.u. speed to 1.038 at time 7 s., with the main capacitance of 25.6 µF and auxiliary capacitance of 7.64 µF.

Figures 10 and 11 illustrate that the load voltage and frequency are almost constant regardless of the values of load impedance, load power factor and operating speed, which is a proof of the validity and accuracy of the proposed technique for calculations of suitable main and auxiliary capacitances.

Table 2 summaries the other cases of dynamic response of the studied no-load TWSPSEIG subjected to a sudden connection of different resistive loads. Table 3 summarizes the other cases of dynamic response of the studied no-load TWSPSEIG subjected to a sudden connection of different inductive loads with different power factors. It is observed that the speed decreases as the load current decreases at constant power factor.

To evaluate the new connection and prove its advantages, the performance of the new connection is compared with that of the traditional connection. In the traditional connection, a TWSPSEIG with an excitation capacitor “CA” connected across the auxiliary winding and a compensation capacitor “CM” shunted with the load across the main winding has been studied. The required set values of capacitors (main and auxiliary) have been calculated at different load conditions and operating speed using the same technique. In both connections, the load voltage is maintained constant at 230 V with the rated frequency. Table 4 shows the comparison of the two connections subjected to a sudden connection of different resistive loads, while Table 5 shows the comparison of two connections subjected to a sudden connection of different inductive loads.

The surveillance of the results in Tables 4 and 5 yields the following significant points:

1. 1.

The operating p.u. speed of the proposed connection is higher than that of the traditional connection at the same load.

2. 2.

The required capacitors of the proposed connection have always lower values than that of the traditional connection at the same load.

3. 3.

The main and auxiliary winding voltages of the proposed connection have always lower values than that of the traditional connection at the same load.

4. 4.

The main winding current of the proposed connection has low values when compared with those of the traditional connection at the same load.

5. 5.

The auxiliary winding current of the proposed connection has high values, but is still in the safe range at the same load.

6. 6.

The efficiency of the proposed connection is lower than that of the traditional connection at the same load, since the operating speed of the proposed connection is higher than that of the traditional connection.

## Availability of data and materials

The data for the machine used was obtained from Ref.: .

## Abbreviations

SPSEIG:

single-phase self-excited induction generators

SWSPSEIG:

single-winding single-phase self-excited induction generators

TWSPSEIG:

two-winding single-phase self-excited induction generators

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## Funding

Not applicable, no funding was received.

## Author information

Authors

### Contributions

HH proposed the research point and contributed in the survey and data collection, in deriving the mathematical analytical model, building the steady-state models and dynamic models for the proposed method and the method found in literature, simulation and derivation of results, analyzing the results, and writing the manuscript and the technical review. HS contributed in the survey and data collection, in deriving the mathematical analytical model, building the steady-state models and dynamic models for the proposed method and the method found in literature, running the simulation and derivation of results, analyzing the results and writing the manuscript and the technical review. AS contributed in deriving the mathematical analytical model, building the steady-state models and dynamic models for the proposed method and the method found in literature, running the simulation and derivation of results, analyzing the results, writing the manuscript, editorial and grammatical review, technical review, submission of paper and applying the received review comments. ME contributed in the editorial and grammatical review and the technical review. All authors read and approved the manuscript.

### Corresponding author

Correspondence to Amr A. Saleh.

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### Competing interests

The authors declare that they have no competing interests. 