3.1.1 Application Example 3.1: Steady-State Operation of Induction Motor at Undervoltage

A P_out = 100 hp, V_{L-L_rat} = 480 V, f = 60 Hz, p = 6 pole, three-phase induction motor runs at full load and rated voltage with a slip of 3%. Under conditions of stress on the power system, the line-to-line voltage drops to V_{L-L_low} = 430 V. If the load is of the constant torque type, compute for the lower voltage:

a) Slip s_low (use small-slip approximation).

b) Shaft speed n_{m_low} in rpm.

c) Output power P_{out_low}.

d) Rotor copper loss P_{cur_low} = (I_r′)²R_r′ in terms of the rated rotor copper loss at rated voltage.

Solution to Application Example 3.1

a) Based on the small-slip approximation $s\propto \frac{1}{V^2}$, the new slip at the low voltage is

$s_{\mathrm{low}}=0.03{\left(480/430\right)}^2=0.0374\text{.}$

b) With the synchronous speed $n_s=\frac{120f}{p}=\frac{120\left(60\right)}{6}=1200\mathrm{rpm}$ one calculates the shaft speed at low voltage as

${n_m}_{\_\mathrm{low}}=\left(1–s_{\mathrm{low}}\right)n_s=\left(1–0.0374\right)\left(1200\right)=1155.14\mathrm{rpm}\text{.}$

c) The output power at low voltage is P_out = ω_m · T_m. For constant mechanical torque one gets $P_{\mathit{out}\_\mathit{low}}=\frac{\left(1-0.0374\right)}{\left(1-0.03\right)}\left(100\mathit{hp}\right)=99.23\mathit{hp}$

d) Due to the relation $T=\frac{P_g}{{\omega }_{s1}}$ (where P_g is the air-gap power), the torque is proportional to P_g, and therefore, for constant torque operation the air-gap power is constant. Thus the rotor loss is $P_{cur\_\mathit{low}}=s_{\mathit{low}}{P}_g=\frac{0.0374}{0.03}{P}_{\mathit{out}\_\mathit{rat}}=1.247P_{\mathit{out}\_\mathit{rat}}$

One concludes that a decrease of the terminal voltage increases the rotor loss (temperature) of an induction motor.

3.1.2 Application Example 3.2: Steady-State Operation of Induction Motor at Overvoltage

Repeat Application Example 3.1 for the condition when the line-to-line voltage of the power supply system increases to V_{L-L_high} = 510 V.

Solution to Application Example 3.2

a) Based on the small-slip approximation $s\propto \frac{1}{V^2}$ the new slip at the high voltage is

$s_{\mathrm{high}}=0.03{\left(480/510\right)}^2=0.02657\text{.}$

b) With the synchronous speed $n_s=\frac{120f}{p}=\frac{120\left(60\right)}{6}=1200\mathrm{rpm}$ one calculates the shaft speed at high voltage as n_{m_high} = (1 – s_high)n = (1 – 0.02657)(1200) = 1168.11 rpm.

c) The output power at high voltage is P_out = ω_m · T_m. For constant mechanical torque one gets $P_{\mathrm{out}\_\mathrm{high}}=\frac{\left(1-0.02657\right)}{\left(1-0.03\right)}\left(100\mathrm{hp}\right)=100.35\mathrm{hp}\text{.}$

d) Due to the relation $T=\frac{P_g}{{\omega }_{s1}}$, where P_g is the air-gap power, the torque is proportional to P_g, and therefore, for constant torque operation the air-gap power is constant. Thus the rotor loss is P_{cur_high} = s_highP_g = $\frac{0.02657}{0.03}$ P_{cur_rat} = 0.886 P_{cur_rat}.

One concludes that an increase of the terminal voltage decreases the rotor loss (temperature) of an induction motor.

3.1.3 Application Example 3.3: Steady-State Operation of Induction Motor at Undervoltage and Under-Frequency

Repeat Application Example 3.1 for the condition when the line-to-line voltage of the power supply system and the frequency drop to V_{L-L_low} = 430 V and f_low = 59 Hz, respectively.

Solution to Application Example 3.3

The small-slip approximation results in the torque equation $T=\frac{q_1\cdot V_s^2\cdot s}{{\omega }_{s1}\cdot R_r^{\prime }}\text{.}$

a) Applying the relation to rated conditions (1) one gets $T^{\left(1\right)}=\frac{q_1\cdot V_s^{2\left(1\right)}\cdot s^{\left(1\right)}}{{\omega }_{s1}^{\left(1\right)}\cdot R_r^{\prime }}\text{.}$
Applying the relation to low-voltage and low-frequency conditions (2) one gets $T^{\left(2\right)}=\frac{q_1\cdot V_s^{2\left(2\right)}\cdot s^{\left(2\right)}}{{\omega }_{s1}^{\left(2\right)}\cdot R_r^{\prime }}\text{.}$
With T⁽¹⁾ = T⁽²⁾ and the relations ${\omega }_{s1}^{\left(1\right)}=\frac{2\pi f^{\left(1\right)}}{p/2}\mathrm{and}{\omega }_{s1}^{\left(2\right)}=\frac{2\pi f^{\left(2\right)}}{p/2}$
one obtains $s^{\left(2\right)}=s^{\left(1\right)}\frac{{\left(V_s^{\left(1\right)}\right)}^2}{{\left(V_s^{\left(2\right)}\right)}^2}\frac{f^{\left(2\right)}}{f^{\left(1\right)}}=0.0367\text{.}$

b) The synchronous speed n_s⁽²⁾ at f⁽²⁾ = 59 Hz is $n_s^{\left(2\right)}=\frac{120\left(59\right)}{6}=1180\mathrm{rpm}$, or the corresponding shaft speed $n_m^{\left(2\right)}=n_s^{\left(2\right)}\left(1-s^{\left(2\right)}\right)=1137\mathrm{rpm}\text{.}$

c) The torque-output power relations are $T^{\left(1\right)}=\frac{P_{\mathrm{out}}^{\left(1\right)}}{{\omega }_m^{\left(1\right)}}\mathrm{and}{T}^{\left(2\right)}=\frac{P_{\mathrm{out}}^{\left(2\right)}}{{\omega }_m^{\left(2\right)}}\text{.}$
With T⁽¹⁾ = T⁽²⁾ one gets $\frac{P_{\mathrm{out}}^{\left(1\right)}}{{\omega }_m^{\left(1\right)}}=\frac{P_{\mathrm{out}}^{\left(2\right)}}{{\omega }_m^{\left(2\right)}}$ or $\frac{P_{\mathrm{out}}^{\left(1\right)}}{{\omega }_{s1}^{\left(1\right)}\left(1-s^{\left(1\right)}\right)}=\frac{P_{\mathrm{out}}^{\left(2\right)}}{{\omega }_{s1}^{\left(2\right)}\left(1-s^{\left(2\right)}\right)}$ or $\frac{P_{\mathrm{out}}^{\left(1\right)}}{\frac{120f^{\left(1\right)}}{p}\left(1-s^{\left(1\right)}\right)}=\frac{P_{\mathrm{out}}^{\left(2\right)}}{\frac{120f^{\left(2\right)}}{p}\left(1-s^{\left(2\right)}\right)}\text{.}$ For the reduced output power it follows $P_{\mathrm{out}}^{\left(2\right)}=\frac{\left(1-s^{\left(2\right)}\right)}{\left(1-s^{\left(1\right)}\right)}\frac{f^{\left(2\right)}}{f^{\left(1\right)}}{P}_{\mathrm{out}}^{\left(1\right)}=97.65\mathrm{hp}\text{.}$

d) The copper losses of the rotor are $P_{\mathrm{c}\mathrm{u}\mathrm{r}}=q_1{\left(I_r^{\prime }\right)}^2{R}_r^{\prime }$ and the output power is $P_{\mathrm{out}}=q_1{\left(I_r^{\prime }\right)}^2{R}_r^{\prime }\frac{\left(1-s\right)}{s}\text{.}$ The application of the relations to the two conditions (1) and (2) yields

$\begin{array}{l}{P}_{\mathrm{out}}^{\left(1\right)}=q_1{\left({I_r^{\prime }}^{\left(1\right)}\right)}^2{R}_r^{\prime }\frac{\left(1-s^{\left(1\right)}\right)}{s^{\left(1\right)}}\mathrm{and}\\ P_{\mathrm{out}}^{\left(2\right)}=q_1{\left({I_r^{\prime }}^{\left(2\right)}\right)}^2{R}_r^{\prime }\frac{\left(1-s^{\left(2\right)}\right)}{s^{\left(2\right)}}\mathrm{or}\\ \frac{{\left({I_r^{\prime }}^{\left(2\right)}\right)}^2}{{\left({I_r^{\prime }}^{\left(1\right)}\right)}^2}=\frac{P_{\mathrm{out}}^{\left(2\right)}}{P_{\mathrm{out}}^{\left(1\right)}}\frac{s^{\left(2\right)}}{s^{\left(1\right)}}\frac{\left(1-s^{\left(1\right)}\right)}{\left(1-s^{\left(2\right)}\right)}=1.20\text{.}\end{array}$

The rotor loss increase becomes now $\frac{P_{\mathrm{c}\mathrm{u}\mathrm{r}}^{\left(2\right)}}{P_{\mathrm{c}\mathrm{u}\mathrm{r}}^{\left(1\right)}}=\frac{{\left({I_r^{\prime }}^{\left(2\right)}\right)}^2}{{\left({I_r^{\prime }}^{\left(1\right)}\right)}^2}=1.20\text{.}$

One concludes that the simultaneous reduction of voltage and frequency increases the rotor losses (temperature) in a similar manner as discussed in Application Example 3.1. It is advisable not to lower the power system voltage beyond the levels as specified in standards.

3.3.1 Application Example 3.4: Unstable and Stable Steady-State Operation of Induction Machines

Figure E3.4.1 shows the torque–angular velocity characteristic of an induction motor (T – ω_m) supplying a mechanical constant torque load (T_L). Determine the stability of the operating points Q₁ and Q₂.

Solution to Application Example 3.4

The stability conditions at the two points where the two curves intersect are

Q1:	$\left.\begin{array}{l}\frac{\Delta T_L}{\Delta {\omega }_m}=0\\ \frac{\Delta T}{\Delta {\omega }_m}>\mathrm{positive}.\end{array}\right\}0<\mathrm{positive}>>>>>>\mathbf{unstable}\text{.}$
Q2:	$\left.\begin{array}{l}\frac{\Delta T_L}{\Delta {\omega }_m}=0\\ \frac{\Delta T}{\Delta {\omega }_m}=\mathrm{negative}.\end{array}\right\}0>\mathrm{negative}>>>>>>\mathbf{stable}\text{.}$

Therefore, operation at Q₁ is unstable and will result in rapid speed reduction. This will stop the operation of the induction machine, while operating point Q₂ is stable.

3.3.2 Application Example 3.5: Stable Steady-State Operation of Induction Machines

Figure E3.5.1 shows the torque–angular velocity relation (T – ω_m) of an induction motor supplying a mechanical load with nonlinear (parabolic) torque (T_L). Determine the stability of the operating point Q₁.

Solution to Application Example 3.5

The stability conditions at the point where the two curves intersect is

Q1:	$\left.\begin{array}{l}\frac{\Delta T_L}{\Delta {\omega }_m}=\mathrm{positive}\\ \frac{\Delta T}{\Delta {\omega }_m}=\mathrm{negative}.\end{array}\right\}>>>>>>\mathbf{stable}\text{.}$

Therefore, an induction machine connected to a mechanical load with parabolic torque-speed characteristics as shown in Fig. E3.5.1 will not experience any steady-state stability problems.

3.3.3 Resolving Mismatch of Wind-Turbine and Variable-Speed Generator Torque–Speed Characteristics

It is well known that the torque–speed characteristic of wind turbines (Fig. 3.18) and commonly available variable-speed drives employing field weakening (Fig. 3.19a) do not match because the torque of a wind turbine is proportional to the square of the speed and that of a variable-speed drive is inversely proportional to the speed in the field-weakening region. One possibility to mitigate this mismatch is an electronic change in the number of poles p or a change of the number of series turns of the stator winding [21–23]. The change of the number of poles is governed by the relation

$\frac{E\cdot p}{f}=4.44B_{\text{max}}\cdot N_{\mathrm{rat}}\cdot 4\cdot R\cdot L\text{,}$

where B_max is the maximum flux density at the radius R of the machine, N_rat is the rated number of series turns of a stator phase winding, and L is the axial iron-core length of the machine.

The change of the number of turns N is governed by

$\frac{E}{f\cdot N}=4.44B_{\text{max}}\cdot 4\cdot R\cdot L/p\text{.}$

The introduction of an additional degree of freedom – either by a change of p or N – permits an extension of the constant flux region. Adding this degree of freedom will permit wind turbines to operate under stalled conditions at all speeds, generating maximum power at a given speed with no danger of a runaway. This will make the blade-pitch control obsolete – however, a furling of the blades at excessive wind velocities must be provided – and wind turbines become more reliable and less expensive due to the absence of mechanical control. Figure 3.19b illustrates the speed and torque increase due to the change of the number of turns from N_rat to N_rat/2, and Fig. 3.19c demonstrates the excellent dynamic performance of winding switching with solid-state switches from number of poles p₁ to p₂ [21]. The winding reconfiguration occurs near the horizontal axis (small slip) at 2400 rpm.

3.6.1 The Fundamental Slip of an Induction Machine

The fundamental slip is

$s_1=\frac{{\omega }_{s1}-{\omega }_m}{{\omega }_{s1}}$

or

$s_1=1-\frac{{\omega }_m}{{\omega }_{s1}}\Rightarrow \frac{{\omega }_m}{{\omega }_{s1}}=1-s_1$

where ${\omega }_{s1}=\frac{{\omega }_1}{p/2}$ is the (mechanical) synchronous fundamental angular velocity and ω_m is the mechanical angular shaft velocity.

The fundamental torque referred to fundamental slip s₁ is shown in Fig. 3.26 where T_e1 is the machine torque and T_L is the load torque. If R_s = 0 then T_e1 is symmetric to the point at (T_e1 = 0/s₁ = 0) or (T_e1 = 0/ω_s1).

3.6.2 The Harmonic Slip of an Induction Machine

The harmonic slip (without addressing the direction of rotation of the harmonic field) is defined as

$s_h=\frac{h{\omega }_{s1}-{\omega }_m}{h{\omega }_{s1}}\text{,}$

where ω_s1 = (ω₁)/(p/2) and ω₁ is the electrical angular velocity, ω₁ = 2πf₁ and f₁ = 60 Hz.

To include the direction of rotation of harmonic mmfs, in the following we assume that the fundamental rotates in forward direction, the 5th in backward direction, and the 7th in forward direction (see Table 3.2 for positive-, negative-, and zero-sequence components).

For motor operation ω_m < ω_s1; thus for the 5th harmonic component, one obtains

$s_5=\frac{-5{\omega }_{s1}-{\omega }_m}{-5{\omega }_{s1}}\text{,}$

where (–5ω_s1) means rotation in backward direction, or

$s_5=\frac{5{\omega }_{s1}+{\omega }_m}{5{\omega }_{s1}}\text{.}$

Note that in this equation 5ω_s1 is the base (reference) angular velocity.

Correspondingly, one obtains for the forward rotating 7th harmonic:

$s_7=\frac{7{\omega }_{s1}-{\omega }_m}{7{\omega }_{s1}}\text{,}$

where 7ω_s1 is the base.

Therefore, the harmonic slip is

$s_h=\frac{h{\omega }_{s1}-\left(\pm \right){\omega }_m}{h{\omega }_{s1}},\begin{array}{c}\hfill \left(+\right):\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{forward}\mathrm{rotating}\mathrm{field}\hfill \\ \hfill \left(-\right):\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{backward}\mathrm{rotating}\mathrm{field}\hfill \end{array}\text{.}$

Harmonic torques T_e7 and T_e5 (as a function of the angular velocity ω_m and the harmonic slips s₇ and s₅) are depicted in Figs. 3.27 and 3.28, respectively.

3.6.3 The Reflected Harmonic Slip of an Induction Machine

If ω_s1 is taken as the reference angular velocity, then for the backward rotating fifth harmonic one obtains

$s_5^{\left(1\right)}=\frac{5{\omega }_{s1}+{\omega }_m}{{\omega }_{s1}}\text{,}$

where s₅⁽¹⁾ is the reflected fifth harmonic slip.

Correspondingly, if one takes ω_s1 as reference then for the forward rotating seventh harmonic one gets

$s_7^{\left(1\right)}=\frac{7{\omega }_{s1}-{\omega }_m}{{\omega }_{s1}}\text{,}$

where s₇⁽¹⁾ is the reflected seventh harmonic slip.

In general, the reflected harmonic slip is

$s_h^{\left(1\right)}=\frac{h{\omega }_{s1}-\left(\pm \right){\omega }_m}{{\omega }_{s1}},\begin{array}{c}\hfill \left(+\right):\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{forward}\mathrm{rotating}\mathrm{field}\hfill \\ \hfill \left(-\right):\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{backward}\mathrm{rotating}\mathrm{field}\hfill \end{array}\text{.}$

To understand the combined effects of fundamental and harmonic torques, it is more convenient to plot harmonic torques (T_eh) as a function of the reflected harmonic slips (s_h⁽¹⁾):

• The harmonic torque T_e7 reflected to harmonic slip $s_7^{\left(1\right)}=\frac{7{\omega }_{s1}-{\omega }_m}{{\omega }_{s1}}$ is depicted in Fig. 3.29; and

• The harmonic torque T_e5 reflected to harmonic slip $s_5^{\left(1\right)}=\frac{5{\omega }_{s1}-{\omega }_m}{{\omega }_{s1}}$ is shown in Fig. 3.30.

3.6.4 Reflected Harmonic Slip of an Induction Machine in Terms of Fundamental Slip

The relation between s₁ (fundamental slip) and s_h⁽¹⁾ (reflected harmonic slip) is

$\begin{array}{c}{s}_h^{\left(1\right)}=\frac{h{\omega }_{s1}-{\omega }_m}{{\omega }_{s1}}=\frac{{\omega }_{s1}-{\omega }_m}{{\omega }_{s1}}+\frac{\left(h-1\right){\omega }_{s1}}{{\omega }_{s1}}\\ =1-\frac{{\omega }_m}{{\omega }_{s1}}+h-1=h-\left(1-s_1\right)\text{.}\end{array}$

Therefore, the reflected harmonic slip, in terms of fundamental slip, is

$s_h^{\left(1\right)}=s_1+\left(h-1\right)\text{.}$

For the backward rotating 5th harmonic one gets

$s_5^{\left(1\right)}=-s_1+6\mathrm{thus}{s}_1=-s_5^{\left(1\right)}+6\text{,}$

and for the forward rotating 7th harmonic

$s_7^{\left(1\right)}=s_1+6\mathrm{thus}{s}_1=s_7^{\left(1\right)}-6\text{.}$

Note that

$s_5^{\left(1\right)}=\frac{5{\omega }_{s1}+{\omega }_m}{{\omega }_{s1}}\text{.}$

In general, the reflected harmonic slip s_h⁽¹⁾ for the forward and the backward rotating harmonics is a linear function of the fundamental slip s₁:

$\begin{array}{l}{s}_h^{\left(1\right)}=s_1+\left(\mathrm{h}-1\right)\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{forward}-\mathrm{rotating}\mathrm{field}\\ s_h^{\left(1\right)}=-s_1+\left(\mathrm{h}+1\right)\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{backward}-\mathrm{rotating}\mathrm{field}\text{.}\end{array}$

Superposition of fundamental (h = 1), fifth harmonic (h = 5), and seventh harmonic (h = 7) torque T_e = T_e1 + T_e5 + T_e7 is illustrated in Fig. 3.31. At ω_m = ω_mrated the total electrical torque T_e is identical to the load torque T_L or T_e1 + T_e5 + T_e7 = T_L, where T_e1 is a motoring torque, T_e7 is a motoring torque, and T_e5 is a braking torque. Figure 3.32 shows the graphical phasor representation of superposition of torques as given in Fig. 3.31.

3.6.5 Reflected Harmonic Slip of an Induction Machine in Terms of Harmonic Slip

In this section, the relation between the reflected harmonic slip s_h⁽¹⁾ and the harmonic slip s_h is determined as $s_h^{\left(1\right)}=f\left(s_h\right)\text{.}$

The harmonic slip (without addressing the direction of rotation of the harmonic field) is

$s_h=\frac{h{\omega }_{s1}-{\omega }_m}{h{\omega }_{s1}}=\frac{\left(h-1\right){\omega }_{s1}}{h{\omega }_{s1}}+\frac{{\omega }_{s1}-{\omega }_m}{h{\omega }_{s1}}$

with

$s_1=\frac{{\omega }_{s1}-{\omega }_m}{{\omega }_{s1}}\text{.}$

It follows that

$s_h=\frac{\left(h-1\right){\omega }_{s1}}{h{\omega }_{s1}}+\frac{s_1}{h}\text{.}$

With $s_h^{\left(1\right)}=s_1+\left(h-1\right)$, one obtains for forward rotating harmonics

$s_1=s_h^{\left(1\right)}-\left(h-1\right)$

or

$\begin{array}{c}{s}_h=\frac{\left(h-1\right){\omega }_{s1}}{h{\omega }_{s1}}+\frac{s_h^{\left(1\right)}-\left(h-1\right)}{h}=\frac{\left(h-1\right)}{h}+\frac{s_h^{\left(1\right)}}{h}-\frac{\left(h-1\right)}{h}\\ =\frac{s_h^{\left(1\right)}}{h}\text{.}\end{array}$

Similar analysis can be used for backward-rotating harmonics. Therefore, s_h⁽¹⁾ is a linear function of s_h:

$s_h^{\left(1\right)}=h{s}_h\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{forward}\mathrm{and}\mathrm{backward}\mathrm{rotating}\mathrm{fields}\text{.}$

3.7.1 Measurement of Nonlinear Circuit Parameters of Single-Phase Induction Motors

The optimization of single- and three-phase induction machines is based on sinusoidal quantities neglecting the influence of spatial and time harmonics in flux densities, voltages, and currents [14,15,34,35]. At nonsinusoidal terminal voltages, the harmonic losses of single-phase induction motors impact their efficiency [25]. The computer-aided testing of [24] will be relied on to measure nonlinear circuit parameters and flux densities in stator teeth and yokes as well as in the air gap of permanent-split-capacitor (PSC) motors. Detailed test results are presented in [36]. One concludes that tooth flux density wave shapes can be nearly sinusoidal although the exciting or magnetizing currents are nonsinusoidal.

A computer-aided testing (CAT) circuit and the computer-aided testing of electrical apparatus program (CATEA) are used in [24] to measure nonlinear parameters. The CAT circuit is shown in Fig. 3.35, where four-channel signals v_m, i_m, v_a, and i_a corresponding to main (m) and auxiliary (a) phase voltages and currents, respectively, are sampled by a personal computer via a 12-bit A/D converter.

Three (see Table 3.3) fractional horsepower PSC single-phase induction motors serving as prime movers for window air conditioners are tested and their parameters listed.

All motors to be tested are operated at minimum load, where input and output lines at the compressor are open to air. Measurements are conducted at an ambient temperature of 22°C. The linear stator leakage impedance of a motor is measured by removing the rotor from the stator bore and applying various terminal voltage amplitudes to the stator windings. When the main-phase parameters are measured, the switch leading to the auxiliary phase must be open, and vice versa. From the sampled data one obtains the fundamental voltages and currents of the main- and auxiliary-phase windings through Fourier analysis.

3.7.1.2 Measurement of Flux-Density Harmonics in Stator Teeth and Yoke (Back Iron)

Flux densities in the teeth of rotating machines are of the alternating type, whereas those in the yoke (back iron) are of the rotating type. The loss characteristics for both types are different and these can be requested from the electrical steel manufacturers. The motors to be tested are mounted on a testing frame, and measurements are performed under no load or minimum load with rated run capacitors. Four stator search coils with N = 3 turns each are employed to indirectly sense the flux densities of the teeth and yoke located in the axes of the main- and auxiliary-phase windings (see Fig. 3.36). The induced voltages in the four search coils are measured with two different methods: the computer sampling and the oscilloscope methods.

Measurement via Computer Sampling

The computer-aided testing circuit and program [24] are relied on to measure the induced voltages of the four search coils of Fig. 3.36. The CAT circuit is shown in Figs. 3.36 and 3.37, where eight channel signals (v_in, i_in, i_m, i_a, e_mt, e_at, e_my, and e_ay corresponding to input voltage and current, main- and auxiliary-phase currents, main- and auxiliary-phase tooth search-coil induced voltages, and main- and auxiliary-phase yoke (back iron)-search coil induced voltages, respectively), are sampled [36].

The flux linkages of the search coils are defined as

$\lambda \left(t\right)={\displaystyle \int e\left(t\right)\mathrm{d}t\text{,}}$

  (3-38)

and numerically obtained from

$\begin{array}{l}{\lambda }_0=0\mathrm{and}{\lambda }_{i+1}={\lambda }_i+\mathrm{\Delta }t\left(e_{i+1}+e_i\right)/2,\\ \mathrm{f}\mathrm{o}\mathrm{r}i=0,1,2\dots ,\left(n-1\right)\text{.}\end{array}$

  (3-39)

The average or DC value is

${\lambda }_{\mathrm{a}\mathrm{v}\mathrm{e}}=\frac{1}{n}\left({\displaystyle \sum _{i=0}^{n-1}{\lambda }_i}\right)\text{,}$

  (3-40)

or the AC values are

${\lambda }_i={\lambda }_i-{\lambda }_{\mathrm{a}\mathrm{v}\mathrm{e}},\mathrm{f}\mathrm{o}\mathrm{r}i=0,1,2,3,\dots ,n\text{,}$

  (3-41)

where e(t) of Eq. 3-38 is the induced voltage measured in a search coil and n in Eqs. 3-39 to 3-41 is the number of sampled points. The maximum flux linkage λ_max can be obtained from λ_i. The maximum flux density is given by

${\mathrm{B}}_{\text{max}}={\lambda }_{\text{max}}/\left(N\cdot s\right)\text{,}$

  (3-42)

where the number of turns of each search coil is N = 3 and s is given for a stator tooth and yoke by the following equations:

$s=k_{\mathrm{f}\mathrm{e}}{w}_{t}l\text{,}$

  (3-43)

$s=k_{\mathrm{f}\mathrm{e}}{h}_{y}l\text{,}$

  (3-44)

where w_t, h_y, l, and k_fe are the width of the (parallel) stator teeth (where two search coils reside), the height of the yoke or stator back iron (where two search coils reside), the length of the iron core, and the iron-core stacking factor, respectively. Measured data of PSC motors at rated voltage are presented in Section 3.7.2.

The maximum flux densities in stator teeth and yoke at the axes of the main- and auxiliary-phase windings are measured for various voltage amplitudes by recording the induced voltages in the four search coils. A digital oscilloscope is used to plot the induced voltages of the search coils. These waveforms are then sampled either by hand or by computer based on 83 points per period. Complex Fourier series components ${\tilde{C}}_h$ are computed – taking into account the Nyquist criterion – and the integration is performed in the complex domain as follows:

$e\left(t\right)={\displaystyle \sum _{h=1}^{27}{\tilde{C}}_h{e}^{\mathrm{j}\mathrm{h}\omega \mathrm{t}}}\text{,}$

  (3-45)

$\lambda \left(t\right)={\displaystyle \sum _{h=1}^{27}\frac{1}{jh\omega }{\tilde{C}}_h{e}^{\mathrm{j}\mathrm{h}\omega \mathrm{t}}}\text{.}$

  (3-46)

The maximum flux linkage λ_max is found from λ(t) and the maximum flux densities are determined from Eqs. 3-42 to 3-44.

3.7.2 Application Example 3.6: Measurement of Harmonics within Yoke (Back Iron) and Tooth Flux Densities of Single-Phase Induction Machines

The three permanent-split capacitor (PSC) motors (R333MC, Y673MG1, and 80664346) of Table 3.3 are subjected to the tests described in the prior subsections of Section 3.7, and pertinent results will be presented. These include stator, rotor, and magnetizing parameters, and tables listing no-load input impedances of the main and auxiliary phases (Tables E3.6.1 and E3.6.2). The bold data of the tables correspond to rated main- and auxiliary-phase voltages at no load. The rated auxiliary-phase voltage is obtained by multiplying the rated main-phase voltage with the measured turns ratio $\left|{\tilde{k}}_{\mathit{am}}\right|$. L_m and L_a are inductances of the main- and auxiliary-phase windings, respectively.