Introduction

In a present scenario, the demand for scarce energy resources had necessitated energy-efficient appliances [1]. In all domestic appliances, either universal motor, DC motor, permanent magnet synchronous motor or induction motor is used [2]. In a mixer grinder appliance, a universal motor is used due to its higher starting torque and operating speed than the induction motor [3]. The universal motor operates same as DC series motor. The rotor and stator windings are in series, and the commutator and brush arrangements supply power to the rotor [4]. The design of the universal motor is cheaper to achieve greater speed for smaller devices [5]. However, the wear-out of the commutator and brush in the universal motor can cause sparking and electromagnetic interference [6]. So, the BLDC motor is suitable for the mixer grinder because of its easy speed torque control and compact size [7]. In a few cases, the quality of the mixing material will affect when the mixer grinder is operated at low speed (3000 rpm), which is possible in the BLDC motor [8]. As the torque vs. efficiency characteristics of the BLDC motor is almost flat, the mixer operating at low speed and part load also save energy [9]. In rural India, the availability of power is poor and the load connected is around 300 W. So the proposed BLDC motor power output was fixed at 200 W and the power input was limited to 250 W. The 200 W output power of a BLDC motor is equivalent to existing 500 W universal motor of the mixer grinder.

This paper contributes in an application of a mixer grinder to the performance and losses analysis of a conventional universal motor and proposed BLDC motor. The proposed BLDC motor has lower power consumption compared to the existing universal motor available in a market.

Mathematical Modelling of a Universal Motor

In a universal motor’s mathematical modelling, with the following assumptions voltage equations are given as:

  1. a)

    Magnetic field shows harmonic layout in an air gap;

  2. b)

    Perfect commutating armature; and

  3. c)

    The effect of commutation is ignored [10].

Figure 1 describes the two pole circuit equivalent of a AC universal motor. R1 is resistance of stator, R2 is resistance of rotor machine, L1 is stator inductance, L2 is machine rotor inductance, Φ is the magnetic flux in the stator, and ui is an induced internal armature voltage [11].

Fig. 1
figure 1

AC operating universal motor equivalent circuit

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Figure 2 represents a DC equivalent circuit of an universal motor. Vt is a supply voltage, E is a back emf, ra is a resistance of the stator, rf is the resistance of the rotor, Ia is the current in the armature, If is an excitation current, Vf is the excitation voltage, Lff is self-inductance of the rotor, and Laa is self-inductance of the stator. The mathematical modelling equations are given as [12]:

$$I_{{\text{f}}} = I_{{\text{a}}}$$
(1)
$$V_{{\text{f}}} = r_{{\text{f}}} I_{{\text{a}}} + L_{{{\text{ff}}}} \left( {\frac{{{\text{d}}I_{{\text{a}}} }}{{{\text{d}}t}}} \right)$$
(2)
$$E = L_{{{\text{af}}}} \omega I_{{\text{a}}}$$
(3)
$$V_{{\text{a}}} = r_{{\text{a}}} I_{{\text{a}}} + L_{{{\text{aa}}}} \left( {\frac{{{\text{d}}I_{{\text{a}}} }}{{{\text{d}}t}}} \right) + E$$
(4)
$$V_{{\text{a}}} = r_{{\text{a}}} I_{{\text{a}}} + L_{{{\text{aa}}}} \left( {\frac{{dI_{a} }}{dt}} \right) + (L_{{{\text{af}}}} \omega I_{{\text{a}}} )$$
(5)
$$V_{{\text{t}}} = V_{{\text{a}}} + V_{{\text{f}}}$$
(6)
$$V_{{\text{t}}} = I_{{\text{a}}} (r_{{\text{f}}} + r_{{\text{a}}} + L_{{{\text{af}}}} \omega ) + \left( {\frac{{{\text{d}}I_{{\text{a}}} }}{{{\text{d}}t}}} \right)(L_{{{\text{ff}}}} + L_{{{\text{aa}}}} )$$
(7)
Fig. 2
figure 2

Equivalent DC circuit in universal motor operation [5]

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The mechanical equations of the motor are:

$$T_{{\text{e}}} = L_{{{\text{af}}}} I_{{\text{a}}} I_{{\text{f}}}$$
(8)
$$T_{{\text{e}}} = J\frac{{{\text{d}}\omega }}{{{\text{d}}t}} + B\omega + T_{1}$$
(9)
$$\frac{{{\text{d}}\omega }}{{{\text{d}}t}} = \left( { - \frac{B}{J}} \right)\omega + \frac{{(T_{{\text{e}}} - T_{{\text{L}}} )}}{J}$$
(10)
$$\omega = \int {\left( {\left( {\frac{ - B}{J}} \right)\omega + \frac{{(T_{e} - T_{L} )}}{J}} \right){\text{d}}t}$$
(11)

where J is the coefficient of inertia, TL is a load torque, Te is an electromagnetic torque, \(\omega\) is a rotor mechanical speed, B is the coefficient of friction and Ω is angular velocity [13].

Losses of Universal Motor Equations

Per slot copper loss:

$$J = \frac{IN}{A}$$
(12)
$$P_{{{\text{Cu}}..{\text{slot}}}} = I^{2} R = J^{2} A\rho l_{{{\text{turn}}}}$$
(13)

where J is a current density, I is current, A is a slot, Lturn is a winding turn length, ρ is the copper specific resistance and N is a number of turns [14].

Hysteresis loss:

$$P_{{\text{h}}} = k_{{\text{h}}} B^{n}_{\max } f$$
(14)

where kh is the constant of hysteresis (ranges from 0.1 to 0.6), f is the frequency of flux reversal in the rotor (50 Hz), B is a maximum flux density (2–4 Tesla) and n is a material-dependent exponents ranges from 1.5 to 2.5.

Eddy current losses:

$$P_{{\text{e}}} = k_{{\text{e}}} B^{2} f^{2} m$$
(15)

where ke is the Eddy current constant (almost equal to 1, i.e. 0.8), f is the frequency of flux reversal in a rotor (50 Hz) and B is a maximum flux density (2–4 Tesla).

The square of frequency influences the eddy current losses, and hysteresis losses rise linearly with the frequency.

Field copper loss:

$$P_{{{\text{fieldcu}}}} = I_{{{\text{se}}}}^{2} R_{{{\text{se}}}}$$
(16)

where Ise is the field winding current in series and Rse is the field winding resistance in series [15].

Mathematical Modelling of a BLDC Motor

Considering a cylindrical stator and rotor in three phases, with windings a, b and c. The rotor consists of a permanent magnet with air gap uniform as stator consists of three phase bounded by star connected [16]. For phase a, phase b and phase c, the dynamic equations are:

$$V_{{{\text{an}}}} = R_{{\text{S}}} i_{{\text{a}}} + L\frac{{{\text{d}}i_{{\text{a}}} }}{{{\text{d}}t}} + M\frac{{{\text{d}}i_{{\text{b}}} }}{{{\text{d}}t}} + M\frac{{{\text{d}}i_{{\text{c}}} }}{{{\text{d}}t}} + e_{{\text{a}}}$$
(17)
$$V_{{{\text{bn}}}} = R_{{\text{S}}} i_{{\text{b}}} + L\frac{{{\text{d}}i_{{\text{b}}} }}{{{\text{d}}t}} + M\frac{{{\text{d}}i_{{\text{c}}} }}{{{\text{d}}t}} + M\frac{{{\text{d}}i_{{\text{a}}} }}{{{\text{d}}t}} + e_{{\text{b}}}$$
(18)
$$V_{{{\text{cn}}}} = R_{{\text{S}}} i_{{\text{c}}} + L\frac{{{\text{d}}i_{{\text{c}}} }}{{{\text{d}}t}} + M\frac{{{\text{d}}i_{{\text{a}}} }}{{{\text{d}}t}} + M\frac{{{\text{d}}i_{{\text{b}}} }}{{{\text{d}}t}} + e_{{\text{c}}}$$
(19)

where M is armature mutual inductance, R is armature resistance, Van, Vbn and Vcn are the voltage of the terminals, L is armature self-inductance, and ia, ib and ic are input currents of the motor [17].

$$\left[ {\begin{array}{*{20}c} {V_{{{\text{an}}}} } \\ {V_{{{\text{bn}}}} } \\ {V_{{{\text{cn}}}} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {R_{{\text{S}}} } \\ 0 \\ 0 \\ \end{array} \begin{array}{*{20}c} 0 \\ {R_{{\text{S}}} } \\ 0 \\ \end{array} \begin{array}{*{20}c} 0 \\ 0 \\ {R_{{\text{S}}} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {i_{{\text{a}}} } \\ {i_{{\text{b}}} } \\ {i_{{\text{c}}} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} L \\ M \\ M \\ \end{array} \begin{array}{*{20}c} M \\ L \\ M \\ \end{array} \begin{array}{*{20}c} M \\ M \\ L \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\frac{{{\text{d}}i_{{\text{a}}} }}{{{\text{d}}t}}} \\ {\frac{{{\text{d}}i_{{\text{b}}} }}{{{\text{d}}t}}} \\ {\frac{{{\text{d}}i_{{\text{c}}} }}{{{\text{d}}t}}} \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {e_{{\text{a}}} } \\ {e_{{\text{b}}} } \\ {e_{{\text{c}}} } \\ \end{array} } \right]$$
(20)

The rotor position function in a BLDC motor is related to the back emf. Each back emf phases has a difference of 120° angle of phase [18]. Hence, the equations for each phase are:

$$e_{{\text{a}}} = K_{{\text{a}}} f_{{\text{a}}} (\theta )\omega_{{\text{r}}}$$
(21)
$$e_{{\text{b}}} = K_{{\text{b}}} f_{{\text{b}}} \left( {\theta + \frac{2\pi }{3}} \right)\omega_{{\text{r}}}$$
(22)
$$e_{{\text{c}}} = K_{{\text{c}}} f_{{\text{c}}} \left( {\theta - \frac{2\pi }{3}} \right)\omega_{{\text{r}}}$$
(23)
$$\left[ {\begin{array}{*{20}c} {V_{{{\text{an}}}} } \\ {V_{{{\text{bn}}}} } \\ {V_{{{\text{cn}}}} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {R_{{\text{s}}} } \\ 0 \\ 0 \\ \end{array} \begin{array}{*{20}c} 0 \\ {R_{{\text{s}}} } \\ 0 \\ \end{array} \begin{array}{*{20}c} 0 \\ 0 \\ {R_{{\text{s}}} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {i_{{\text{a}}} } \\ {i_{{\text{b}}} } \\ {i_{{\text{c}}} } \\ \end{array} } \right] + L_{s} \left[ {\begin{array}{*{20}c} {\frac{{{\text{d}}i_{{\text{a}}} }}{{{\text{d}}t}}} \\ {\frac{{{\text{d}}i_{{\text{b}}} }}{{{\text{d}}t}}} \\ {\frac{{{\text{d}}i_{{\text{c}}} }}{{{\text{d}}t}}} \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {e_{{\text{a}}} } \\ {e_{{\text{b}}} } \\ {e_{{\text{c}}} } \\ \end{array} } \right]$$
(24)

The total torque is given as:

$$e_{{\text{a}}} = K_{{\text{a}}} \omega_{{\text{r}}}$$
(25)
$$P_{{\text{m}}} = e_{{\text{a}}} i_{{\text{a}}} + e_{{\text{b}}} i_{{\text{b}}} + e_{{\text{c}}} i_{{\text{c}}}$$
(26)
$$T_{{\text{e}}} = \frac{{P_{{\text{m}}} }}{{\omega_{{{\text{rm}}}} }} = \frac{{(e_{{\text{a}}} i_{{\text{a}}} + e_{{\text{b}}} i_{{\text{b}}} + e_{{\text{c}}} i_{{\text{c}}} )}}{{\omega_{{\text{r}}} }}\frac{P}{2}$$
(27)
$$T_{{\text{e}}} = \frac{{P(K_{{\text{a}}} i_{{\text{a}}} + K_{{\text{b}}} i_{{\text{b}}} + K_{{\text{c}}} i_{{\text{c}}} )}}{2}$$
(28)

Mechanical part:

$$T_{{\text{e}}} - T_{{\text{L}}} = J\frac{{{\text{d}}\omega_{{{\text{rm}}}} }}{{{\text{d}}t}} + B\omega_{{{\text{rm}}}}$$
(29)
$$\frac{{{\text{d}}\omega_{{{\text{rm}}}} }}{{{\text{d}}t}} = \frac{P}{2J}\left( {T_{{\text{e}}} - T_{{\text{L}}} - \frac{2B}{P}\omega_{{\text{r}}} } \right)$$
(30)

where TL is a load torque, J is a current density, Te is an electromagnetic torque and B is a flux density [7].

Losses Equations of the BLDC Motor

Hysteresis loss:

$$W_{{\text{h}}} = k_{{\text{h}}} B_{{{\text{max}}}}^{\alpha } f$$
(31)

where f is the frequency of flux reversal in the rotor (50 Hz), Bmax is the maximum flux density (0.638 Tesla), kh is the hysteresis constant (ranges from 0.1 to 0.6) and \(\propto\) is the constant (1.5–2.5).

Copper loss:

$$W_{{{\text{cu}}}} = I_{{{\text{rms}}}}^{2} R_{{\text{a}}}$$
(32)

where Irms is the current RMS value and Ra is the armature resistance.

Eddy current loss:

$$W_{{\text{e}}} = k_{{\text{e}}} B_{\max }^{2} f^{2}$$
(33)

where ke is the eddy current constant (almost equal to 1, i.e. 0.8), f is the frequency of flux reversal in a rotor (50 Hz) and B is the maximum flux density (0.638 Tesla) [20, 21].

Simulink Model of an Universal Motor

Table 1 presents Simulink model specification of an AC operating universal motor. The conventional mixer grinder of 600 W power rated with voltage of 230 V AC and rated speed of 20,000 rpm was considered. The operating torque is based on the size of a jar (0.4 Litre, 1.3 Litres and 1.8 Litres) and the materials used. As per the market available mixer grinders, the required torque is in the range of 0.2–0.4 Nm. Figure 3 demonstrates Simulink model of an universal motor in state of AC operation.

Table 1 Specifications of the AC universal motor Simulink model [5]
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Fig. 3
figure 3

Simulink model of an AC operating universal motor

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Table 2 depicts parameters of Simulink model of a DC operating universal motor. A mixer grinder speed of 20,000 rpm, 600 W rated power, current of 2.73 A and voltage of 220 V DC was considered. Figure 4 depicts the Simulink model of a DC operating universal motor. Due to the presence of resistance, the current gets restricted, and there is no effect of saturation.

Table 2 Specifications of the DC universal motor Simulink model [5]
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Fig. 4
figure 4

Simulink model of the DC operating universal motor

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Figure 5 shows the load torque vs. speed characteristics of AC and DC operating the universal motor. For different speeds, the load torque at DC operating condition was more than the AC operating condition. Due to the presence of reactance in the AC operating condition of the universal motor, there was a drop in the speed curve.

Fig. 5
figure 5

Load torque versus speed characteristics of AC and DC operating a universal motor

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Table 3 presents a Simulink model parameters of BLDC motor. A mixer grinder with a speed of 10,000 rpm, 200 W rated power and voltage of 48 V was considered. Figure 6 shows the Simulink model of a BLDC motor.

Table 3 Parameters of BLDC motor’s Simulink model
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Fig. 6
figure 6

Simulink model of 48 V BLDC motor

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Figure 7 represents the load torque vs. speed characteristics of a 48 V BLDC motor. The torque decreases with an increase in speed. It can be observed that the starting torque was higher in the BLDC motor than in the universal motor.

Fig. 7
figure 7

Load torque versus speed characteristics of 48 V BLDC motor

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Figure 8 shows the efficiency vs. speed characteristics of AC and DC operating the universal motor and BLDC motor for different speeds. It can be seen that the efficiency of an AC operating universal motor is 52.16%, and the DC operating universal motor efficiency is 54.5%, while that of the 48 V BLDC motor is 81.15%. The efficiency of 48 V BLDC motor is higher than 230 V AC and 220 V DC operating universal motor.

Fig. 8
figure 8

Efficiency versus speed characteristics of a universal motor and BLDC motor

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Losses Analysis of the Universal Motor and BLDC Motor

Efficiency is the ratio of power output to power input, depending on the different power losses, i.e. iron loss, ventilation loss, copper loss, friction loss, mechanical loss and so on. The losses in the 230 V and 220 V universal motor and 48 V BLDC motor are depicted in Tables 4, 5 and 6.

Table 4 Universal motor of 230 V AC operating losses
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Table 5 Losses of 220 V DC universal motor
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Table 6 Losses of 48 V BLDC motor
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Tables 4, 5 and 6 show that the losses in the 48 V BLDC motor are lesser than those in the 230 V AC and 220 V DC universal motor. The loss in the 220 V DC universal motor is lesser than in the 230 V AC universal motor.

Philips HL 1643/04 model is configured to prove experimental. The model specifications are 600 W, 230 V and speed of 20,000 rpm. The energy and power quality analyser of the Flux 435 series II is used to take the readings as depicted in Fig. 9.

Fig. 9
figure 9

Experimental test of the conventional mixer grinder

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Comparison of proposed simulated 48 V BLDC motor and experimental conventional mixer grinder is presented in Tables 7 and 8. Compared to the proposed system, the amount of distortion is high in the existing one. When the crest factor is 1.42, no distortion exists, and if the crest factor is greater than 1.8, then high distortion occurs. The total harmonic distortion of a voltage and current is high in the conventional compared to the proposed system. So, it can be concluded that the proposed 48V BLDC motor has a reduced amount of distortion than the existing in the application of a mixer grinder.

Table 7 Comparison of proposed simulated BLDC motor and experimental conventional mixer grinder under no-load condition
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Table 8 Comparison of proposed simulated 48 V BLDC motor and experimental conventional mixer grinder at load condition
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Conclusions

In a MATLAB/Simulink, a proposed BLDC motor and existing universal motor for the application of a mixer grinder were simulated according to the specifications. The comparative efficiency and loss analysis were determined for the experimental convectional universal motor and BLDC motor. The starting torque in the BLDC motor can be observed to be higher than in the universal motor. Due to the absence of brushes friction, the 48 V BLDC motor had a higher efficiency than the universal motor working 230 V AC and 220 V DC. The losses in the 48 V BLDC motor were lesser than in the universal 230 V AC motor and the universal 220 V DC motor. In an existing system, the total harmonic distortion and crest factor are higher than the proposed system.