When the electromagnetic brake is working, a DC chopper serves to change the conduction time of the electromagnetic brake and power supply. The braking torque required by the controller is adjusted by changing the equivalent voltage on the exciting coil (Zhang et al., 2014). According to the Eq. (14), the exciting current of single exciting coil can be written as: B=8πρFeTNpkSp2Δhω2I=Bμ0lg+f1(B1)l+f2(B2)Δh+2SpBωΔh8πρN Where T is the braking torque required by the controller.

The resistance of the exciting coil is related to the diameter of the copper wire and the size of the coil skeleton, the diameter of the iron core and the electrical resistivity of copper wire. The resistance of an exciting coil can be described as: R0=ρCub×[a(d+2δ)π+a(a-1)πd0]14πd02 Here a is the number of turns in radial direction, a=fix(D-d-2δ2d0). b is number of turns in axial direction, b=fix(H-2δd0). H, D and δ represent the width, external diameter of the skeleton and the thickness of the skeleton, respectively. d0 is the diameter of the copper wires.

The numbers of turns can be obtained as: N=ab The quantity of exciting coil of an electromagnetic brake is 4Np. These two opposite exciting coils are cascaded as an exciting winding. Exciting windings are parallel to each other. The power consumption of electricity can be drawn up as: P=4NpI2R0

The principle prototype of electromagnetic brake which has been designed by Jiangsu Province Key Laboratory of Automotive Engineering is treated as example and the hardware in the loop simulation platform of electromagnetic and friction integration braking system is used to do experimental verification as shown in Fig. 5 (He et al., 2013). Structural parameters of electromagnetic brake is shown in Table 1. In this paper, the dichotomy is used to obtain the air-gap magnetic flux density. In order to solve the nonlinear Eq. (12), a new function is given as follow: P(B)=NI-Fe-Bμ0lg-f1(B1)l-f2(B2)Δh

Braking torque of electromagnetic brake increases as the wheel speed increases if the wheel speed is less than 500 r min-1. When the wheel speed reaches 500 r min-1, braking torque peaked. And if the wheel speed continues to rise, the braking torque really starts to drop. Theoretical calculation curve can be a good approximation of the experimental curve especially in the high-speed region and also follow the downward trend of the torque-speed curve which indicates that this prediction model of braking performance is in a position to articulate the impact of eddy current MMF on air gap MMF. Meanwhile, both curves show the same critical speed reflecting this model can well express the influence of magnetic saturation characteristics of soft magnetic materials. There are differences of amplitude of the experimental curve and the theoretical curves as shown in Fig. 6 which may be caused by subtle changes of the air gap length in the process of installing electromagnetic brake.

Firstly, friction brake continues working several times so that the brake disc elevates its temperature for absorbing the braking energy sufficiently. At different temperature of the disc brake, electromagnetic brake work several times to obtain braking torque–temperature experimental curve which showed in Fig. 7a. The theoretical calculation curve of braking torque changes with temperature of the brake disc agrees well with the experimental curve. But experimental curve can only express the temperature of the brake disc between 20–250∘ due to the difficulty of the experiment operability. With increasing temperature of the brake disc, the decrease of braking torque is limited which indicates its strong fade resistance.

Conducting power supply and excitation coil for a long time so that copper wire temperature rises to 80∘. And then controlling the motor driven flywheel rotates to 300 r min-1 to get braking torque–temperature experimental curve as showed in Fig. 7b. Theoretical calculation curve of braking torque changing with temperature of copper wire goes well with the experimental curves. Figure 3b shows that with increasing temperature of copper wire its resistance becomes large. So the constant excitation current voltage becomes smaller. The Eq. (15) demonstrates that the intensity of the excitation current is proportionate to the air gap magnetic induction. The temperature of copper wire changing from 20 to 80∘, braking torque of electromagnetic brake reduces from 210 to 130 N m which indicates that the temperature of copper wire has a greater effect on the braking torque.

As shown in Fig. 8, the power consumption of electromagnetic energy is larger when the vehicle speed is low. With the rapid decrease of the car speed, the small slope increases slowly, and the power of electromagnetic energy consumption reaches a minimum when the vehicle speed is 50 km h-1. When the vehicle speed is in the range of 30 to 80 km h-1, the electromagnetic brake energy consumption of the electromagnetic brake principle prototype is relatively low, so this vehicle speed interval is called “low power consumption area”; When the vehicle speed is 10 km h-1, the power of electromagnetic energy is more than the minimum, and when the vehicle speed is less than 10 km h-1, the electromagnetic energy consumption will increase with a large slope. Therefore, the electromagnetic braking should be prohibited when the vehicle speed is less than 10 km h-1. Comparing with Fig. 8a and b, it can be found that the change trend of the power consumption of the electromagnetic brake that varies with the vehicle speed has nothing to do with the braking strength that vehicles demand.

There is a “low power consumption area” corresponding to the vehicle speed range 30–80 km h-1 in which the electromagnetic brake works efficiently. Figure 9a shows the ECE–EUDC driving condition ,whose the average braking intensity is 0.079 and the biggest braking intensity is 0.139. The UDDS driving condition is shown in the Fig. 9b, whose the average braking intensity is 0.058 and the biggest braking intensity is 0.148. The average velocity of those two typical city driving conditions is 32 km h-1. According to Figs. 8 and 9, brake speed range of typical city driving conditions is 20–40 km h-1 which gets a little probability to be in the “low power consumption area”. As a result, the principles prototype of the electromagnetic brake consumes much electric energy under the typical city driving conditions. Its structure parameters should be optimized to make sure its ?low power consumption area? can include the brake speed range of typical city driving conditions to decrease its energy consumption considering the main purpose of the electromagnetic brake is to instead the friction brake under city driving conditions.

The main optimization design variables for structural parameters of the electromagnetic brake are as follows (Xia et al., 2013):

The structure parameters of the exciting coil: the width and external diameter of the coil skeleton and the diameter of the iron core;

According to the above analysis, the total energy consumption under ECE–EUDC driving cycle is chosen as the energy saving index and could be written as E=∑i=1n4NpR0∫ω1ω2I2(ω) Where n, ω1 and ω2 represent the brake times under ECE–EUDC driving cycle, the initial angular velocity of the wheel when the single brake starts and finishes, respectively.

Installing an electromagnetic brake will increase the unsprung mass which not only has an influence on the automobile dynamics performance but also leads to additional energy consumption. So that the total weight of the electromagnetic brake is required to be lighter. For the total weight of the electromagnetic brake, the weight of the iron core, the exciting coin and the brake disc account for a large part. As a result, the total weight of the electromagnetic brake can be obtained ignoring the weight of bracket: m=md+2Np(mi+mc) Where, mi, mc and md represent the weight of the iron core, the weight of the exciting coin and the weight of the brake disc, respectively.

Taking such difference of the importance of these two index into consideration, the weighting coefficients are employed. The objective function can be written as: f(x)=a1E+a2m Where, a1 is the weighting coefficient of the power consumption of electromagnetic brake. a2 is the weighting coefficient of the total weight of electromagnetic brake.

The Genetic Algorithm is utilized to obtain the global optimal solution and the constraint condition is dealt by the penalty function method. The comparison between the original parameters and the optimized parameters is shown in Table 2. The center radius of the brake disc and the external diameter of the coil skeletons are increased to the limitation of the constraint condition. Diameter of copper wire and iron core are increased to be bigger in a slight degree. The result of energy saving optimization design shows the total weight of the electromagnetic brake decreases. However, this change is insignificant which is related to the weighting coefficient of the total weight of electromagnetic brake.

As showed in Fig. 10, the optimization results of power consumption of electricity under different speeds are better than the experimental results which also verify the energy consumption of optimization result under UDDS driving circle is lower than the principle prototype as showed in Table 2. The “low power consumption area” of the energy saving optimization design moves to the left and changes from 30–80 to 20–60 km h-1. Meanwhile, as is shown in Fig. 11, the curve of peak braking torque only increased 5∘ N m. However, critical speed changes from 500 to around 350 r min-1.

Involving in the control of the electromagnetic brake, there are two problems needing to be solved. One is the way that how to control the exciting windings when braking, namely the brake task allocation between the excitation windings. The other is the distribution of electromagnetic braking torque in the automobile front and rear axles.

It can be pointed out in Eq. (3) that the resistance value of the excitation winding increases because of its temperature rise during a long time working. And the additional electricity consumption will be generated to ensure the demanded exciting current to be a constant basing on Eqs. (16) and (18). If the other conditions are unchanged, the output braking torque of electromagnetic brake would have a certain degree of attenuation as the temperature of brake disc rises. In order to maintain the demand braking torque, more energy is needed for the electromagnetic brake. Therefore, it is necessary to study the influence of the thermodynamic properties on the energy consumption of electromagnetic brake.

The energy into the control body are required to be equal to the sum of the energy which leaves the control body and its storage energy per unit time according to the energy conservation law (Agrawal, 2004). In the thermodynamics analysis, the following assumptions should be made to simplify the analysis: (1) The air temperature around the electromagnetic brake is consistent with the environment temperature (Limpert, 1975). (2) Due to the low electromagnetic braking torque, it is assumed that there is no slip between the tire and the road. (3) Heat conduction occurs only inside the braking parts and the external cooling could be negligible in the continuous working process of the electromagnetic brake. (4) The temperature distribution in the brake disc is assumed to be uniform for the strong thermal conductivity of the brake disc.

When the electromagnetic brake works, the vehicle's kinetic energy is converted into thermal energy to be dissipated. Part of the thermal energy is stored in the brake disc causing the rise of brake disc temperature and another part of the thermal energy is sent out into the air by the constantly rotating brake disc (Artus et al., 2005). Q=Pe-Pd Where, Pe is the absorbed heat energy of the electromagnetic brake per unit time, namely the electromagnetic braking power. Pd is the heat energy released by the brake disc per unit time. Q is the stored thermal energy of the brake disc per unit time.

The heat energy released by brake disc per unit time can be calculated by the following equation (Ali and Mostefa, 2014): Pd=hdAd(Td1-Td0) Where hd is the sum of the convective heat transfer coefficient and the radiation heat transfer coefficient of the brake disc; Ad is the effective heat dissipation area of the brake disc. Td0 and Td1 are the brake disc temperature before and after braking respectively.

For solid brake disc, when Re>2.4×105, the convective heat transfer coefficient can be expressed as: hcon=0.04KaDRe0.8 Where D is the diameter of the brake disc, Ka is the thermal conductivity of air

When Re≤2.4×105, the convective heat transfer coefficient can be expressed as: hcon=0.7KaDRe0.05 The Reynolds number Re is defined as the ratio of the inertial force and friction force for the fluid flow with the expression of: Re=vρaμaLd Where v is the kinematic viscosity of air, ρa is the air density. μa is the air viscosity, Ld is a defined characteristic length defined by Reynolds number.

The radiation heat transfer coefficient of the brake disc is determined by the following equation: hrad=εσ(Td14-Td04)Td1-Td0 Where σ is the Stephen-boltzmann constant; ε is the radiation rate of the brake disc.

The heat stored by the brake disc per unit time can be calculated by the following equation: Q=Mdcd(Td(t+Δt)-Tdt)Δt Where Md is the mass of brake disc; cd is the specific heat of brake disc; Td(t+Δt) is the temperature of the brake disc at time t+Δt; Tdt is the temperature of the brake disc at time t.

If Δt→0, the Eq. (34) would be transferred into: Q=Mdcdlim⁡Δt→0(Td(t+Δt)-Tdt)Δt=McpTd Substituting Eqs. (14), (29) and (35) into Eq. (28), it is available to get the transient temperature expression of the brake disc: MdcdT˙d=kSp2Δhω28πρFeB2-hdAd(Td-Td0) The transient temperature analysis of the exciting winding is similar to the analysis above and does not have to be repeated here. According to the analysis above, the temperature of the exciting winding and the brake disc increases with the raise of braking time and braking intensity which further improves the exciting winding resistance and current leading to more energy consumption of the electromagnetic brake. In this case, the study on the electromagnetic brake control method to make it use minimal power for the same brake task is becoming very meaningful.

In this paper, two sets of experiments have been designed. Firstly, we use two groups of exciting windings and four groups of exciting windings respectively to ensure the same braking torque as shown in Figs. 12 and 13. The curve 1 is the experimental results using all four groups of the exciting windings. And the curve 2 is the experimental results only using two groups of exciting windings. During these two situations, the output braking torque of electromagnetic brake is kept in 100 N m and the wheel speeds are 400 r min-1. The braking power of using all four groups of the exciting winding is less than that of only using two groups as is shown in Fig. 13. The absorption braking energy and temperature rise of the brake disc are almost the same for the braking torque and rotational speed are the same. While the Fig. 12 shows that the excitation winding temperature of using all the exciting winding increases more slowly which has lower increasing amplitude than that of only using two groups too which suggests that the higher excitation winding temperature is a major cause leading to the poor energy consumption characteristics for using only two groups of excitation windings.

Then the braking force would be distributed to the front axle's electromagnetic brake or allocated equally to the front and rear axle's electromagnetic brakes. The total power consumptions of these two situations are shown in Fig. 14 with the wheel speed remaining at 400 r min-1 and the braking torque keeping 100 and 200 N m respectively. Obviously, for the temperature of both the excitation winding and the brake disc, the larger the output brake torque is, the higher the corresponding temperatures as shown in Figs. 15 and 16. It can be seen that the power consumption in the situation by distributing all the braking force on one axle is more than distributing equally to the front and rear axles. The higher excitation winding and brake disc temperatures are still the major cause leading to excessive energy consumption.

According to the conclusions analyzed above, distributing the brake task equally can prevent the temperature of the exciting windings and the brake disc to rise too high which could reduce the energy consumption in the process of using the electromagnetic brake. The corresponding energy saving control methods of electromagnetic brake are that all the exciting windings should be controlled to work with the same excitation current and when distributing the electromagnetic braking force to the front and rear axles, the electromagnetic braking torque should be distributed according to the ideal braking force distribution curve considering the limit of the braking regulations since the main purpose of electromagnetic brake is to instead the friction brake under the city driving conditions.