Bus Capacitors

The bus capacitors are the most important passive component in a motor controller. They source and sink high instantaneous currents to and from the inverter, in affect shielding the rest of the controller from the high frequency switching transients. They also accommodate for battery (or power supply) cable inductance, absorbing energy that would otherwise create damaging voltage spikes when current is suddenly switched off [15]. Figure 57 shows the placement of the bus

capacitor; it is crucial that the capacitor be physically located as close as possible to the inverter MOSFETs, so that the resistance and inductance of the line between the capacitor and the MOSFETs is very low. Figure 59 shows the physical placement of the bus capacitors in this controller. Each inverter gets a separate bus capacitor, located immediately adjacent to the MOSFET module’s power terminals.

Figure 55: The bus capacitor is placed across the DC lines, as close as possible to the inverter MOSFETs.

The bus capacitor is typically an electrolytic capacitor, or a group of electrolytic capacitors in parallel. In high-current controllers, it can occupy as much or more volume than the MOSFETs themselves. It can also dissipate as much or more heat, meaning it contributes significantly to the overall efficiency of the controller. Sizing the bus capacitor is critical to a motor controller design. There are two main considerations for sizing: voltage ripple and heating.

Voltage ripple is easy to predict by making the worst-case assumption that the bus capacitor supplies all of the switching current at the PWM frequency. In this case, the power supply provides only a steady DC current. When the PWM is off, the bus capacitor is charged by the power supply current. When the PWM is on, the bus capacitor is discharged, sourcing current (in addition to the power supply current) to the MOSFETs. Table 18 lists the currents present in these two states.

Table 18: Current flow during the PWM on and off times, assuming the power supply provides only a DC average current (high power supply inductance and/or high frequency).

Power Supply Current MOSFET Current Capacitor Current

PWM Off (1-D) (D)Imotor 0 (D)Imotor

PWM On (D) (D)Imotor Imotor -(1-D)Imotor

The duty cycle, D, is the fraction of time the PWM is on. (1-D) is t off time. By taking the time-weighted sum of the currents in Table 18, the average current in from the power supply over one switching cycle is the same as the average current out to the MOSFET. Additionally, the average capacitor current is zero, a necessary condition for steady-state.

The voltage ripple depends on the current, the duty cycle, and the switching frequency. Starting from the constitutive equation for a capacitor, it can be calculated as follows:

 

This is the calculation for charging. Discharging will yield the same equation, but with a negative change in voltage. (So that over one switching cycle in steady-state, the capacitor returns to the same voltage.) From this, it is easy to see that the maximum ripple voltage occurs at 50% duty cycle, where D(1-D) = 0.25.

The capacitors for each inverter in the case study controller are 330μF, 63V electrolytic

capacitors. The switching frequency is 14.5kHz. The peak motor current is 30A. Carrying out the voltage ripple calculation with these values yields ΔV = 1.57V at 50% duty cycle. This would not make a very good switching power supply, but for a motor controller it is acceptable. The important criterion is that the voltage does not exceed any hard limits (such as MOSFET

breakdown voltage or power supply low voltage cutoff). For the RC car, which has peak currents of 40A and a voltage supply that is only 1.8V higher than the cutoff, a set of two 560μF/35V capacitors is used instead. In general, the lower the system voltage, the higher capacitance is available in the same form factor.

The other important design consideration in sizing the bus capacitor is heat generation.

Calculating the capacitor heat loss is impossible without knowing the ripple current more exactly. The worst-case assumption, though, is the same as in the voltage ripple calculation: the bus capacitor supplies the entire AC component of current, at 50% duty cycle. One way to very roughly estimate capacitor heating is using the “loss tangent,” which is defined as follows:

CR

 

tan .

The value tan δ is often published in capacitor datasheets, and is assumed to be constant for all frequencies ω. C is the capacitance and R is an equivalent series resistance. From this, it is clear that that equivalent series resistance decreases with increasing frequency. Using tan δ = 0.1, from the datasheet for the capacitors in this controller, the equivalent series resistance at 14.5kHz is:

  

This is likely a low estimate of the equivalent series resistance, since in reality the assumption that tan δ = constant does not hold true at high frequency. Instead, the equivalent resistance settles out to some minimum value. The crossover is typically below 10kHz [16]. The full curve is not usually published in capacitor datasheet. Knowing that the resistance value predicted by the low tangent is a low estimate is still useful. The order of magnitude starting here and going up (to 33mΩ in this case) can be analyzed. Figure 56 shows a plot over this range of resistances of capacitor heat generation with a motor current of 30A and 50% duty cycle (15A in/out).

Figure 56: Capacitor power dissipation over a range of ESR starting at the one predicted using loss tangent and going up one order of magnitude.

The power dissipation in the capacitor is definitely high enough to significantly influence the overall controller efficiency. It may also be high enough to heat the capacitor above its maximum temperature. (Unlike the MOSFETs, the capacitor is not attached to a heat sink.) Capacitor heat generation at points other than maximum current, 50% duty cycle is lower, so depending on duty cycle the actual performance may be significantly better. After a significant amount of controller testing, the bus capacitors seem adequate and heating seems low. However, to increase the current handling capability of the controller, the capacitors would be the first “upgrade.” (There is still some overhead in the MOSFETs.)