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# Principle of Operation - Dynamic Braking

# What is Dynamic Braking

###### By: Jeff Theisen, Rockwell Automation

###### Posted By: Michael Cromheecke, Rockwell Automation

When an induction motor’s rotor is turning slower than the synchronous speed set by the drive’s output power; the motor is transforming electrical energy obtained from the drive into mechanical energy available at the drive shaft of the motor. This process is referred to as ‘motoring’. When the rotor is turning faster than the synchronous speed set by the drive’s output power, the motor is transforming mechanical energy available at the drive shaft of the motor into electrical energy that can be transferred back into the utility grid. This process is referred to as ‘regeneration’. On most AC PWM drives, the AC power available from the fixed frequency utility grid is first converted into DC power by means of a diode rectifier bridge or controlled SCR bridge, before being inverted into variable frequency AC power. These diode or SCR bridges are very cost effective, but can handle power in only one direction, and that direction is the motoring direction. If the motor is regenerating, the bridge is unable to conduct the necessary negative DC current, and the DC bus voltage will increase until the drive trips off due to a Bus Overvoltage trip. There are bridge configurations, using either SCRs or Transistors that have the ability to transform DC regenerative electrical energy into fixed frequency utility electrical energy but are expensive. A much more cost effective solution is to provide a Transistor Chopper on the DC Bus of the AC PWM drive that feeds a power resistor which transforms the regenerative electrical energy into thermal heat energy which is dissipated into the local environment. This process is generally called ‘Dynamic Braking’, with the Chopper Transistor and related control and components called the ‘Chopper Module’, and the power resistor called the ‘Dynamic Brake Resistor’. The entire assembly of Chopper Module with Dynamic Brake Resistor is sometime referred to as the ‘Dynamic Brake Module’.

Chopper Modules are designed to be applied in parallel if the current rating is insufficient for the application. One Chopper Module is the designated ‘Master’ Chopper Module, while any other Modules are the designated ‘Follower’ Modules. Two lights have been provided on the front of the enclosure to indicate Chopper Module operation – the ‘DC Power’ light and the ‘Brake On’ light. The DC Power light will be lit when DC power has been applied to the Chopper Module. The Brake On light will be lit when the Chopper Module is operating or ‘chopping’ and will be a flickering type of indication.

### How it works

There are two different types of control for dynamic braking, hysteretic control and PWM control. Each used by themselves in a standard stand alone product has no advantage over the other. The preferred control would be the PWM method when the application is common dc bus. This advantage is decribed below.

#### Hysteretic Control

The hysteretic method of dynamic braking uses a voltage sensing circuit to monitor the dc bus. As the dc bus volage increases to the

*Vdc_on*level the brake IGBT is turned on and is left on until the voltage drops to the

*Vdc_off*level (which is not so desirable in common dc bus applications - see below). Some of the Powerflex drives

^{[1]}allow the

*Vdc_off*level, [DB Threshold],to be adjusted if the application required it. Setting this level lower will make the dynamic braking more responsive but could lead to excessive DB activation.

**References**

- ↑ PF40, PF40P

#### PWM Control

This type of control to operate the brake IGBT is similar to the way output voltage to the motor is controlled. As the dc bus voltage increases and hits some predetermined limit the brake IGBT is turned on/off according to a control algorithm switch at 1khz. This type of control virtually eliminates bus ripple. The big advantage is when this type of control is in a common bus configuration.

##### Duty Cycle

#### Common DC Bus Applications

In a common bus configuration when a dynamic braking resistor is installed on each drive sharing the dc bus, it’s possible that the brake IGBT in some drives may not turn on, giving the impression that the drive is not functioning correctly or seeing one drive’s brake IGBT failing consistently while the other drives are fine. Looking at the below diagram, it shows the dc bus level for two drives on common bus. The delta between these voltages are exaggerated for clarity. As the voltage increases, drive 1’s IGBT turns on and decreases the voltage level before drive 2 sees voltage high enough to be told to turn on. This results in drive 1 doing all the “DB work” and drive 2 taking a powder. Now this situation could be ok as long as the minimum ohmic value for resistance is not violated and the regen event isn’t so great that a single resistor can’t handle the power. Of course if there is a large regen event where the voltage continues to rise after drive 1 has “turned on”, drive 2 will fire it’s IGBT when it reaches the voltage limit.

Here are two drives with PWM DB control on a common bus. Since one drive will turn on at a certain duty cycle the bus voltage will likely continue to rise guaranteeing that the other drive’s IGBT will turn on (at a different duty cycle).

### How to Select A Chopper Module and Dynamic Brake Resistor

In general, the motor power rating, speed, torque, and details regarding the regenerative mode of operation will be needed in order to estimate what Chopper Module rating and Dynamic Brake Resistor value to use. A rule of thumb to use is that a Dynamic Brake Module can be specified when regenerative energy is dissipated on an occasional or periodic basis. When a drive is consistently operating in the regenerative mode of operation, serious consideration should be given to equipment that will transform the electrical energy back to the fixed frequency utility.

The peak regenerative power of the drive must be calculated in order to determine the maximum Ohmic value of the Dynamic Brake Resistor and to estimate the minimum current rating of the Chopper Module. The Rating of the Chopper Module is chosen from the Brake Chopper Module manual. Once the Chopper Module current rating is known, a minimum Dynamic Brake Resistance value is also known. A range of allowable Dynamic Brake Ohmic values is now known. These values exist from the minimum value set by the Chopper Transistor current rating to a maximum value set by the peak regenerative power developed by the drive in order to decelerate or satisfy other regenerative applications. If a Dynamic Brake Resistance value less than the minimum imposed by the choice of the Chopper Module is made and applied, damage can occur to the Chopper Transistor. If a Dynamic Brake Resistance value greater than the maximum imposed by the choice of the peak regenerative drive power is made and applied, the drive can trip off due to transient DC Bus overvoltage problems. Once the choice of the approximate Ohmic value of the Dynamic Brake Resistor is made, the wattage rating of the Dynamic Brake Resistor can be made.

The wattage rating of the Dynamic Brake Resistor is estimated by applying the knowledge of the drive motoring and regenerating modes of operation. The average power dissipation of the regenerative mode must be estimated and the wattage of the Dynamic Brake Resistor chosen to be slightly greater than the average power dissipation of the drive. If the Dynamic Brake Resistor has a large thermodynamic heat capacity, then the resistor element will be able to absorb a large amount of energy without the temperature of the resistor element exceeding the operational temperature rating. Thermal time constants in the order of 50 seconds and higher satisfy the criteria of large heat capacities for these applications. If a resistor has a small heat capacity, the temperature of the resistor element could exceed the maximum temperature limits during the application of pulse power to the element and could exceed the safe temperature limits of the resistor.

The peak regenerative power can be calculated in English units (Horsepower), in The International System of Units (SI) (Watts), or in the per unit system (pu) which is dimensionless for the most part. In any event, the final number must in Watts of power to estimate Dynamic Brake Ohmic value. Calculations in this page will be demonstrated in SI units.

### Speed, Torque, Power Profile

The following figure is a typical dynamic braking application. The top trace represents speed and is desigated by the omega symbol. In the profile the motor is accelerated to some speed, it holds that speed for a period of time and is then decelerated. This deceleration is not necessarily to zero speed. The cycle is then repeated.

The middle trace represents motor torque. Torque starts out high as the motor is accelerated then drops down to maintain the commanded speed. Then the torque turns negative as the motor is decelerated. The cycle is then repeated.

The bottom trace represents motor power. Power increases as the motor speed increases. Power decreases some to maintain the commanded speed then goes negative when deceleration starts. (this point called -Pb is the first value that needs to be calculated). The cycle is then repeated.

## Dynamic Braking Module

Figure 1 shows a simplified schematic of a Chopper Module with Dynamic Brake Resistor. The Chopper Module is shown connected to the positive and negative DC Bus conductors of an AC PWM Drive. The two series connected Bus Caps are part of the DC Bus filter of the AC Drive. The significant power components of the Chopper Module are the protective fusing, the Crowbar SCR, the Chopper Transistor (an IGBT), the Chopper Transistor Voltage Control (hysteretic voltage comparator), and a freewheel diode for the Dynamic Brake Resistor.

The protective fuse is sized to work in conjunction with the Crowbar SCR. Sensing circuitry within the Chopper Transistor Voltage Control determines if abnormal conditions exist within the Chopper Module. One of these abnormal conditions is a shorted Chopper Transistor. If this condition is sensed, the Chopper Transistor Voltage Control will fire the Crowbar SCR, shorting the DC Bus, and melting the fuse links. This action isolates the Chopper Module from the DC Bus until the problem can be resolved.

The Chopper Transistor is an Isolated Gate Bipolar Transistor (IGBT). There are several transistor ratings that are used in the various Chopper Module ratings. The most important rating is the collector current rating of the Chopper Transistor that helps to determine the minimum Ohmic value used for the Dynamic Brake Resistor. The Chopper Transistor is either ‘ON’ or ‘OFF’, connecting the Dynamic Brake Resistor to the DC Bus and dissipating power, or isolating the resistor from the DC Bus.

The Chopper Transistor Voltage Control regulates the voltage of the DC Bus during regeneration. The average value of DC Bus voltage is 375 Volts dc (for 230 Vac input), 750 Volts dc (for 460 Vac input), and 937.5 Vdc (for 575 Vac input). The voltage dividers reduce the DC Bus voltage to a low enough value that is useable in signal circuit isolation and control. The DC Bus feedback voltage from the voltage dividers is compared to a reference voltage to actuate the Chopper Transistor.

The Freewheel Diode (FWD) in parallel with the Dynamic Brake Resistor allows any magnetic energy stored in the parasitic inductance of that circuit to be safely dissipated during turn off of the Chopper Transistor.

### Sizing the Dynamic Brake Module

Gather the following information:

1. The nameplate power rating of the motor in watts, kilowatts, or horsepower.

2. The nameplate speed rating of the motor in rpm, or rps.

3. The motor inertia and load inertia in kilogram-meters2, or lb-ft2.

4. The gear ratio, if a gear is present between the motor and load, GR.

5. Review the Speed, Torque Power profile of the application.

Equations used for calculating Dynamic Braking values will use the following variables.

ω(*t*) = The motor shaft speed in Radians/second, or

N_{(t)} = The motor shaft speed in Revolutions Per Minute, or RPM

T_{(t)} = The motor shaft torque in Newton-meters, 1.01 lbft - 1.355818Nm

P_{(t)} = The motor shaft power in Watts, 1.0HP = 746 Watts

-P_{b} = The motor shaft peak regenerative power in Watts

## Step 1 – Determine the Total Inertia

J

_{T}= J_{m}+ GR^{2}X J_{L}J

_{T}= Total interia reflected to the motor shaft, kilogram-meters^{2}, kg-m^{2}, or pound-feet^{2}, lb-ft^{2}J

_{m}= Motor inertia, kilogram-meters^{2}, kg-m^{2}, or pound-feet^{2}, lb-ft^{2}GR = The gear ratio for any gear between motor and load, dimentionless

J

_{L}= Load inertia, kilogram-meters^{2}, kg-m^{2}, or pound-feet^{2}, lb-ft^{2}-- 1 lb-ft^{2}= 0.04214011 kg-m^{2}

## Step 2 – Calculate the Peak Braking Power

J

_{T}= Total inertia reflected to the motor shaft, kg-m^{2}ω = rated angular rotational speed,

N = Rated motor speed, RPM

t

_{3}- t_{2}= total time of deceleration from rated speed to 0 speed, in secondsP

_{b}= peak braking power, watts ( 1.0 HP = 746 Watts)

Compare the peak braking power to that of the rated motor power, if the peak braking power is greater that 1.5 times that of the motor, the deceleration time, (t3-t2), needs to be increased so that the drive does not go into current limit. Use 1.5 times because the drive can handle 150% current maximum for 3 seconds.

Peak power can be reduced by the losses of the motor and inverter.## Step 3 – Calculating the Maximum Dynamic Brake Resistance Value

V

_{d}= The value of DC Bus voltage that the chopper module regulates at and will equal 375Vdc, 750Vdc, or 937.5VdcP

_{b}= The peak braking power calculated in step 2R

_{db1}= The maximum allowable value for the dynamic brake resistor

The choice of the Dynamic Brake resistance value should be less than the value calculated in step 3. If the value is greater than the calculated value, the drive can trip on DC Bus overvoltage. Remember to account for resistor tolerances.

## Step 4 – Choosing the correct Dynamic Brake Module

Catalog No. Resistance Wattage 240 VoltKA005 28 ohms 666 watts KA010 13.2 ohms 1650 watts KA050 na na 460 VoltKB005 108 ohms 1500 watts KB010 52.7 ohms 2063 watts KB050 10.5 ohms 7000 watts 600 VoltKC005 108 ohms 1500 watts KC010 52.7 ohms 2063 watts KC050 15.8 ohms 8000 watts

In the Table above choose the correct Dynamic Brake Module based upon the resistance value being less than the maximum value of resistance calculated in Step 3. If the Dynamic Brake Resistor value of one Dynamic Brake Module is not sufficiently low, consider using up to three Dynamic Brake Modules in parallel, such that the parallel Dynamic Brake resistance is less than Rdb1 calculated in Step 3. If the parallel combination of Dynamic Brake Modules becomes too complicated for the application, consider using a Brake Chopper Module with a separately specified Dynamic Brake Resistor.## Step 5 – Estimate average power

It is assumed that the application exhibits a periodic function of acceleration and deceleration. If (t3-t2) = the time in seconds necessary for deceleration from rated speed to 0 speed, and t4 is the time in seconds before the process repeats itself, then the average duty cycle is (t3-t2)/t4. The power as a function of time is a linearly decreasing function from a value equal to the peak regenerative power to 0 after (t3-t2) seconds have elapsed. The average power regenerated over the interval of (t3-t2) seconds is Pb/2. The average power in watts regenerated over the period t4 is:

P

_{av}= Average dynamic brake resistor dissipation, in wattst

_{3}- t_{2}= Elapsed time to decelerate from rated speed to 0 speed, in secondst

_{4}= Total cycle time or period of process, in secondsP

_{b}= Peak braking power, in watts

The Dynamic Brake Resistor power rating of the Dynamic Brake Module (singly or two in parallel) that will be chosen must be greater than the value calculated in Step 5. If it is not, then a Brake Chopper Module with the suitable Dynamic Brake Resistor must be specified for the application.## Step 6 – Calculate Percent Average Load

The calculation of AL is the Dynamic Brake Resistor load expressed as a percent. Pdb is the sum of the Dynamic Brake Module dissipation capacity and is obtained from the Table in step 4. This will give a data point for a line to be drawn on the curve in Figure 3. The number calculated for AL must be less than 100%. If AL is greater than 100%, an error was made in a calculation or the wrong Dynamic Brake Module was selected.

AL = Average load in percent of Dynamic Brake Resistor

P

_{av}= Average dynamic brake resistor dissipation calculated in Step 5 (Watts)P

_{db}= Steady state power dissipation capacity of resistors obtained from the table in step 4 (Watts)

## Step 7 – Calculate Percent Peak Load

The calculation of PL in percent gives the percentage of the instantaneous power dissipated by the Dynamic Brake Resistors relative to the steady state power dissipation capacity of the resistors. This will give a data point to be drawn on the curve of Figure 3. The number calculated for PL will commonly fall between 300% and 600% for the Dynamic Brake Modules. A calculated number for PL of less than 100% indicates that the Dynamic Brake Resistor has a higher steady state power dissipation capacity than is necessary.

PL = Peak load in percent of Dynamic Brake Resistor

P

_{av}= Peak braking power calculated in Step 2 (Watts)P

_{db}= Steady state power dissipation capacity of resistors obtained from the table in step 4 (Watts)

## Step 8 – Plot PL and AL on Curve

Draw a horizontal line equal to the value of AL (Average Load) in percent as calculated in Step 6. This value must be less than 100%. Pick a point on the vertical axis equal to the value of PL (Peak Load) in percent as calculated in Step 7. This value should be greater the 100%. Draw a vertical line at (t3 - t2) seconds such that the line intersects the AL line at right angles. Label the intersection point “Point 1”. Draw a straight line from PL on the vertical axis to Point 1 on the AL line. This line is the power curve described by the motor as it decelerates to minimum speed.

If the line you drew lies to the left of the constant temperature power curve of the Dynamic Brake Resistor, then there will be no application problem. If any portion of the line lies to the right of the constant temperature power curve of the Dynamic Brake Resistor, then there is an application problem. The application problem is that the Dynamic Brake Resistor is exceeding its rated temperature during the interval that the transient power curve is to the right of the resistor power curve capacity. It would be prudent to parallel another Dynamic Brake Module or apply a Brake Chopper Module with a separate Dynamic Brake Resistor.

## Chopper and Resistors

### Sizing the Chopper and Resistors

Sizing the chopper module is the same as the dynamic brake module with a couple of added steps. Since the chopper is seperate from the resistors, an additional calculation for current needs to be made. Additionally a calculation for watt-seconds or joules needs to be made for resistor sizing.

## Step 1 – Determine the Total Inertia

J

_{T}= J_{m}+ GR^{2}x J_{L}J

_{T}= Total inertia reflected to the motor shaft, kilogram-meters^{2}, kg-m^{2}, or pound-feet^{2}, lb-ft^{2}J

_{m}= motor inertia, kilogram-meters^{2}, kg-m^{2}, or pound-feet^{2}, lb-ft^{2}GR

^{2}= the gear ratio for any gear between motor and load, dimensionlessJ

_{L}= load inertia, kilogram-meters^{2}, kg-m^{2}, or pound-feet^{2}, lb-ft^{2}(1.0 lb-ft^{2}= 0.04214011 kg-m^{2})

## Step 2 – Calculate the Peak Braking Power

J

_{T}= Total inertia reflected to the motor shaft, kg-m^{2}ω = rated angular rotational speed,

N = Rated motor speed, RPM

t

_{3}- t_{2}= total time of deceleration from the rated speed to 0 speed,secondsPb = peak braking power, watts (1.0HP = 746 Watts)

Compare the peak braking power to that of the rated motor power, if the peak braking power is greater that 1.5 times that of the motor, then the deceleration time, (t3-t2), needs to be increased so that the drive does not go into current limit. Use 1.5 times because the drive can handle 150% current maximum for 3 seconds.

Peak power can be reduced by the losses of the motor and inverter.

## Step 3 – Calculating the Maximum Dynamic Brake Resistance Value

V

_{d}= The value of DC Bus voltage that the chopper module regulates at and will equal 375Vdc, 750Vdc, or 937.5VdcP

_{b}= The peak braking power calculated in step 2R

_{db1}= The maximum allowable value for the dynamic brake resistor

The choice of the Dynamic Brake resistance value should be less than the value calculated in step 3. If the value is greater than the calculated value, the drive can trip on DC Bus overvoltage. Remember to account for resistor tolerances.## Step 4 – Choosing the Chopper Module

I

_{dl}= The minimum current flowing through the chopper module transistorV

_{d}= The value of DCBus voltage chosen in step 3R

_{dbl}= The value of the dynamic brake resistor calculated in step 3

The value of I_{d1}sets the minimum value of current rating for the Chopper Module. When the Chopper Module choice has been made, the current rating of the Module Transistor must be greater than or equal to the calculated value for I_{d1}. See the table below for rating values.

Drive Voltage

(Volts AC)Turn-On Voltage

(Volts DC)Catalog

NumberPeak Transistor

Current Rating

(Amps)Minimum DB

Resistor Value

(Ohms)230 375 WA018 50 9.0 WA070 200 2.3 WA115 400 1.25 460 750 WB009 25 37 WB035 100 9.0 WB110 400 2.5 575 935 WC009 25 46 WC035 75 15.5 WC085 400 3.0 ## Step 5 – Determine the Minimum Resistance

Each chopper module in the table above has a minimum resistance associated with it. If a [[[resistance]] lower than the value show in the table is connected to the chopper module, the brake transistor will most likely be damaged.

## Step 6 – Choosing the Dynamic Brake Resistance Value

To avoid damage to this transistor and get the desired braking performance, select a resistor with a resistance between the maximum resistance calculated in step 3 and the minimum resistance of the selected chopper module.

## Step 7 – Estimating the Minimum Wattage requirements for the Dynamic Brake Resistor

It is assumed that the application exhibits a periodic function of acceleration and deceleration. If (t3-t2) = the time in seconds necessary for deceleration from rated speed to 0 speed, and t4 is the time in seconds before the process repeats itself, then the average duty cycle is (t3-t2)/t4. The power as a function of time is a linearly decreasing function from a value equal to the peak regenerative power to 0 after (t3-t2) seconds have elapsed. The average power regenerated over the interval of (t3-t2) seconds is Pb/2. The average power in watts regenerated over the period t4 is:

P

_{av}= average dynamic brake resistor dissipation, wattst

_{3}- t_{2}= Elapsed time to decelerate from rated speed to 0 speed, secondst

_{4}= Total cycle time or period of process, secondsP

_{b}= Peak braking power, wattsThe Dynamic Brake Resistor power rating in watts that will be chosen should be equal to or greater than the value calculated in step 7.

## Step 8 – Calculate the requires Watt-Seconds (joules) for the resistor

In order the ensure that the resistors thermal capabilities are not violated, a calcualtion to determine the amount of energy dissipated into the resistor will be made. This will determine the amount joules the resistor must be able to absorb

P

_{ws}= Required watt - seconds of the resistort

_{3}-t_{2}= Elapsed time to decelerate from ω_{b}speed to ω_{0}speed, secondsP

_{b}= Peak braking power, watts

## Internal Brake IGBT

### Sizing Resistors for an internal DB IGBT

Sizing resistors for the internal DB IGBT Uses the same formula's as previous, and is very similar to the Chopper Module sizing.

## Step 1 – Determine the Total Inertia

J

_{T}= Total inertia reflected to the motor shaft, kilogram-meters^{2}, kg-m^{2}, or pound-feet^{2}, lb-ft^{2}J

_{m}= motor inertia, kilogram-meters^{2}, kg-m^{2}, or pound-feet^{2}, lb-ft^{2}GR = The gear ratio for any gear between motor and load, dimensionless

J

_{L}= load inertia, kilogram-meters^{2}, kg-m^{2}, or pound-feet^{2}, lb-ft^{2}(1.0 lb-ft^{2}= 0.04214011 kg-m^{2})## Step 2 – Calculate the Peak Braking Power

J

_{T}= Total inertia reflected to the motor shaft, kg-m^{2}ω = rated angular rotational speed,

N = Rated motor speed, RPM

t

_{3}- t_{2}= total time of deceleration from the rated speed to 0 speed,secondsPb = peak braking power, watts (1.0HP = 746 Watts)

Compare the peak braking power to that of the rated motor power, if the peak braking power is greater that 1.5 times that of the motor, then the deceleration time, (t3-t2), needs to be increased so that the drive does not go into current limit. Use 1.5 times because the drive can handle 150% current maximum for 3 seconds.

Peak power can be reduced by the losses of the motor and inverter.## Step 3 – Calculating the Maximum Dynamic Brake Resistance Value

V

_{d}= The value of DC Bus voltage that the chopper module regulates at and will equal 375Vdc, 750Vdc, or 937.5VdcP

_{b}= The peak braking power calculated in step 2R

_{db1}= The maximum allowable value for the dynamic brake resistorThe choice of the Dynamic Brake resistance value should be less than the value calculated in step 3. If the value is greater than the calculated value, the drive can trip on DC Bus overvoltage. Remember to account for resistor tolerances.

## Step 4 – Determine the Minimum Resistance

Each drive with an internal DB IGBT has a minimum resistance associated with it. If a resistance lower than the minimum value for a given drive is connected, the brake transistor will most likely be damaged.

## Step 5 – Choosing the Dynamic Brake Resistance Value

To avoid damage to this transistor and get the desired braking performance, select a resistor with a resistance between the maximum resistance calculated in step 3 and the minimum resistance of the selected chopper module.

## Step 6 – Estimating the Minimum Wattage requirements for the Dynamic Brake Resistor

It is assumed that the application exhibits a periodic function of acceleration and deceleration. If (t3-t2) = the time in seconds necessary for deceleration from rated speed to 0 speed, and t4 is the time in seconds before the process repeats itself, then the average duty cycle is (t3-t2)/t4. The power as a function of time is a linearly decreasing function from a value equal to the peak regenerative power to 0 after (t3-t2) seconds have elapsed. The average power regenerated over the interval of (t3-t2) seconds is Pb/2. The average power in watts regenerated over the period t4 is:

P

_{av}= Average dynamic brake resistor dissipation, in wattst

_{3}- t_{2}= Elapsed time to decelerate from rated speed to 0 speed, in secondst

_{4}= Total cycle time or period of process, in secondsP

_{b}= Peak braking power, in wattsThe Dynamic Brake Resistor power rating in watts that will be chosen should be equal to or greater than the value calculated in step 6.

## Step 7 – Calculate the requires Watt-Seconds (joules) for the resistor

In order the ensure that the resistors thermal capabilities are not violated, a calcualtion to determine the amount of energy dissipated into the resistor will be made. This will determine the amount joules the resistor must be able to absorb

P

_{ws}= Required watt - seconds of the resistort

_{3}-t_{2}= Elapsed time to decelerate from ω_{b}speed to ω_{0}speed, secondsP

_{b}= Peak braking power, watts

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