John Sussmeier
Engineering Fellow,
Systems Engineering
Pitney Bowes Inc.
Danbury, Conn.

Engineering has just created yet another radical prototype mechanism that must start and stop aggressively under ac or dc servomotor control. As with so many other motioncontrol applications prior, you are prepared and have done your homework. You have sized the motor and amplifier appropriately, selected a suitable coupling ratio between motor and mechanical load and have verified there is enough voltage and current headroom available to do the job.
The engineering team in the lab stands anxiously as you are about to breathe life into their mechanism by commanding its first actual incremental motion profile. Then the moment of truth: You press the key, the mechanism jolts and then nothing, just the sound of the cricket that has set up residence in the wall dividing the lab from the machine shop. The team sighs as you observe that the amplifier fault LED is on and is laughing at you.
Did you do your homework and still get an "F"? Or is the problem a bad connection, bad amplifier, bad motor, bad encoder, improper servotuning, a bound mechanism, or perhaps a lunar eclipse?
Congratulations are in order, as this time it's none of the above. But you have stumbled onto a brand new problem that has been around since the invention of the electric motor. Like so many other motioncontrol engineers before you, you have just been foiled by the ill effects of regenerative power.
Your task now is to tame this power by caging it in the form of a regeneration circuit. Fortunately there are straightforwardand practical procedures for engineers to predetermine if, and how much, regenerative power will need to be dissipated in a regenerative circuit. A few supporting equations help to accomplish this rapidly.
As engineers create machines that must operate ever faster, motioncontrol engineers are increasingly faced with servomotor applications that demand aggressive decelerations. Such needs often require the use of a regenerative circuit.
To accommodate the trend, many new commercial amplifiers include the circuitry to connect an external regeneration resistor. While decelerating a mechanical load, mechanical energy is converted into electrical energy, known as regeneration. Some of this regenerated energy is lost to friction in the mechanical system. And some is converted into heat within the motor through I ^{2 }R losses. The remainder is reflected back to the internal bus capacitance of the amplifier, resulting in a higher bus voltage.
If too much energy reflects into the amplifier, the bus voltage will rise high enough to trip the amplifier's overvoltage protection, shutting it down and causing a loss of control over the mechanism. To prevent this, a regenerative (or regen) circuit shunts some or most of this excess energy into an external resistor.
Application notes for commercial amplifiers typically neglect the effect of friction. Thus they assume more energy is directed to a regen circuit during a deceleration segment of a motion profile than really may be the case. Though this method provides conservative results, it can lead designers of highfriction applications to erroneously conclude that a regen circuit is necessary when it may not be.
However, inaccurate characterization of large friction loads can also have the opposite result, where calculations undersize the dissipative energy capacity of the regen circuit. Case in point: Motioncontrol engineers at Pitney Bowes Inc., a company that designs and manufactures highvolume production mail inserting equipment, recently had difficulty sizing the regen circuits on two separate ac servo applications. The confusion that followed warranted further investigation and resulted in the procedure outlined below that we now use.
SIZING UP FRICTION
First consider only systems that have high friction loads. Not surprisingly, determining the right way to use a regen circuit and its dissipativeenergy capability greatly depends on accurately characterizing the friction load over the life of the mechanism. The torque a motor generates can be given by a linear secondorder differential equation:
where p = generated torque of the motor, Nm; C_{1}= J = total reflected inertia as seen by the motor, including the motor inertia, kgm^{2};q
= angular acceleration of the motor, rad/sec^{2};C_{2} = viscous friction coefficient, Nmsec/rad; q
= angular velocity of the motor rad/sec; and C_{3} = Coulomb friction coefficient, Nm.
The viscousfriction component of equation (1) can be quite high for systems containing machine elements that significantly deform during operation. These elements include components like elastomeric belts and drive roller pairs that have high normal forces between them for conveying material. Paperhandling mechanisms used in production mailinserting equipment can often see a viscous friction torque component that exceeds all other torque components.
In practice, you can determine C_{1}, C_{2} and C_{3} empirically by measuring the realtime current delivered to the motor windings. We have found the easiest way to do this is by monitoring the realtime input voltage (typically 0 to 10 Vdc) to an amplifier configured for torque mode. Knowing this voltage at any loop closure period lets you calculate the generated instantaneous torque of the motor. The torque is simply this input voltage multiplied by the
transconductance of the amplifier, multiplied by the torque constant of an ac or dc servomotor. In equation form:
where p = generated torque of the motor, Nm; V_{in} = commanded input voltage to the amplifier, Vdc; K_{trans} = transconductance of the amplifier,A/Vdc; and K_{t} = torque constant of the motor, Nm/A.
One precaution: Verify the specified units of current for both the amplifier transconductance and the motor torque constant. Current can be specified in rms amps, peak amps, or average amps and is quite often not explicitly specified for both constants. The appropriate manipulation must be made in equation (2) to ensure the units of amps, A, cancel out.
You can determine C_{3} from equation (1) and by operating the mechanism of interest at very low rpm. Here the first two terms in the equation are effectively zero. Next, you can determine C_{2} from the same equation by operating the mechanism at its peak constant velocity. Under these conditions the first term in the equation is still zero.
Finally, the equation can again determine C_{1} or J at any point during a constant acceleration segment of a motion profile as long as the instantaneous velocity is known at the selected point. In practice, we have access to the commanded torque at every loop closure period. So we average several data points surrounding the selected point to minimize the measurement error caused by servo jitter.

Engineers should exercise caution, however, when determining the friction coefficients. Over time many mechanisms will "breakin." We have found that C_{2} can drop slightly and C_{3} can fall significantly. When C_{2} and C_{3} decrease, so does the total friction torque. This introduces the possibility that the system might need a regen circuit only after "breakin" or a bigger regen circuit than initially required. This is a good reason to break in mechanisms for at least 24 hr of continuous operation before making the measurements that define coefficients C_{2} and C_{3}.
The energy returned by the mechanism during a constant deceleration segment of a motion profile can be divided into three components, analogous to the components in equation (1):
where E_{mech} = mechanical energy returned by the mechanism, J; E_{1} = inertial energy, J; E_{2} = viscous friction work, J; and E_{3} = Coulomb friction work, J.
For a rotary system, energy U put into that system is
Therefore energy E returned by the system to an amplifier, say,
Thus to determine E_{1}:
where w_{1} = initial angular velocity of the motor, rad/sec; and w_{2} = final angular velocity of the motor, rad/sec. To determine E_{2},
wherea
= angular acceleration of the motor, a negative value for deceleration, rad/sec_{2}. Then to determine E_{3},
whereq
= total angular displacement traveled during the deceleration segment, rad. Therefore, the total energy returned by the mechanism during a constant deceleration segment of a motion profile is:
Now we can decide if we need a regen circuit, knowing the mechanical energy returned by the mechanism during a deceleration segment. We can determine the excess energy that needs to be dissipated to avoid tripping the amplifier overvoltage protection as follows:
where E_{excess} = excess energy that needs to be dissipated, J; E_{motor} = amount of energy dissipated by the motor windings during the deceleration segment, J; and E_{capmax} = amount of energy that can be absorbed by amplifier bus capacitance before tripping the overvoltage limit of the amplifier, J.
The amount of energy dissipated by the motor during the deceleration segment is an I ^{2 }R power calculation multiplied by time and can be determined as follows:
where p_{min} = minimum torque needed to decelerate the motor, Nm; K_{t} = torque constant of the motor, Nm/A; R = phasetophase resistance of the motor, ohms; T_{decel} = the total time of the deceleration segment, sec.
This equation is a conservative approximation andtmin
should be evaluated as follows, at the beginning of a deceleration segment where the viscous friction loss is at a maximum:
where q_{decel} = angular deceleration of the motor, rad/sec^{2}, a negative value; q_{max } = maximum angular velocity of the motor,rad/sec. We have found, however, that typically the value of E_{motor} is small compared to the other terms found in equation (10) and can usually be neglected.
The amount of energy that can be absorbed by bus capacitance before tripping the amplifier overvoltage protection, E_{capmax}, can be determined as follows:
where C = bus capacitance, F; V_{limit} = bus overvoltage limit of the amplifier, Vdc; V_{mains} = mains voltage applied to the amplifier, Vac.
The value of E_{excess} can now be computed according to equation (10). If E_{excess} is calculated to be a negative value, then the application needs no regen circuit. If Eexcess is calculated to be a positive value, a regen circuit is required. Note that the energy that the capacitance can absorb depends heavily on the value of V_{mains} and does not depend on the backEMF of the motor, a common misconception.
Therefore, one should always use in equation (13) the maximum anticipated value of V_{mains} that the system is ever expected to see, particularly if the system is to be shipped and installed abroad. We use a value of 264 Vac to satisfy all our anticipated applications.
Once calculations have shown the necessity of a regen circuit, one next determines how much dissipative energy capacity the circuit needs. The regen circuit must trip before the bus voltage reaches the amplifier overvoltage limit. Thus the regen circuit trips at a bus voltage that must be lower than this value. The energy that the regen circuit must dissipate can be determined from:
where E_{regen} = amount of actual energy that must be dissipated by the regen circuit, J; E_{motor} = amount of energy dissipated by the motor windings during the deceleration segment, J; E_{cap} = amount of energy that can be absorbed by amplifier bus capacitance before the regen circuit trips, J.
The actual amount of energy that is absorbed by bus capacitance before the regen circuit trips is determined as follows:
where C = bus capacitance, F; V_{regen} = bus trip voltage of the regen circuit, Vdc; V_{mains} = mains voltage applied to the amplifier, Vac.
Note again that the calculation should use the highest possible mains voltage that the system will ever experience. The actualenergydissipative capacity of the regen circuit can now be determined using equation (14).
The peak and continuous power that the regen circuit must accommodate can now be determined as follows:
where P_{peak} = peak pulse power dissipation during the deceleration segment, W; P_{cont} = average power dissipation for a continuously repeating incremental motion profile, W; T_{decel} = the total time of the deceleration segment, sec; and T_{cycle} = the period between a continuously repeating incremental motion profile, sec.
The maximum resistance of the regen resistor can now be determined as follows:
The actual resistance value used should be less than the calculated value. But it must also exceed the minimum regen resistor value specified by the manufacturer of the amplifier or regen circuit. We have had good success using a 20 o regen resistor and then selecting its continuous power rating to the next commercially available size that exceeds that calculated by equation (17).
Many new amplifiers now available commercially include circuitry to connect an external regeneration resistor. There are often offtheshelf regenerative circuit packages for those that do not, some with the regen resistor built in.
THE FINALE
The engineering team is now applauding as they watch their newly created mechanism come to life. Though the mechanism is heavy, it cycles rapidly and authoritatively, completely under precise servoposition control. As it moves, it also converts a regen resistor into a space heater.
The motioncontrol engineer has succeeded once again and has earned yet another paycheck. The applause builds to pandemonium as the mechanism vibrates its way across the floor like an out of balance washing machine. As this distracts the engineering team to attack a new problem, the motioncontrol engineer seizes the opportunity for a temporary reprieve, and slips out of the lab unscathed and rides off into the sunset. Fade to black.
MAKE CONTACT:
Pitney Bowes Inc.,
(203) 3565000,
www.pb.com.