Inertia Ratio revisited


Thread Starter

Guy H. Looney

In late April I posted questions about the importance of motor to load inertia ratio. I received numerous responses & sincerely thank
everyone who contributed. I would like to offer a summary of those responses, my belief in the matter, and finally end with a question (or two).

It seems everybody agreed on a couple themes:
1) Bandwidth (system response) is a function of system inertia, load to motor inertia ratio, and the drivetrain/mechanics.
2) A 1:1 ratio is ideal.
3) A system's bandwidth can be no better than the
mechanical bandwidth the motor is attached to.
4) If an infinitely stiff system is coupled to a motor and it does not require a high degree of responsiveness, it can handle a higher load
to rotor inertia ratio.
5) The inertia reflected to the motor is critical for proper sizing of the motor.
6) Most loads are not infinitely stiff. Stiffness can be a very subjective/relative term; ie, what one person considers stiff, another
may consider sloppy.
7) When elements are added to reduce reflected inertia (gearheads, belt & pullies, etc.), the stiffness of the system is reduced. Therefore these elements reduce the maximum bandwidth of the system.
8) Higher inertia mismatches are harder to properly tune. When they are tuned properly, the gains are lower than they would be with a 1:1 ratio.
9) As our servo systems get higher in performance, a mechanical system's resonant frequency will play more of a factor.
Many manufacturers have begun introducing low-pass & notch filters to address the possibility of instability caused by exciting resonant frequencies.

There were a couple of comments that I feel were incorrect:
1) Inertia ratios that are not equal to 1:1 make tuning a nightmare.
The desired response dictates how tightly the system has to be tuned. One could literally spend hours or days on tuning a system (I've done it). Many times however, default gains give the desired response. The application dictates how difficult tuning is.
2) Inertia isn't dependant upon where devices are mounted, it's how they are shaped.
The shape is critical to the inertia of a load, but so is the position of the load. The position is also used to calculate the reflected inertia. The parallel axis theorem is used to reflect inertias that are rotating in a different plane than the motor's shaft.
3) Elements added to reduce reflected inertia (gearheads, belt & pullies, etc.) dramatically reduce the systems performance.
When they are added, slop is introduced & therefore the system repeatability, response, & longevity had been sacrificed.
Many precision gearheads that are available have efficiencies higher than 95% & backlash less than 2 arc-minutes. Certainly
using a device such as this will be completely different than using a belt & pulley.
4) The only reason inertia is important is to size the motor.
That's definitely one aspect of inertia. However, there are several other reasons. Regen calculations and how responsive the system can be are two obvious factors.
5) Inertia ratio is a guideline that says a motor's design torque is probably capable of driving an assembly of rotating elements whose load inertia is "x" times the motor's rotor inertia.
This can't be used reliably, because motors have different design characterstics. It is very common for manufacturers to have a high inertia motor & a low inertia motor that produce similar torques. I can't recall ever seeing a manufacture spec out an inertia ratio that relates to the torque it can produce. I have
seen it spec'd for regen purposes however.

There were a couple of comments that were only made by one person, but I found them to be extremely valuable & agree completely:
1) A motor's bandwidth is a function of the motor's torque to motor's inertia ratio. A higher ratio means higher bandwidth.
Therefore motors w/ a higher bandwidth can tolerate higher load to rotor inertia ratio with good response.
2) The goal of tuning is maximize responsiveness & minimize instability
3) The ideal solution for tuning would be to include more parameters in the PID loop structure. If we could accurately determine a particular element's characteristics (stiffness as an example), it would allow us to better model the system. The only draw back is, the more complex we make a PID loop the longer it would take to tune. We have seen more "tools" added in for tuning however: feedforward, integral limits, system bandwidth, etc.

My take on the matter is as follows:
Inertia ratio is important. However, the importance of the ratio is relative to how responsive the system needs to be and how stiff the system is. Obviously a directly driven load could respond faster w/ a 10:1 ratio than a chain & sprocket could.
Inertia ratio should be considered when extremely high accelerations & decelerations are necessary; settling times & regen could comprimise such a system. Guidelines are there for a reason, but there are exceptions to them. 10:1 for servos
& 5:1 for steppers are the maximum ratios that I've always gone by and I see no reason to change unless that range can't be achieved.

One question still hasn't been answered to my satisfaction: Why do some manufacturers offer inertia slugs on motors?
There were several attempts to address the question. I think we all agree that these slugs do increase the system's inertia (motor inertia plus load inertia). I think most of us would also agree that this increase in inertia lowers the system's bandwidth. The disagreement still remains though, does this lower the motor to load inertia ratio? In other words, is the slug part of the motor inertia? The manufacturers that offer these slugs certainly state that it is. Those of you that responded were split on this one.
Personally, I do not believe it does add to the motor's rotor inertia. The motor's rotor inertia is just what it says......ROTOR INERTIA. The slug's coupling may be infinitely stiff, but it is
still a load and therefore not part of the motor's rotor.

Please let me know what you guys think. Does a slug help? If so, why? If not, why offer it?


Guy H. Looney
Sales Engineer

Regan Controls, Inc.
475 Metroplex Dr.
Suite 212
Nashville, TN 37211
phone: (615) 333-1940
fax: (615) 333-1941
email: [email protected]
In my experience, no other issue in motion control is as confused as the issue of “inertia mismatch”. I think this is because there are a lot of competing and overlapping ideas, which few people separate out properly.

One must be careful not to read too much into the “matched inertia” doctrine, even for the issue of optimizing power transfer. The statement that “the matched inertia case optimizes power transfer to the load” can easily be misinterpreted, leading people to believe that adding motor inertia to a system can improves power transfer, which is absolutely incorrect.

Start with the equation of Newton’s second law (rotary format) in a motor/load system with a gear ratio:

Tm = (Jm + Jl/n^2) * Am = (n*Jm + Jl/n) * Al

Tm is the torque generated by the motor
Jm is the motor moment of inertia
Jl is the load moment of inertia
n is the gear ratio (the motor spins ‘n’ times as fast as the load)
Am is the motor acceleration
Al is the load acceleration (= Am/n)

Note that this equation implicitly assumes the whole system is treated as a rigid body (infinite stiffness).

If you ask the question, “For a given motor inertia and a given load inertia, what is the gear ratio that maximizes the ratio of load acceleration to motor torque?”, you create a classic “max/min” problem by taking the derivative of motor torque over load acceleration with respect to gear ratio and setting this to zero:

d(Tm/Al)/dn = (Jm – Jl/n^2) = 0

n^2 = Jl / Jm

Since the “reflected” inertia of the load (the equivalent amount of directly coupled inertia) is Jl/n^2, the optimal gear ratio is the one that “matches” load inertia to motor inertia. (For this to be strictly true, one must ignore the fact that changing the gearing ratio often changes the inertia on the motor and load “sides” of the gearing, but the principle can still be important.) This is a neat, mathematically elegant result, and it gets way too much attention. Remember that there are many other, usually more important, considerations that go into selecting a gear ratio.

However, if you ask, “For a given load inertia and gearing ratio – i.e. a given physical system –what is the motor inertia that maximizes the power transfer to the load?”, you should take the derivative of motor torque over load acceleration with respect to motor inertia:

d(Tm/Al)/dJm = n

This derivative is positive for all real-world situations, so the optimal motor inertia is zero, or at least the minimal possible inertia. This should be an intuitive result, but many people have convinced themselves otherwise! In the matched inertia case, fully half of the input power goes toward accelerating the motor inertia. If you truly could produce a zero-inertia motor, you could accelerate the load twice as fast as the matched inertia case. In this respect, the “optimum” inertia ratio (reflected load inertia to motor inertia) is infinite, and this has been what has pushed the development of low-inertia brushless motors with rare-earth magnets.

Note that for these purposes, anything on the “motor” side of the gearing, such as encoders or inertia slugs, is considered motor inertia; anything on the “load” side is considered load inertia.

On the other hand, there are other considerations for which the “optimum” inertia ratio is zero! That is, for these considerations, things keep getting better as the effective load inertia is reduced, even when it is less than the motor inertia. These considerations include:

- Changes in load inertia: The lower the inertia ratio is, the less a given percentage change in load inertia changes the overall inertia, and therefore the less it affects the tuning. A big problem with direct-drive motors is that tuning constants often have to be changed on the fly as parts are picked up and dropped off. If gearing has backlash, a large inertia mismatch can be problematic. In a system with a 10:1 ratio, the motor sees only 9% of the total load while reversing!

- Systems with compliance: For many systems, we have to abandon the rigid-body assumptions inherent in the equations above, and deal with systems with a finite stiffness “k”. Usually we can deal with these systems pretty well as “two-body” problems – two rigid bodies, motor and load, connected by a compliant link – the coupling or belt. In these systems, you have to deal with natural frequencies determined by the square root of the quantity {stiffness divided by inertia}.

More specifically, in a two-body motor-load system, there is a resonant frequency (pole pair) at:

wp = sqrt (k * (1/Jm + 1/Jle))

where Jle is the effective load inertia (= Jl/n^2)

Note that for these purposes, anything on the “motor” side of the compliant coupling, such as encoders or inertia slugs, is considered motor inertia; anything on the “load” side is considered load inertia. The split between “motor” and “load” here may not be exactly the same as the split for the gearing calculations.

If feedback on the motor is used, there is also an anti-resonant frequency (pair of zeros) at:

wz = sqrt (k / Jle)

The higher the frequency of the resonant poles – due to greater stiffness or lower inertia – the less they are a problem, because the whole system reacts less at high frequencies. Anti-resonant zeros aren’t usually considered a problem, but if their frequency gets down within the desired bandwidth of the system, they can wreak havoc, because they cause the system to fail to respond to desired actions.

Also, however, the closer in frequency the anti-resonance is to the resonance, the less of a problem the resonance is (the smaller the magnitude of this oscillatory mode) because of the tendency of zeros to “cancel” poles. The greater the inertia ratio (even if less than 1:1), the greater the difference in frequency between the anti-resonance and the resonance, and the greater the magnitude of the oscillatory mode.

Remember that in a system with just motor feedback and a “loose” coupling to the load, the control system itself can appear well behaved, but the load itself (which is what we really care about) can be poorly behaved. Especially with some friction (a non-linear effect), the motor can be locked or nearly locked, so the control loop is happy, but the load can be oscillating vigorously. In this mode, known as “cantilevering” by analogy to a flexible beam with one end fixed, the load resonates at the anti-resonance frequency shown above.

The purpose of the “inertia slugs” some vendors add is to increase the motor inertia. While this actually decreases the resonant frequency (a bad thing), it brings it much closer to the anti-resonant frequency (a good thing that usually more than compensates for the decrease), providing better cancellation of its effects. For example, Kollmorgen’s inertia slugs increase their “low-inertia” motor’s inertia by a factor of 7 to create their “medium-inertia” motors. Taking a system with one of these low-inertia motors and a 14:1 inertia ratio, the resonant frequency is almost 4 times the anti-resonant frequency. The identical system with the medium-inertia motor has a 2:1 inertia ratio, and the resonant frequency is reduced so that it is only about 1.6 times the anti-resonant frequency. Effective total system inertia is increased less than 50%.

Even though the “matched inertia” case is not the optimum here, as you get close to it, these problems tend to become manageable. This is why you get the rules of thumb as to maximum tolerable mismatch. Ultimately, this issue, as with most engineering design issues, is one of tradeoffs and balancing of different desires. Not an easy issue, but it’s why we get paid the big bucks, right?

Johan Bengtsson

I think it might be useful some times, this is the reason for those circumstances where it will be useful:

1. You have a high inertia load but don't need to change the speed that fast.
2. The coupling are not very stiff.

If you add inertia to this system and place it near the motor with a much stiffer coupling that you have for the load then this extra inertia will:
1. Count as motor inertia.
2. Make the whole system slower.

This might make the tuning a lot easier and robustness of the system much better, at the cost of speed changes.

/Johan Bengtsson

P&L, the Academy of Automation
Box 252, S-281 23 H{ssleholm SWEDEN
Tel: +46 451 49 460, Fax: +46 451 89 833
E-mail: [email protected]

Brandon Ellis

While I am probably in no position to even create replies to theoretical discussions of this magnitude (that and the fact that I normally don't
have time), must drop a short comment.

I have only had time to briefly peruse all of these responses to Mr. Looney's email, and therefore must refrain from agreeing or disagreeing with anyone. I would like to say this, however, I am not a mechanical expert by any stretch of the imagination, but I am an electrical engineer and motor theory is both electrical and mechanical.

If I understand correctly, Guy Looney has pointed out that companies such as Kollmorgen and Compumotor are pressing what they call inertial
slugs onto the motor shafts and claiming a resulting "high inertia motor" by including the inertia of the pressed on fly-wheel with the
rotor inertia. While the mechanical folks may agree or disagree with this logic, my common electrical sense says it's a crazy claim.

From an electrical viewpoint, the rotor is the section of the motor (AC or DC) containing the necessary parts (e.g. permanent magnets or coils)
for creating magnetic and, therefore, mechanical movement (e.g. an rotating electro-magnetic field). Using a brushless servo motor as an
example, the rotor truly consists of the permanent magnets. The shaft is normally included in the rotor's inertia number since it must be there to support the magnets (and usually it's inertial component is small since it resides close to axial center). Nevertheless, it is the rotor /stator combination that create the mechanical torque of motor, and the torque is what drives the load. If don't have enough torque (e.g. power), you won't have a good system response time.

If all these companies are doing is adding a fly-wheel assembly with no electro-magnetic characteristics to a fixed performance stator /rotor assembly then all they have added is load inertia. From the electrical aspect, this has simply increased the current / torque requirement of the system.

Either this is fundamentally correct or I am completely crazy.

Brandon S. Ellis
Sales Manager
Robotic Control Group
700 S. Illinois Ave.
Suite A104
Oak Ridge, TN 37830
Tel: (865) 425-0301, Ext. 160
Fax: (865) 425-0268

> Either this is fundamentally correct or I am completely crazy

Sorry, but I'm afraid I think its the latter. ;-)

The best electrical analogy I can think of off the top of my head is two AC circuits coupled by a transformer with a significant step-up or step down. The effect of a given resistor added to primary or secondary side of the transformer is significantly different. In one sense, the resistor always adds "loading" to the circuit, but it is very valid for many purposes to make the distinction between "primary" and "secondary" resistors.

In these mechanical systems, what counts as "motor" inertia and what counts as "load" inertia depends on what is being analyzed. For the purpose of analyzing gear ratios, what matters is what side of the gearing the inertia is. Everything on the motor side of the gearing, including the gear wheel attached to the motor, counts as motor inertia; everything on the load side, including the gear wheel attached to the load, counts as load inertia.

For the purposes of analyzing compliance (assuming only one significantly compliant member in the system), everything on the motor side of the compliance, whether or not it plays a direct role in generating torque or not, counts as motor inertia; everything on the load side of the compliance counts as load inertia.

Imagine you have two brushless servo motors in front of you that look identical from the outside. One uses heavy ferrite magnets on the rotor; the other uses lightweight rare-earth magnets and has an inertia slug stiffly installed on the shaft. You try to figure out which is which without disassembling them. Accelerating them from an external source, you see that their rotor moments-of-inertia are the same. Pumping current into them and measuring their torque outputs, you see that they have the same torque constant and torque capability.

For all meaningful purposes of analysis as to how these motors work in a system, they are identical. The inertia slug has to be treated as "motor" inertia. Of course it does add to total system inertia, but for any purpose of making a distinction between motor and load inertia, it has to count as motor inertia, the same as the "extra" inertia of the heavier ferrite magnets.

Curt Wilson
Delta Tau Data Systems
Fundamentally correct, however as Mr. Curt Wilson pointed out, very beneficial when applied in certain dynamic applications with closed loop control on the motor feedback.


Brandon Ellis

While my intent is certainly not to argue (like I had stated in my previous email, I am by no means an expert in mechanics), I am afraid that some folks may have missed my point. In short, based
on the aforementioned mechanical equations and theories presented by many folks involved in this discussion, the derived result could be that if one could produce a motor with infinite power, yet infinitely small inertia, it could NOT maintain/turn a load, despite the load's inertial characteristics.

I am not one to argue as to whether system inertia, fore or aft of the gearhead, should be counted as load or rotor inertia, but, as Curt
pointed out it is definitely system inertia. With this, I say again, it comes down to power and as we all know, when an inertial load is added to an electro-mechanical system consisting of a rotating
electro-magnetic field (e.g. a motor), the amount of current required to produce the extra torque needed to maintain the load must increase. If this current is not available (or more than the motor's coils can handle which results in smoke) then system stability is compromised.

Again, I am not trying to argue and only refer to Mr. Wilson because I would love to hear have his input. If I am wrong, yet I learn, than I have gained. I just wanted to make sure everyone
understood my angle of this subject.

Brandon S. Ellis
Sales Manager
Robotic Control Group
700 S. Illinois Ave.
Suite A104
Oak Ridge, TN 37830
Tel: (865) 425-0301, Ext. 160
Fax: (865) 425-0268
To some extent this is a semantic issue. If someone is asking whether a given body should be counted as "motor inertia" or "load inertia", you have to look at in what context -- why they are asking the question.

Almost always this question has to do with gearing or compliance. If it is in the context of gearing, the only place to draw the line is at the gear. To give an example, take a motor with a 10-to-1 reduction to the load shaft. An inertia slug pressed on the motor shaft directly adds to the total system inertia that the motor "sees". The same inertia slug pressed onto the load shaft only creates 1% as much of an addition to the inertia that the motor sees -- a critical difference.

If the question is in the context of compliance, the only reasonable place to draw the line between motor and load inertia is at the compliant link. Take a motor and load shaft tied together through a compliant coupling (no gearing). The identical inertia slug pressed onto a shaft will have very different effects on the dynamic equations and the stability of the system depending on which side of the coupling it is added.

The companies that add an inertia slug onto the motor shaft explicitly say that the purpose is to improve the stability of compliant systems with high "load inertias", by bringing up the motor inertia. Obviously, they are talking about inertia levels in the context of compliance, so what matters is which side of the compliant link the inertia is.

Some companies produce ferrite-magnet motors for their higher-inertia motors and rare-earth-magnet motors for their lower-inertia motors. Other companies, for production reasons, have decided to produce rare-earth magnet motors, and add these inertia slugs when higher inertia is required. My point is that in any meaningful context, these two types of higher-inertia motors behave identically as to their basic physical performance, and the inertia slug should no more be considered "load inertia" than the fraction of the mass of the ferrite magnet that is not producing useful magnetic field (or the extra structural mass required to attach the bigger magnet).

Of course, the inertia slug contributes to total system inertia, and not at all to the motor's generated torque -- so in this context I suppose it could be considered load. But this is true of lots of other parts of the motor rotor -- and these parts are never considered "load inertia". Hypothetically, compare the inertia of a steel motor-rotor shaft to a lighter-weight titanium shaft (we don't see titanium -- that I know of -- because no one's willing to pay for it). Should the extra inertia of the steel shaft be considered load inertia?

My fundamental point is, that in the key contexts that it is important to make a distinction between motor and load inertia, an inertia slug on the motor shaft must be considered as part of the "motor inertia".

Curt Wilson
Delta Tau Data Systems

Bruce Durdle

I hate to prolong this discussion more than necessary, but surely there is no effect of inertia on the power needed to MAINTAIN the load? There will be more power needed to accelerate the system at a given rate, and if speed changes are required to be carried out in a given time, then certainly more power is needed. But if speed is maintained constant then the addition of inertia will not affect the power needed.

Regard the effect of inertia as the mechanical equivalent of reactance in an electrical circuit - it will have an effect on the current or voltage
while the circuit is changing (AC or transient DC) but no effect on steady DC.


>...the amount of current required
> to produce the extra torque needed to maintain the load must
> increase. If this current is not available (or more than the motor's
> coils can handle which results in smoke) then system stability is
> compromised.

Brandon Ellis


You are certainly correct. When I used the term "maintained" I was thinking in terms of the constant adjustments being made by the closed
loop system especially following a shock to the system (e.g. a fluctuation of the load). As I am sure you know, in an high inertia, high gain system, a shock to the system can result in a diverging oscillation if the motor is undersized / the inertial mis-match is large.

Thanks for pointing out the error of my general statement. A new perception is as good as gold.

Brandon S. Ellis
Sales Manager
Robotic Control Group
700 S. Illinois Ave.
Suite A104
Oak Ridge, TN 37830
Tel: (865) 425-0301, Ext. 160
Fax: (865) 425-0268

Johan Bengtsson

Of course that is fundamentally correct, but it wasnt that the question was about in the first place (if I remember it correctly). Of course you need to have torque enough to move the load, of
course extra inertia don't help the torque at all, and of course you only need more torque if inertia is increased (regardless of where).
But then: whay should the motor have any inertia at all? Depending on type of motor you could make it without almost anything in the rotor but some wires and get a very low inertia motor.
Depending on how you make this motor it could still produce a very high torque, and that mean a very high acceleration.
The servo manufacturers however state something like:
"The load should not have a inertia more than X times the motor torque" (recalculated according to gearing and so). Torque is what makes things move (and stops them), inertia is what keep them
moving at constant speed. If you are not concerned with changing the speed that fast, and don't need a lot of torque to hold or move the load at constant speed (hanging load / friction)
Then the question is: can I use a smaller motor (with less inertia) since that one produce enough torque for my needs?
Some says: No way! Keep the inertia ratio between motor and load. Ok, then the question is: Can I increase the motor inertia in order to get a better ratio. - I still have enough torque on this smaller motor to get the torque I need.
Some says: The inertia ratio is needed for easier tuning, it could be done with other ratios, but you have to spend more time with the tuning.
Some says: No way No way, buy a bigger motor and servo amplifier!

/Johan Bengtsson

P&L, the Academy of Automation
Box 252, S-281 23 H{ssleholm SWEDEN
Tel: +46 451 49 460, Fax: +46 451 89 833
E-mail: [email protected]
All depends on the mechanical configuration. If it is a lift, the load is hanging off of the motor and so the holding torque increases with the load. If it is a horizontal conveyor belt, then there is no holding torque.

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Cameron Anderson

I am a motion control specialist and size thousands of servo/stepper applications a year. Here is quickly what I have to say about Load to Rotar Inertia Ratio:

1. Never ever size an application where the Load to Rotor Inertia Ratio is greater than 10:1. Some manufacture's say "With our high performance servo algorithims, we can handle up to 50:1 mismatch." Yeah, we can all handle 50:1, but will it perform like 10:1? NO! You are asking for trouble above 10:1. Don't get sucked into the BS sales and marketing scams.

2. 10:1 or less for servo's and steppers. 1:1 May be perfect, but I find that having a 1:1 or a less than 1:1 ratio is not good. You want some inertia out there to dampen you system. Make the motor work for its living. Keep at least 2:1 to 10:1 range. If it's less than 1:1, then you paid too much for your motor.

3. Again, Never ever size an application where the Load to Rotor Inertia Ratio is greater than 10:1.

4. Torque = Inertia x Rotational Acceleration
The more intertia, the more torque the motor needs to produce.

5. To reduce inertia reflected to your motor use a gear box. The formula to reduce inertia is <i>load inertia/ratio^2</i>

6. Don't forget about the Inertia of the gearbox and the inertia of the coupler when sizing your motor.

7. Once again, Never ever size an application where the Load to Rotor Inertia Ratio is greater than 10:1.

Cameron Anderson - Power/mation division
Motion Control Specialist - St. Paul, MN
API Motion - Baldor - Bayside - CT
DSPCG - Emerson - IDC - Normag
Superior - Warner Electric
Phone: 651-605-4437 or 800-843-9859
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Gannel Leonid

Dear Mr. Looney,
Please find an answer on your question below.
The motor slug allows increasing the motor moment of inertia only and thereby decreasing of mismatch ratio. This assumption is true due to very high stiffness of slug attachment to motor shaft. And as a rule the motor slug needs for applications with high mismatch more than 1:5…10 especially with brushless motor owing to its relative lower moments of inertia. Operation with a reasonable (optimal) mismatch ratio (1:2…5) is a good practical solution not only from energy dissipation point of view. It also allows damping of mechanical vibrations by servo amp tuning only.
Best regards,
Leonid Gannel
R&D Servo Control Engineer
NUR Macroprinters Ltd.
Tel: 972-3-908 76 76, Ext. 353
Fax: 972-3-908 65 68
Email: [email protected]

Gannel Leonid

You might add one more suggestion: “Choose motor with higher moment of inertia”. Some motor producers as for example MPC, Baldor, etc. allow ordering the same motor with low (standard) or high moment of inertia. As a rule the difference is 2...4 times. For Baldor we might find A (low moment of inertia) and B (high moment of inertia) series of motors.

Leonid Gannel
R&D Servo Control Engineer
NUR Macroprinters Ltd.
Tel: 972-3-908 76 76, Ext. 353
Fax: 972-3-908 65 68
Email: [email protected]