# Basic Physics behind the system

The basic HO locomotive system is show in the following figure:

From this the equations of motion can be readily derived. Schematically this is shown in the following Figure.

This is the classic definition for a train system. There are several references on-line that show this or a similar schematic. The tractive effort is the resulting pulling force of the locomotive. This includes the drag on the loco.
The resistance is the resistive force of the train. The resistance typically is defined using the ‘Davis’ equation from 1924. This equation is shown in the following figure.

For almost all situations term e is orders of magnitude larger than all the other terms.

As sown in the figure above, the grade in term e is defined the same way we define it in model railroad work. It is basically the tangent function that will increase asymptotically as it approaches 90 degrees. So it gets severe very fast. Typical grades are no more than 3 percent.

This chart shows the resistance for a typical 100 Ton freight car.

This shows that a 1% grade increases the resistance by roughly a factor of 8 where increasing the speed up to 70 mph has a factor of 3 or less. This series of charts are actually for full size railroad systems, but the equations and the trends will be the similar for model trains.

Traditionally Locomotive performance has been defined using the term of Tractive Effort.

This is the force which a locomotive can exert when pulling a train, and depends on various factors. For electric locomotives, which obtain their power by drawing current from an external source, the most important are:
weight- the adhesion between the driving wheels and the track depends on the weight per wheel, and determines the force that can be applied before the wheels begin to slip.
speed- up to a certain speed, the tractive effort is almost constant. As speed increases further, the current in the traction motor falls, and hence so does the tractive effort. While the model motors do not act the same as real traction motors, the characteristic reacts in a similar fashion.

The typical shape of the Tractive Effort as a function of speed is shown in the following figure.

This curve is for a specific horsepower. As power is increased, the curve moves out in the direction shown in the figure. As power is reduced, the curve collapses along a similar slope. The maximum speed is limited by the design of the trucks and the speed capability of the traction motor. This is essentially a no load condition and the load of the locomotive will be balance against the motor capability to define this speed.
The maximum Tractive Effort has three limits in real railroads. These are discussed in the following figure.

For the maximum Tractive Effort, three limits come into play for train locomotives. The first is the starting TE. This is dominated by the capability of the traction motors before they stall. In the case of the model, it will be dependent on the motors performance before stall at a given power. It has been observed by several authors that the maximum Tractive Effort for full sized locomotives is a constant for speeds under about 8 miles per hour.

A second limiting factor is the maximum continuous point where the system just starts to overheat. Operating any mechanical system near its stall line is generally inefficient and will cause thermal issues at the very least. It is a common practice to insist that some margin is included for a long time operation. This point occurs below the maximum TE line. A similar practical restriction should be applied to model motors. Operating them in a near stall condition will limit their useful life.

The third factor is the locomotive wheel adhesion line. This line is the point where the torque produced is equal to the wheel friction. Above this line the wheels will slip. More force is produced, but the wheels begin to spin rapidly. In general, it is not a very desirable situation over long periods of time. Excessive heat will lead to wheel and track damage.
The position of this line is dependent primarily on locomotive weight. The number of drive wheels and the roughness of the wheels also play a role in this lines location on the chart.
These charts are useful in understanding the balanced system at a given speed for a specific resistance (train load and grade). This is shown in the following figure.

Here the Tractive Effort curve is for a specific throttle power setting. The train load, resistance. is plotted versus speed. The speed that the train will balance out at is shown by the arrow.

For this power setting, the speed V1 would be satisfied if the train load were heavier (more car weight or higher grade). In the case of the higher speed, V2, the load would have to be reduced.
This is the characteristic representation that is the objective of the test series. But before elaborating of how to define this in a test series, the basic physics of the power generation line for the HO mode locomotive motors needs to be examined.
The typical Model locomotive motor is shown in the following figure:

The general equations for this motor are as follows:
Electric Power
P = V I
P = R I^2
P = V^2/ R

where
P = power (watts, W)
V = voltage (volts, V)
I = current (amperes, A)
R = resistance (ohms, Ω)
Electrical Motor Efficiency
μ = 746 Php / Pinput_w
where
μ = efficiency
Php = output horsepower (hp)
Pinput_w = input electrical power (watts)
or alternatively
μ = 746 Php / (1.732 V I PF)
Electrical Motor – Power
P3-phase = (V I PF 1.732) / 1,000
where
P3-phase = electrical power 3-phase motor (kW)
PF = power factor electrical motor
Electrical Motor – Amps
I3-phase = (746 Php) / (1.732 V μ PF)
where
I3-phase = electrical current 3-phase motor (amps)
PF = power factor electrical motor

What this means is that by providing a certain electrical power, VI, the motor will rotate and produce a certain output horsepower, Php. In actuality, there is a balancing act that is required, because even though the supply has the current capacity, the motor won’t draw it unless it needs to. Thus when you apply voltage, V, to a free-standing motor like shown in the figure above, the motor starts rotating. If you had the proper dynamic response sensors, you would see the rpm quickly increase to a peak value, then it would go through a sinusoidal oscillation until it settled at the particular motors rpm value for the power added. At this stabilized point, the motor is demanding a specific current draw, I. This is the current required for the electrical power to provide the horsepower to overcome the mechanical losses at that rpm. As long as the power supply has the capacity, then the motor will run stably at this rpm value. By increasing or decreasing the voltage, the same balancing act is repeated until the rpm settles out at a new value. Again along the motors rpm power curve.

This rpm can be expected to vary between motors of the same model because they will not have the same efficiency. In brand new motors, the manufacturing tolerance will impose different mechanical and electrical losses, thus different efficiencies. As the motor is used or just sits, one can expect these mechanical and electrical losses to increase due to dirt, oxidation, and increased resistance just to name a few possibilities.

Typical electric motor characteristics are shown in the following figure

This is typical and not for the motor shown in the previous figure. Electric motors vary by design. The power to torque relationship is tailored to fit the application. In the case of full size railroads, freight engines are designed to have a maximum torque at low speed. Loco’s used primarily for passenger service tend to push the power to higher speeds because the trains are not as heavy and the emphasis is on having power available at high speed.

The actual situation for model locomotives is illustrated in the following figure:

In this case there are additional losses involved, relative to the motor only example On the electrical side, those losses take the form of voltage drops as follows:

a) From the power supply to the track. These are the Buss losses. These can be minimized by using a heavier gauge wire.
b) Along the track. The Nickel Silver is an average conductor. The joiners are a high resistance connection. These can be minimized by connecting every section, ie 3 foot of track as is the recommendation for DCC operation.
c) The rolling contact between the track and wheels leaves the opportunity for loss in voltage. Here, the best thing to do is have like materials on both the track and wheels and have clean contact surfaces.
d) The various contact joints from the wheel surface to the motor. Dirt and oxidation will have an adverse impact here as well.

On the other side of the issue, the horsepower delivered to the track is also impacted by the mechanical losses. These power losses include:

a) Spline vibration. Excessive oscillation will reduce the torque delivered to the gear box.
b) Drive resonance frequencies. All mechanical systems have natural frequencies where they have a significant resonance with very little energy input. While it is unlikely that a first order resonance will exist in the operating range, second and third order will likely occur. These will rob power from the system when they occur.
c) Gear box losses. The truck towers no matter what the design are reduction gear boxes. In every case there is some power lost in the gear box process. Dirt, grime and oxidation will make these losses grow rapidly.
d) The power is transferred to the track through the friction forces exerted by the wheels on the track.
1- Clearly the number of wheels and the weight per wheel will impact this rail frictional force.
2- The roughness of the surface of the rails and the wheels will also impact the force transfer. In this case rough surfaces provide more force and smooth or polished surfaces will reduce the force.
3- Water, ice or oil and similar substances on the rail will limit the force transferred.

The test results will be implicitly examining the performance of the motor in combination with these losses. So two loco’s using the same motor will have significantly different results depending how these additional losses are handled.