## Thursday, September 22, 2011

### What I know about gears, RPM and speed

I was always curious about cars maximum speed, not the one shown in the speedometer, but the real one. So, after some research, this came up to me that the speed is related to 3 simple things.

- RPM: The speed the engine can turn itself in revolutions per minute.
- The gears ratio: How the engine speed can be transferred to tires
- The tires's size: Yes, size matters!

So, lets do the magic.

1 - Gears: The gear is a toothed wheel that when coupled to another gear can alter the relation between the speed of a driving mechanism. This relation about teeth and speed, can be expressed in the following formula:

Where:
V1 is the speed at the first gear.
V2 is the speed at the second gear.
G1 is the number of teeth in the first gear.
G2 is the number of teeth in the second gear.

Looking at the example below we can see another gears property. Coupled gears work in inverse rotational directions, when gear A is turning clockwise, gear B is turning counter clockwise. If the same rotational direction is needed another gear can be added to the mechanism or pulleys can be used instead.

Using the same example, we can determine what will be the speed at both gears if one of them starts turning. Notice gear A has 12 teeth and gear B has 18 teeth. Applying the formula given above, let's suppose that gear A suddenly starts turning at the speed of 30 RPM, then:

A small gear coupled to a bigger one will decrease the speed of its counterpart (and increase the power of it), which brings us to the following question: What if the gear B was the one that started turning at 30 RPM? So, lets see what happens:

When this speed is applied to gear B, the gear A will start turning at 45 RPM. Therefore, a bigger gear coupled to a small one will increase the speed of the mechanism. Two important things: Gears in the same axis have the same RPM no matters the size of them. When the speed is increased, the power is decreased and vice-versa.

2 - Ratio: Another way of calculate the speed in each gear is use the ratio between the first and second gear. As you will discover, all the car gears are expressed using this number. This can be easily done applying this formula:

Where:
R is the gear ratio
G1 is the number of teeth in gear A
G2 is the number of teeth in gear B
V1 is speed at gear A (in RPM)
V2 is speed at gear B (in RPM)

3 - Tires. There is a very good article about tire codes at Wikipedia. I got the picture below from there which I will use to do a little summary about the parts that is related to our job.

From the picture above we are interested in 3 things:

a - Nominal width: This is the width of the tire in millimeters, using the example above the tire's width is 215mm.

b - Ratio of the height to width: The height of the tire is a percentage from its width. Using the example above, this will be 65% of the width: 0.65 x 215 = 139.75mm.

c - Rim diameter code: This is the diameter of the wheel, this is expressed in inches. As all the calculations are done using the metric system, we are going to convert it to meters. Again using the example, the wheel diameter is 15 inches, or 381 millimeters.

We are going to use all these number to discover one simple thing, the circumference of the whole wheel (wheel + tire). From the picture above we can clearly see that the mounted wheel diameter is the wheel size + 2 times the tire height. So, the full diameter will be 381 + 2 * 139.75 = 660.5 mm. Now we just multiply by π and we get the circumference: 660.5 mm * 3.1416 = 2075.03 mm, or approximately 2.07 meters.

3 - Remembering: Let's make a summary about we got until now:

a - We know the engine speed is measured in RPM (a common RPM speed for cruise cars is about 5000).

b - We know that car have gears and these gears have ratios, some of them increase the speed of the engine and some of them decrease it. Some of them are fixed gears (called differentials) and some of them can be changed while you are driving. This is done using the clutch on manual cars, and they are automatically changed on automatic cars.

c - We know that gears on the same axis will turn at the same speed, doesn't matter their sizes.

4 - Speed:  How about after all the gearing the very last gear is attached in the same axis with the wheel? It will just be a matter of calculate all the speed transmission among the gears to discover the RPM at the last gear and as the last gear is in the same axis with the wheel, every time the gear turns, the whole will turns as well, or, every time the last gear turns, the car moves 2.07 meters using the tires from our example (I told you size matters!). Supposing that after all the coupling the last gear is turning at the speed of 10 RPM, the car will be moving at 20.7 meters per minute, or 3.45 meters per second or 12.42 km/h or 7.72 mph.

5 - Common ratios: After a little research in the net I found some gear ratios regarding commercial bikes (B), cars (C) and SUVs (S). See below the table.

VehicleRPM Start Differential Final Differential
 Gear ratio 1st 2nd 3rd 4th 5th 6th
(C) Volkswagen Golf57503.0871.003.362.09 1.471.191.150.98
Speed (km/h) with default tire: 1.81m60.1697.31137.60169.96175.93206.45
(B) Suzuki GSX-R 1000108001.5532.472.687 2.0521.6811.451.3041.208
Speed (km/h) with default tire: 2.00m125.76164.64201233.04259.08279.72
(S) Ranger 3.0 Electronic38003.541.004.082.291.471.000.72-
Speed (km/h) with default tire: 2.19m34.5661.6395.92140.99195.92-
(B) BMW F650 GS Dakar65001.9462.93752.751.751.311.050.88-
Speed (km/h) with default tire: 2.18m54.0288.94118.77148.33176.84-
(B) BMW K 1200 S102501.5592.822.521.841.451.281.141.01
Speed (km/h) with default tire: 1.94m107.67147.48187.17211.96238.04268.65

To calculate the maximum hypothetical speed at any gear just do the math. For instance, to calculate the VW Golf speed at the 3rd gear with default tires:

The ratio is including fixed gears in its calculation (differentials), we just need to multiply them together. Notice these values are expressed as maximum possible values and will rarely be obtained due to external factors, like friction and engine efficiency, which can literally blow if this is kept on the maximum RPM for long periods.

Have fun!

## Tuesday, September 20, 2011

For a while a question has been hunting me: How the transformers really work under the hoods? Below I present some gathering I got from the internet, including concepts, formulas and some explanation about it. Have fun!

1 - What is a transformer? According to wikipedia:
“A transformer is a device that transfers electrical energy from one circuit to another through inductively coupled conductors—the transformer's coils.”

In other words, the transformer changes the voltage applied to a new value, smaller or greater than the original value. The applied voltage is commonly called the primary voltage or primary winding, and the converted value is called the secondary voltage or secondary winding. NOTICE transformers are by their nature bi-directional, “primary and secondary” concepts are dependent of the way you are going to use it. If you get a transformer 110/220, this can be used to convert 110 (primary) to 220 (secondary), or backwards, 220 (primary) to 110 (secondary).

2 - How does this work? When one alternating current is applied on the primary coil of the transformer this generates a magnetic field due to the inductance. This magnetic field then inducts a current on the secondary coil of the transformer. The relation of the applied and the converted voltages depends on the wire turns in both coils as in the following formula:

Where Vs is the voltage in the secondary, Vp is the voltage on the primary, Ns is the number of turns in the secondary and Np is the number of turns in the primary. This formula will be used and explained later on this post.

3 - Power and current: According to the physics's law the power in the primary and in the secondary must remain the same. As the power of the circuit is voltage multiplied by the current, the current in the secondary coil will be proportional to the current and inversely proportional to the voltage in the primary coil (as bigger is the voltage in the primary, as smaller is the current in the secondary). This relation is best described in the following formula:

Where Vs is voltage in the secondary, Vp is the voltage on the primary, Ip is the current on the primary and Is is the current on the secondary. As an example, if we have 2 amperes in the primary with a voltage of 220 and we want to transform this to 110 volts, the current in the secondary will be 4 amperes, as following:

4 - Defining current and wiring: As mentioned before, the current is related to the wiring used, as bigger the current as thick the wire must be. Before knowing the thickness, we need to calculate how many turns of wire the coil must have. The primary step for these calculations is to define the area of the transformer core. All further calculations are based on this. This is very simple to find and its based on the power in watts the transformer must supply. The area is calculate in square centimeters (cm2).

Where A is the area area in square centimeters (cm2), W is the power (voltage x current) and Q is a material quality factor. This factor is based on the quality of the material used for the core. Typically this is 0.8 for very good materials, 1 for regular materials and 1.2 for poor materials. As a rule of thumb, use 1 for calculations for homemade transformers.

Below we can see an example of what a transformer core looks like. The core is mounted using E plates which can be found in several sizes. During the mount process 2 things must be kept in mind: The area must be as close as possible from the value found in the formula and the core's shape must be as similar as possible from the square shape. The core's area can be calculated from the E plates multiplying its width by its height in centimeters. In the picture below the core's area is shown in blue.

Following our previous examples, lets calculate the area needed for a transformer which has 220V and 2A (440W) on the primary coil.

After calculating the area of the core, now is just calculate how many turns of wire we are going to need in the primary coil. This is done using the following formula:

Where Np is the number of turns in the primary, V is the voltage, f is frequency of the alternating current in hertz, B is the magnetic flux of the core material in Gauss and A is the area of the core in cm2. The magnetic flux of the material is given by the material supplier. When this number is unknown, use 10000 for a approximate calculation. 4.44 is 4 times the alternating current form factor. 108 is derived from the electromotive force principles (e.m.f). Continuing with the previous example, let's calculate the number of turns necessary for 220V, 2A, and the frequency of 60Hz.

Now that we know how many turns we need in the primary coil, we just use the formula given above to calculate the number of turns in the second coil for the voltage of 110V, as follows:

The number of turns in the secondary coil is 197. As stated before, for this example, the current in the primary is 2A and the current in the secondary is 4A. The thickness of the wire is defined by these values and can be calculated using the following formula:

Where d is the diameter of the wire in millimeters (mm), I is the current in amperes and J is current density in the material in amperes per square millimeter (A/mm2). The density is a decimal number commonly in the range from 1 to 2. When using 1 the transformer will flow without extra heating but more material are used. When using 2, less material is needed to build the transformer but this cannot be used for long periods because this can literally burn. A widely used value is 1.5. Below is the calculation for the primary and secondary coils for our example.

For our example the wire thickness for the primary coil must have at least 0.85mm and for the second at least 1.7mm. In stores the wires are sold by the AWG table, see below what is the closest greater value for the diameter found in the formula. For our examples, the wire for the primary coil is AWG 19 (0.91mm) and for the secondary coil is AWG 13 (1,83mm).

Related material can be found in: