## Introduction

In this article I shall attempt to explain, in simple terms, how a dynamo’s output varies with the speed of the bicycle, and how this variation can be limited to prevent blowing bulbs. If you find anything that is incomprehensible, or just plain wrong, please let me know. My electrical machines theory is more than a little rusty.

## Simple model

To predict the output of dynamo lights under various conditions we need a mathematical model of the system, i.e. an equivalent circuit. Figure 1 shows a voltage source *V*_{s}, an internal resistance R_{int}, an internal inductance L_{int} and a load resistance R_{load} connected in series.

To keep things simple, I am making several assumptions:

- the load resistance is constant. In reality, the resistance of a light bulb increases as it warms up.
- the voltage source is sinusoidal. Measurements I did on a Union bottle dynamo show that this is the case, except when there is no load connected.
- the internal resistance & inductance are constant.

As the cycle speed increases, the voltage & frequency of the source also increase:*V*_{s} = k *f*

The reactance of the internal inductance is also proportional to frequency:*X*_{int} = 2π *f* L_{int}

The resistance and reactance form two sides of a right angled triangle. To compute the overall impedance of the circuit we use Pythagoras’ theorem:*Z* = √((R_{int} + R_{load})^{2} + (*X*_{int})^{2})

The current is now*I* = *V*_{s} / *Z*

This is where the notion that bicycle dynamos produce a constant current comes from. If *X*_{int} is much larger than (R_{int} + R_{load}), both *Z* and *V*_{s} are proportional to speed, so *I* is constant. In reality, at 10 mph, *X*_{int} is only twice (R_{int} + R_{load}).

## Measurements

I measured the voltage & frequency produced by a Union bottle dynamo under various load conditions and speeds (by turning my bike upside down and spinning the pedals). I also have a Sturmey-Archer Dynohub, but have not yet made a test rig to enable it to be spun while taking measurements.

With no load connected, the source voltage *V*_{s} is measured directly. From these measurements:

k = 0.053V/Hz.

The internal resistance can be measured directly:

R_{int} = 5.9Ω.

The load resistance of front and rear lights in parallel is calculated from the voltage and power:

R_{load} = V^{2} / P = 6 * 6 / 3 = 12Ω

The internal inductance cannot be measured directly (with my multi-meter) so I inferred it from measurements of load voltage at four different frequencies, using an electrical resistor rather than a light bulb as a load. A good fit is obtained with:

L_{int} = 14mH

Using these values, the output power for various loads & speeds can be calculated. I haven’t yet measured the relationship between road speed and dynamo output frequency. Assuming the system achieves full power (3W) at 10mph, the output at 20mph will be 4W, and at 30mph it will be 4.2W (40% overload). It is no wonder that bulbs blow so often.

It has been suggested that you can get more power from a dynamo by using 12V 6W bulbs instead of 6V 3W. This gives a load resistance of 24Ω. Full power will be reached at 17mph, at 30mph the power will be 7.7W (28% overload).

## Limiting the dynamo’s power

There are two main reasons for premature failure of dynamo light bulbs – loose connections and cycling so fast that the dynamo generates too much power.

If either the front or rear lamp becomes temporarily disconnected, the other lamp will bear the full brunt of the dynamo’s output. This may blow the bulb, leaving the first lamp to bear the full dynamo output when it is reconnected. This bulb may then blow. Hence both bulbs have blown because one lamp had a loose connection.

The high speed riding problem can only be cured by limiting the dynamo’s output in some way. High tech solutions to this involve using the dynamo to charge batteries, and using the batteries to drive the lamps. This is heavy and expensive. The cheaper way is to fit Zener diodes to the dynamo.

Zener diodes are electronic devices that have a fixed “breakdown” voltage. They only conduct when a higher voltage is applied. Two such diodes, connected back to back, can be used to clip the peaks of the alternating voltage produced by the dynamo.

Calculating the output power of a dynamo with Zener diodes is not easy, as the circuit is no longer made up of linear elements. Non-linear circuit analysis is beyond the scope of this document.

A simple approximation is to assume that the diodes merely clip the dynamo’s sinusoidal output wave-form, with no other effect. The power dissipated in the lamps under these conditions depends on the area under the graph of the square of the clipped voltage.

I have computed the lamp power for various dynamo output voltages and values of Zener diode. The results are tabulated below:

unclipped volts | unclipped watts | clipped watts | ||

6.8V diodes | 7.5V diodes | 8.2V diodes | ||

5.5 | 2.52 | 2.46 | 2.52 | 2.52 |

6.0 | 3.00 | 2.69 | 2.93 | 3.00 |

6.5 | 3.52 | 2.87 | 3.20 | 3.45 |

7.0 | 4.08 | 3.01 | 3.40 | 3.74 |

7.5 | 4.69 | 3.13 | 3.56 | 3.97 |

8.0 | 5.33 | 3.23 | 3.70 | 4.16 |

As you can see, none of these provides a perfect solution – the output power can still exceed 3W. The 7.5V diodes have minimal effect on the output at low voltages, but provide some useful power reduction at higher voltages.

## Fitting Zener diodes

Zener diodes are available from electronic component suppliers such as Maplin for a few pence. Unfortunately the minimum postage & packing charges add significantly to the cost. For regulating a normal 6V 3W dynamo lighting system, you need two 7.5V 1.3W Zener diodes. If you can only find higher wattage types, this is not a problem.

Connect the diodes in series “back to back”, then connect this assembly between the dynamo’s output terminal and the cycle frame (assuming the dynamo uses the frame as part of the lighting circuit).

The only difficult part is making the whole assembly resistant to vibration, mud and other environmental hazards. It may be best to fit the diodes inside a small plastic box and screw this to the dynamo mounting bracket. Alternatively, some head lamps may have room to fit the diodes inside. At present I am using a three way terminal block mounted on the dynamo bracket, but this is not really satisfactory.