As a child I enjoyed playing with the power of the sun, wreaking havoc on my plastic soldiers with a magnifying glass. About that time I also discovered the magic of Fresnel lenses but my inability to buy any locally thwarted my budding interest. Once I reached university I started playing around with solar energy again. Parabolic mirrors and Fresnel reflectors caught my interest.

Fresnel reflectors consist of concentric mirrored rings. Each ring is tilted inward by a precise angle. Any light falling on the ring from above is reflected toward a central focal point. The angle of the tilt needs to be quite precise and this makes the construction of a Fresnel reflector a bit tricky.

When a flat ring has a chord section removed, it gains a bevel when rejoined.

At the time I observed that a flat ring of cardboard with a section removed could be joined back together to form a conical ring. This conical section is just what's needed to make a Fresnel reflector. The idea isn't new and others have used this to make their own mirrors but determining the size of rings needed and what sections to remove is difficult. In this post I present a program that you can use to generate an easy cutting guide that will guarantee a working Fresnel reflector.

Download FresnelCalc (VB)

  

Brief Theory

To begin, I specify a value of a which is the inner radius of the current fresnel ring. Ultimately I want to calculate /2. I do this by first calculating and using the fact that = 90-.

Because the focus is not a point, the value of is also dependent on the radius of the focus, r. Calculating analytically is difficult for me so I used an iterative approach.

e = r / sin()     and    = atan (FL / (a + e))

First I guess a value of and use it to calculate e. The I use e to calculate . The process converges after about a half dozen iterations.

Armed with I can now determine /2 and the width of the ring.

RingWidth = 2r/cos(/2)

In order to determine the size of the flat ring needed to make the beveled reflector ring I use the fact that the ratios of the outer vs inner circumferences of the flat vs beveled rings must be the same.

For the beveled ring, the ratio is   b/a   or   (a + 2*r)/a

For the flat ring, the ratio is   (c + 2r/cos(/2))/c    where c is the flat ring inner radius.

Because the ratios must be the same we can write

(c + 2r/cos(/2))/c = (a + 2*r)/a

c = a / cos(/2)

To determine the angular portion of the ring that must be removed, , we remember that the remaining inner circumference of the flat ring must equal the inner circumference of the beveled ring,

remainingCircumference = (2c) * (1-/360) = 2a

= 360 * (1 - 2a/2c)

Using the Program

To validate the program I built a fresnel reflector based off the settings shown above. I am omitted the inner ring since it contributes little to the overall reflector.

The numbers I used are in centimeters but could be any units you like. The 'radius of focus' determines how tightly the light will be focused. It also controls the width of the strips. A small value offers higher concentrating power but produces many thinner rings. You need to consider the material from which you build the reflector. I'm using bristol board and have chosen a value that will create a few rings with about a 1" (2.54cm) width.

The focal length determines how far from the ring the light will converge. Short focal lengths produce rings with steeper bevel angles. This creates more rigid structures. You'll notice that there are gaps between the rings. This prevents inner rings from shading outer rings. At short focal lengths these gaps increase and 'coverage' drops. Coverage refers to how much of the reflector's surface is reflective. Long focal lengths increase coverage but produce rings with shallow angles that are less rigid. If you used a rigid material like 1/8th inch MDF (medium density fibreboard) to make the rings you should be able to use a much longer focal length and get better coverage. But of course you'd probably have to use a router mounted on a swing arm (or a CNC) to cut the rings.

The 'Max Ring Radius' refers to the size of the media from which the flat rings will be cut. You'll notice that the largest ring (29.5) slightly exceeds the stated limit of 27.94. This is because the program allows for the fact that part of the ring is missing (53.8 degrees) and can be slightly larger.

Building the Fresnel Reflector

In my example there are six rings that provide a 59.9% reflective coverage. If built exactly, the light at the focus will be concentrated 117 times (area of the rings as seen from directly above divided by 1/2 the surface area of the sphere). The tilt angle of the final ring is 31.72 degrees and will, I hope, produce a sturdy ring structure.

The printed numbers in the screen shot are the flat ring dimensions. The inner radius, the outer radius, and the size of the chord to remove in degrees. Here's a picture of my rings. As mentioned above I've omitted the smallest as it contributes little. I covered the rings using reflective aluminum tape after I cut them out. Alternately you could use shiny aluminum foil plus glue, or perhaps reflective mylar. Be creative.

The finished rings are mounted on a flat piece of plywood using dabs of hot glue. Foam ceiling tiles or foam insulation would be other good choices. Some large flat screen computer monitors also come in boxes that would be suitable.

I've tried to centre the rings within an 8th of an inch. This is bristol board after all and not the most precise of building materials but a bit of care goes a long way. Maybe a bit more hot glue on the outer ring. I see a gap in the upper right.

Testing the Fresnel Reflector

The problem with solar is of course that you need a sunny day. Luckily the next day was sunny despite a cloudy prediction. It's January in Guelph (latitude N43), Ontario so the sun is not too intense this time of the year. Nonetheless, I did manage to burn holes in some black construction paper I had.

A larger version that I built some 20 years ago (about 36" diameter) melted the plumbing solder that I used in the construction of a focal-point heat exchanger. These mirrors have great potential and I hope some of you will build your own.

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