Quarter-Wavelength Equations

---Introduction---

In microwaves you will soon encounter various structures that focus on a fraction of a wavelength.  In particular, the quarter-wavelength is of much significance, although a number of designs center around 1/8-wavelength, 3/4-wavelength, etc.


---Some variables to define---

C = 300,000,000 m/s (Speed of Light in Vacuum)

L = lambda (I will use "L" but the symbol looks like ,\ ).  This is what we are solving for.

F = Frequency of operation, or center-frequency (in Hz or cyc/s).  From your test spec.

Er = Relative dielectric constant of the material (no units).  From manufacturer datasheet.

V = Velocity modifier (takes the Er into account).


---Equation for obtaining quarter-wavelength based on material and frequency---

First, know the effective dielectric of the material you're using.  It is never less than 1 and can be as high as 10, but is often between 2 and 4 for most substrates.  You can get this from the manufacturer datasheet.  Note that you can use this value directly for stripline, but for microstrip the air will reduce this value (bring closer to 1) and you will need to use a program like AppCAD to obtain the effective value.

Next, compute the velocity modifier as follows:

V = 1 / sqrt(Er) ; This is a unitless number.

Hold onto this number for later.  In air/vacuum, the V = 1, so it will have no effect.  For all other values, the larger the dielectric constant, the shorter the wavelength will be.

Now we are ready to compute the wavelength:

C = L*F ; I always start here because I remember it.

L = C / F ; We want to solve for L (ultimately L/4 but we'll get there).

L = (C / F) * V ; Multiply the velocity modifier here.  The units m/s and /s should divide, cancelling out /s and you are left with m.

L = (C / F) * V * 39370.1 ; But we are in America so we don't use meters.  It is often practical to use mils (1/1,000ths of an inch) so we insert the conversion factor (will show derivation)

L/4 = (C / F) * V * 39370.1 * (1/4) ; Lastly, we want to obtain quarter-wavelength, so we simply divide both sides by 4.

Here is the unit conversion:

(C-m/s / F-/s) * (1,000-mm / 1-m) * (1-in / 25.4-mm) * (1,000-mil / 1-in)


---Notes---

It is important to remember that the quarter wavelength is dependent upon the frequency AS WELL AS the dielectric medium.  It is often easy to forget this.  Additionally, microstrip (single board in air) versus stripline (2+ boards sandwiched) circuits of the same material will behave differently because the effective dielectric is different.  More on this in a future blog post.

---Practical Applications---

I don't know about you, but for me, most of the theory is forgotten the next day without some practical applications, so here they are.

Directional Couplers: A single-section coupler consists of two parallel conductors of certain width and spacing.  The length between the coupled section is a quarter-wavelength.

Power Dividers: For the Wilkinson structure, each section is separated by transmission lines of quarter-wavelength of varying impedance.  They may be cascaded indefinitely.

Impedance Transitions: Used in matching a source of one impedance to a load of a different impedance.  If you have ever built audio amplifiers, you may use a transformer to match the amp output to an 8-ohm load.  Same principle, however at microwave frequencies a quarter-wave section at a certain width is all you need.  Also used in amplifiers to match RF transistors.  Will do a blog post on impedance transitions.