# Re: Bogus proof?

```Hi Jeff,

> I was doing some reading up on transmission line theory and I don't
> understand what the 1/4 wavelength principle of the secondary has to do
> with resonance. Consider this proof:
>
> Velocity of a wave travelling down a transmission line is:
>
> v = l/sqrt(LC)         l = length of transmission line
>                        C = capacitance of the length "l" transmission line
>                        L = inductance " " " "
>
> And we know the resonant frequency of a secondary coil is:
>
> f = 1/(2*pi*sqrt(LC))        eq. 2
>
> also, since
>
> v = f*lamda    lambda = wavelength
>                f = frequency
>
> then:
>
> l/sqrt(LC) = f*lamda         eq. 3
>
> substituting resonant eq. 2 into eq. 3 for "f":
>
> l/sqrt(LC) = lambda/(2*pi*sqrt(LC))
>
> cancelling terms and solving for "l" the length of the transmission line:
>
> l = lambda/(2*pi)
>
> Thus at resonance, the actual physical length of the wire should be 1/2pi
> of the wavelength, and not 1/4. So where is this proof bogus? What exactly
> does the 1/4 wavelength frequency have to do with resonance? I thought
> resonance is only a function of the L and C of the coil. I hope Fr. McGahee
> will include this in the Guide.

It can be viewed from both points of view. Re transmission lines, the
key thing to remember is how they are terminated at each end. At Fr,
the coil is effectively 1/4 wavelength long. One end is open circuit
(top) and the other is short circuited (bottom). Waves reflected off
the o/c end are reflected back inphase with waves travelling towards
that end while waves reflected off the s/c end undergo a 180 degree
phase reversal and are reflected off the s/c end out of phase with
with waves travelling towards that end. The net effect is:
90 degrees (wave travels to top end) + 90 degrees (wave travels from
top to bottom) + 180 degrees (out of phase reflection off the bottom)
giving 360 degrees in total. Alternate wavefronts travelling towards
the top are thus +,-,+,- etc. This matches the oscillation of the
primary tank and in particular, the reversal of current direction in
the primary coil every half cycle.

Transmission lines that are only a fraction of a wavelength long
are wondrous things. By choosing the appropriate length for a given
frequency, you can transform a resistive termination into a pure
capacitance and vice versa. Another length at the same fr transforms a
resistive termination into an inductor. A diferent length again
(1/4) can transform an inductive termination into a capacitor. In
choosing one 1/4 wavelength long, we can transform a very low
resistance (s/c) into a very high one (o/c in effect). However,
things are never quite that simple and losses reduce the o/c to
something less than an infinite impedance. A fractional wavelength
line can be used to match one resistance to another (e.g. 50 Ohm
coax to 75 Ohm coax).

Think I got all that right but stand to be corrected as usual.
Malcolm
```