Download Boundary conditions: reflection, input impedance and standing wave

Boundary conditions: reflection, input impedance and standing wave ratio
˜ we can write the z-dependent impedance
• From the z-dependent solutions for Ṽ (z) and I(z)
(z)
. At z=0 this gives us the load resistance, while at z=-l it gives us the effective
Z(z) = ṼI(z)
˜
input impedance as seen from the generator circuit.
• Setting Z(0) = ZL , the load impedance, we get the voltage reflection coefficient Γ, On the
other hand, Z(−l) gives us the effective input impedance Zin as seen from the generator
end.
ZL − Z0
ZL + Z0
!
ZL + jZ0 tan βl
Z0 + jZL tan βl
Γ=
Zin = Z0
(1)
• The reflection has a magnitude of unity (total reflection) if the load is substantially
different from the channel, such as an open, short or reactive circuit.
• The input impedance is a useful concept to play with if you want to design networks with
desirable properties. For instance, one could adjust the length of the transmission line so
that a generator waveform suffers one reflection at its front end, and another at the back
end (ie, at the load), and the two are exactly separated by half a wavelength so that the
crests and troughs tend to cancel each other out. This gives us 2l = λ/2, i.e., a ‘quarter
wave transformer’. Furthermore, if the strengths of the reflected waves at the front and
back ends are equal (which depend only on the impedance ratio at each end, so that we
need Z0 /Zg = ZL /Z0 ) then the cancellation is complete and you have an antireflection
coating.
• One could get the same results mathematically by requiring that Zin = Zg and working
backwards from there, using the expression for the input impedance (see above).
• One can similarly design a ‘half-wave transformer’ which in essence eliminates the transmission line’s characteristic impedance altogether (Zin = ZL ). Finally, one can add an
extra stub in parallel, so that you play with the length of the transmission line to eliminate
its real part, and with the length of the stub (which ends with a short or open circuit to
make it purely reactive) to eliminate its imaginary part. This is one way of ‘impedance
matching’ a network.
• Average power per cycle is given by Pav = 12 Re(Ṽ · I˜∗ ). The fractional power transmitted
to the load is 1 − |Γ|2 . The rest of the incident power is not lost (we’re dealing with
lossless lines!) but redirected back to the source.
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