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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. 1