# (p.253) E Reflection coefficient in electronic circuits

# (p.253) E Reflection coefficient in electronic circuits

A popular means of measuring the impedance of nanoelectronic samples fast and accurately is to connect the sample as a termination impedance at the end of an electronic waveguide and measure the reflection^{1} coefficient of waves sent to the waveguide. This appendix presents the calculation of the reflection coefficient for a waveguide terminated with impedance *Z*(*ω*).

A waveguide is characterized by an inductance *L* and a capacitance *C* per unit length. We may derive the equation of motion for the voltages and currents in the waveguide by first discretizing it into units of length *h* as in Fig. E.1 The currents *I*(*x* _{i}) = *I* _{i} flowing through inductors and the stray currents *I* _{c}(*x* _{i}) = *I* _{ci} flowing through the capacitors are related with the voltages *V*(*x* _{i}) = *V* _{i} of the nodes of the circuit by

where ${\dot{V}}_{i}(t)$ denotes the time derivative of the voltage. Current conservation into node *i* yields

We can identify the first term as the discretization of the second derivative. Dividing both sides by *h*, taking the limit *h*→ 0 and differentiating once by time yields the wave equation

Here we identify the effective speed of light, $c=1/\sqrt{LC}$. For a given voltage profile, the current can be obtained from eqn (E.1a) in the limit *h* → 0,

Let us assume that the waveguide is terminated at *x* = 0 by an impedance *Z*(*ω*) as in Fig. E.2. There we obtain for the Fourier transformed voltages and currents *V*(0,*ω*) = *Z*(*ω*) *I*(0,*ω*). Combining this
(p.254)
with eqn (E.4) hence gives a boundary condition

where the integral over time is replaced by 1/*iω*.

Now finding the reflection coefficient is rather straightforward. Assume we send a wave with amplitude *V* _{0} and frequency *ω* along the waveguide towards the sample, and get a reflected wave with amplitude *rV* _{0}. The total wave is thus

where the wave number has been chosen so that the wave equation (E.3) is satisfied. The boundary condition (E.5) yields for the Fourier amplitude at frequency *ω*,

Solving this gives the reflection coefficient

where ${Z}_{0}=Lc=\sqrt{L/C}$ is the characteristic impedance of wave-guide the waveguide. Measuring the changes in the amplitude and/or the phase of *r* as a function of some control parameter, the sample impedance can be measured very accurately. For the coaxial cables typically used in microwave experiments, *Z* _{0} is often of the order of 50 Ω.

## Notes:

(^{1})
Or the transmission coefficient, but here we only focus on the reflection coefficient.