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Time Domain Reflectometry
Theory
Application Note 1304-2
                            For Use with
                            HP 54750A and
                            HP 83480A Mainframes
                                                  2




Introduction   The most general aproach to evaluating the time domain response
               of any electromagnetic system is to solve Maxwell's equations in the
               time domain. Such a procedure would take into account all the
               effects of the system geometry and electrical properties, including
               transmission line effects. However, this would be rather involved
               for even a simple connector and even more complicated for a
               structure such as a multilayer high speed backplane. For this
               reason, various test and measurement methods have been used to
               assist the electrical engineer in analyzing signal integrity.

               The most common method for evaluating a transmission line and
               its load has traditionally involved applying a sine wave to a system
               and measuring waves resulting from discontinuities on the line.
               From these measurements, the standing wave ratio () is
               calculated and used as a figure of merit for the transmission
               system. When the system includes several discontinuities, however,
               the standing wave ratio (SWR) measurement fails to isolate them.
               In addition, when the broadband quality of a transmission system
               is to be determined, SWR measurements must be made at many
               frequencies. This method soon becomes very time consuming and
               tedious.

               Another common instrument for evaluating a transmission line is
               the network analyzer. In this case, a signal generator produces a
               sinusoid whose frequency is swept to stimulate the device under
               test (DUT). The network analyzer measures the reflected and
               transmitted signals from the DUT. The reflected waveform can be
               displayed in various formats, including SWR and reflection
               coefficient. An equivalent TDR format can be displayed only if the
               network analyzer is equipped with the proper software to perform
               an Inverse Fast Fourier Transform (IFFT). This method works well
               if the user is comfortable working with s-parameters in the
               frequency domain. However, if the user is not familiar with these
               microwave-oriented tools, the learning curve is quite steep.
               Furthermore, most digital designers prefer working in the time
               domain with logic analyzers and high speed oscilloscopes.

               When compared to other measurement techniques, time domain
               reflectometry provides a more intuitive and direct look at the DUT's
               characteristics. Using a step generator and an oscilloscope, a fast
               edge is launched into the transmission line under investigation.
               The incident and reflected voltage waves are monitored by the
               oscilloscope at a particular point on the line.
                                             3




This echo technique (see Figure 1) reveals at a glance the
characteristic impedance of the line, and it shows both the position
and the nature (resistive, inductive, or capacitive) of each
discontinuity along the line. TDR also demonstrates whether losses
in a transmission system are series losses or shunt losses. All of
this information is immediately available from the oscilloscope's
display. TDR also gives more meaningful information concerning
the broadband response of a transmission system than any other
measuring technique.

Since the basic principles of time domain reflectometry are easily
grasped, even those with limited experience in high frequency
measurements can quickly master this technique. This application
note attempts a concise presentation of the fundamentals of TDR
and then relates these fundamentals to the parameters that can be
measured in actual test situations. Before discussing these
principles further we will briefly review transmission line theory.

                      X
                                                            e x (t)




     Ei              ex        Zo                      ZL
                                                                           Ei +Er
                                                                      Ei
                                          Zo  Z L
                                                                                    t
                      Transmission Line             Load


Figure 1. Voltage vs time at a particular point on a mismatched
transmission line driven with a step of height Ei
                                                             4




Propagation on a    The classical transmission line is assumed to consist of a
Transmission Line   continuous structure of R's, L's and C's, as shown in Figure 2. By
                    studying this equivalent circuit, several characteristics of the
                    transmission line can be determined.

                    If the line is infinitely long and R, L, G, and C are defined per unit
                    length, then



                                                       
                                                               R + j L
                                            Zin = Zo         ----------
                                                               G + jC

                    where Zo is the characteristic impedance of the line. A voltage
                    introduced at the generator will require a finite time to travel down
                    the line to a point x. The phase of the voltage moving down the line
                    will lag behind the voltage introduced at the generator by an
                    amount  per unit length. Furthermore, the voltage will be
                    attenuated by an amount  per unit length by the series resistance
                    and shunt conductance of the line. The phase shift and attenuation
                    are defined by the propagation constant , where


                                          =  + j = (R + jL) (G + jC)

                    and  = attenuation in nepers per unit length
                     = phase shift in radians per unit length

                          ZS                      L      R               L   R




                     ES              C        G              C       G                    ZL




                    Figure 2. The classical model for a transmission line.


                    The velocity at which the voltage travels down the line can be
                    defined in terms of :

                                             
                                    Where  = -- Unit Length per Second
                                             

                    The velocity of propagation approaches the speed of light, c, for
                    transmission lines with air dielectric. For the general case, where
                    er is the dielectric constant:
                                                         c
                                                    = ----
                                                        er
                                       5




The propagation constant  can be used to define the voltage and the
current at any distance x down an infinitely long line by the relations

                    Ex = Eine 



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