Phasor Wave Solutions to the Telegrapher Equations. Termination of TLs
We will continue our TL review by considering the steady state response of TLs to sinusoidal excitation. Consider the following TL in the sinusoidal steady state:
We previously derived the wave equations for the voltage and current as
For sinusoidal steady state, we will employ the phasor representation of the voltage and current as
where V (z) and I (z) are spatial phasor functions. Substituting (3) into (1) gives
We define as the phase constant for reasons that will be apparent shortly. (L and C are the usual TL per-unit-length parameters.)
Substituting this into (5) gives
Similarly, from (4) and (2) we can derive
Equations (7) and (8) are the wave equations for V and I in the frequency domain (i.e., the phasor domain). The solutions to these two second-order ordinary differential equations are
We can confirm the correctness of these two solutions by direct substitution into For example, substituting from (9) into (7) gives
which is indeed true. Therefore, V0 +e-jbz in (9) is a valid solution to (7). The constants I0+ and I0 = in (10) can be expressed in terms of V0+ and V0- . In particular, it can be shown that
If we substitute (11) and (12) into (10) we find that
V(z) = V0+ e -jbz + V0- e -jbz
Both of these equations should be committed to memory. They are the general form of phasor voltages and currents on transmission lines. The first terms in (13) and (14) are the phasor representation of waves propagating in the +z direction along the TL. The second terms in both equations represent waves propagating in the –z direction.
• As stated above, the first terms in (13) and (14) are the phasor representation of waves traveling in the +z direction. To see this, convert the first term in (14) to the time domain:
We can clearly see in this last result that we have a function of time with argument t – z/ vp . From our previous discussions with TLs we recognize that this is a wave that is propagating in the +z direction with speed vp.
• Similarly, we can show that V0 + e +jbz (and I0 + e+jbz ) are waves propagating in the -z direction.
Generality of TL Theory
It was mentioned in the last lecture that transmission lines could be used to model the voltage and current waves on any structure supporting only TEM waves. What changes from structure to structure are the values for L, C,R, and G, as shown below in Table 2.1 for three common TEM structures:
Consequently, the values of Z0, vp, b generally all change from one TL to another. The numerical values can also be changed within a type of TL by varying the dimensions and construction materials.
Termination of Transmission Lines
We will now consider the termination of TLs that are excited by sinusoidal steady state sources.
Adding terminations produces reflections so that the total voltage and current anywhere on the TL are sums of forward and reverse propagating waves. From (13) and (14), the voltage and current on the TL will have the form
V(z) = V0+ e -jbz + V0-e+jbz
The “lumped load” ZL that terminates the TL is considered a boundary condition for the voltage and current in
V (z = 0) = I (z =0) ZL
Therefore, we can solve for V0 in terms of V+ by applying this boundary condition as:
Forming the ratio of these quantities gives
Solving for V0 + / V0 + , and defining this ratio as the voltage reflection coefficient at the load (z = 0), we find
Note that, in general, TL is a complex number since ZL is complex.
Example N3.1: For an open-circuit load on the TL shown above, compute the load reflection coefficient and sketch the voltage and current magnitude on the TL. For an open circuit load ZL = ¥ (i.e., an extremely large impedance), so that from (18) TL = +1 . With this value of then from (15) and (16) the solutions for V (z) and I (z) are
These two equations (19) and (20) are not traveling waves. So, where has the traveling wave behavior in V (z) and I (z) gone The interference between the incident and reflected wavesproduces standing waves, such as these. V (z) and I (z) are shown here for the open-circuit load:
Phasor voltage and current magnitudes vary noticeably along TLs provided the TL length is greater than about 0.05 l or so. Remember, though, that we are plotting the magnitude of phasor voltages and currents. The voltage and current oscillate as functions of time with amplitudes equal to V (z) and I (z) respectively, at each point along the TL.