IDEAL TRANSMISSION-LINE FUNDAMENTALS
3.5 TRANSMISSION-LINE REFLECTIONS
3.5.7 Reflections from Reactive Loads
So what can we learn form this? The answer is:symmetry. Whenever a topol- ogy is considered, the primary area of concern is symmetry. Make certain that the topology looks symmetrical from the point of view of any driving agent. This is usually accomplished by ensuring that the lengths, impedances, and loading are identical for each leg of the topology. The secondary concern is to minimize the impedance discontinuities at the topology junctions, although this may be impossible in some designs.
00 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 1
2 3
Time, ns
Voltage at A
vs 75 Ω
50 Ω A
(a)
tr= 250ps tr= 10ps
tr= 500ps tr= 750ps
(b) 0-2 V
τd= 250 ps
Figure 3-38 (a) Underdriven transmission line; (b) example of how increased rise and fall times mask the reflections.
connectors tend to be inductive. This makes it necessary to understand how these reactive elements affect the reflections in a transmission-line system. In this section we briefly introduce the effect that capacitors and inductors have on reflections.
Reflection from a Capacitive Load When a transmission line is terminated in a reactive element such as a capacitor, the shape of the waveforms at the driver and the load will be dependent on the value of the capacitor, the characteristic impedance of the transmission line, and any resistive terminations that may be present. Essentially, a capacitor is a time-dependent load that will look initially like a short circuit when the signal reaches the capacitor and will then look like an open circuit after the capacitor is fully charged. Let’s consider the reflection coefficient at time t =τd (the delay of the transmission line). At time t=τd, which is the time when the signal has propagated down the line and has reached the capacitive load, the capacitor will not be charged and will look like a short circuit. As described earlier in the chapter, a short circuit will have a reflection coefficient of −1. This means that the initial wave of magnitude vi will be reflected off the capacitor with a magnitude of−vi, yielding an initial voltage of 0 V. The capacitor will then begin to charge at a rate dependent onτ, which is the time constant of an RC circuit, whereC is the termination capacitor value and Ris the characteristic impedance of the transmission line. Once the capacitor is fully charged, the reflection coefficient will be 1 since the capacitor will resemble an open circuit. Equation (3-108), which is the step response of a simple network with a time constant τ, approximates the voltage at the end of a transmission
line terminated with a capacitor beginning at timet =τd, vcapacitor=vss
1−e−(t−τd)/τ]
t > τd (3-108) whereτ =CZ0is the time constant, τd the time delay of the transmission line as given by (3-107), and vss the steady-state voltage determined by the voltage sourcevs and the voltage divider between the source resistanceRs and the ter- mination resistanceRt. Note that (3-108) is an approximation because it assumes a step function with an infinite edge rate (i.e ., the rise time is infinitely fast).
Figure 3-39 shows the response of a line terminated with a capacitive load. The waveform shape at node B follows equation (3-108). Notice that the waveform at the source (node A) dips toward zero att =500 ps (which is 2τd)because the capacitor initially looks like a short circuit, so the reflection coefficient is −1.
Note that the shape of this wave at node A is also dictated by (3-108) when the exponent term ise−[(t−2τd)/τ], which simply shifts the time. The voltage reflected back toward the source is initially vi, where vi is the initial voltage launched onto the transmission line. After the capacitor is fully charged, it will look like an open and have a reflection coefficient of+1. Consequently, the reflected wave- form at the receiver will double. As seen in Figure 3-39b, the waveform at the receiver (B) reaches a steady-state value of 2 V after about three time constants [3τ =3(50)(2 pF=300 ps] after arriving at the receiver, just as circuit theory predicts.
If the line is terminated with a parallel resistor and capacitor, as depicted in Figure 3-40, the voltage at the capacitor will be dependent on the time constant
00 0.25 0.5 0.75 1
0.5 1 1.5 2 2.5
Time, ns
Volts A
B
(b) (a) vs
Rs = 50 Ω
Z0 = 50 Ω B CL = 2pF A
0-2 V
td= 250 ps
Figure 3-39 (a) Transmission line terminated with a capacitive load; (b) step response showing reflections from the capacitor.
vs
Rs Rt
CL Z0
td
Figure 3-40 Transmission line terminated in a parallelRC network.
betweenCLand the parallel combination ofRt andZ0: τ1= CLZ0Rt
Rt+Z0 (3-109)
Reflection from an Inductive Load In the real world, when a transmission line is terminated, there is usually a series inductance caused by the physical connec- tion between the transmission line and the resistor. Some common examples of this inductive connection are bond wires, lead frames, and vias. When a series inductor appears in the electrical pathway of a transmission-line termination, as depicted in Figure 3-41a, it will also act as a time-dependent load. Initially, at
00.2 0.3 0.4 0.5 0.6
0.5 1 1.5 2
Time, ns (b) B
A
Volts
vs
Rs = 50 Ω
Rt = 50 Ω Z0 = 50 Ω
L= 1 nH
(a) 0-2 V
A
τd= 250 ps
Figure 3-41 (a) Transmission line terminated with a seriesLR load; (b) step response showing reflections from the inductor.
timet =τd, the inductor will resemble on open circuit. When a voltage step is applied initially, no currentflows across the inductor. This produces a reflection coefficient of 1, causing an inductive spike seen as a reflection at node A in Figure 3-41b. The value of the inductor will determine how long the reflection coefficient will remain 1. If the inductor is large enough, the signal will double in magnitude. Eventually, the inductor will discharge its energy at a rate dependent on the time constantτ of anLRcircuit. For the circuit depicted in Figure 3-41a, the wave shape of the rising edge at node B is calculated:
vinductor =vss
1−exp
−(t−τd)(Z0+Rt) L
t > τd (3-110a) Note that the wave shape calculated by (3-110a) will also be valid for the falling edge of the inductive spike, shown at node B in Figure 3-41b, ifτd is adjusted to shift the waveform to the correct position in time (2τd), the waveform is inverted, and dc is shifted to the appropriate level:
vinductor=vss
1+exp
−(t−2τd)(Z0+Rt) L
(3-110b)
Filtering Effects of Reactive Components Figures 3-39b and 3-41b show how the series inductor and the shunt capacitor affect the signal integrity. The series inductance will cause an inductive spike, which is seen as a positive reflection, the capacitance will cause a capacitive dip, which is seen as a negative reflection, and both will smooth the rising and falling edges seen at the receiver (node B).
To understand why the edges are smoothed, we must explore how an inductor or a capacitor will filter the harmonics of a digital waveform. In Chapter 8 it will be shown that high-frequency harmonics are associated with the rising and falling edges of a digital waveform. Consequently, if the higher-frequency har- monics arefiltered out by the capacitive or inductive loads, the rising and falling times will be increased. Equation (3-111) shows that the impedance of the shunt capacitor will decrease with frequency, which means that the higher-order har- monics of the digital waveform will be shunted to ground, increasing the rise and fall times:
Zcap= 1
j ωC (3-111)
whereω=2πf.
Similarly, (3-112) shows that the series impedance of an inductor will increase with frequency, which will also tend tofilter out the higher harmonics because they will experience larger impedances than the lower-frequency harmonics.
Zind=j ωL (3-112)
Consequently, for a digital pulse, reactive components such as inductors and capacitors will low-pass-filter the waveform, resulting in increased rise and fall times. The exception to this statement is when specificfilters are constructed using reactive components to equalize a channel, which is described is Chapter 12.