ZS ωp ,()p 1
j ω?p
2
?+:=Source impedance of the driver: parameter vector p must contain: p 0=1 (type), p 1=series resistance (?), p 2=series inductance (H)
You can add parameters to make the source and load impedances as complicated as you like. Properties of the transmission line, source and load
rad/s
Calculation trick: offset the zeroth frequency
slightly from DC to avoid pide-by-zero singularities
rad/s
Hz
Frequencies used to sample Fourier transform functions
Index to frequency points index Horizontal axis for time plots 3.6.6 Lossless Transmission Line
The overall circuit gain G of a long transmission-line configuration can differ substantially from H, where H represents the one-way attenuation of signals moving from end to end across the transmission line. Even in cases where H is relatively well-behaved, the overall circuit performance can sometimes suffer from terrible resonance.
There are three classic ways to stabilize the propagation function of an LC transmission line, eliminating the undesirable resonance. Each case uses a resistive termination to deliver a circuit gain proportional to the one-way transmission line propagation function H(w) (see [3.14]). In the case of a relatively short LC transmission line (such as normally used on a printed-circuit card for high-speed digital signals) the
propagation function H is practically flat with linear phase, indicating the line itself acts as nothing more than a time-delay element with a minor amount of attenuation.
Presuming that the attenuation (and possibly distortion) of H(w) falls within acceptable bounds, a properly terminated LC-mode transmission line may be operated satisfactorily at any length.
Table 4.2 Example code (page 250)
?T 1010
12??:=s Sampling resolution Desired simulation time
Tlen 1010
9
??:=s
Length of sample vector (must be a power of two)integer
n 01,N 1?..:=index Index to time points s
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3.6.6 Lossless Transmission Line.mcd
[3.98]
11S S C L C L 1
122G Z Z Z H H H H Z Z Z ??=
?
+?+++
? ?
The three classic termination styles appear in Figure 3.24. The overall circuit gain G for each is derived from
[3.17], repeated here for convenience.
Equation 3.98, Overall Circuit Gain
3 pF
50 ohms pload 35031012??
:=Impedance of load (in shunt with end of line)
type 3length, m v0, m/s Z0 = 50 ohms θ0 (not used)pline 202π?109?00501.4457108?.12
:=Transmission line parameters
R0 (not used)ω0 (not used)RDC (not used)type 23 nH 10 ohms psource 1103109??
:=Source impedance of the driver
type 1Particular properties of this transmission line example
A shunt load impedance: parameter vector p must contain: p 0=3 (type), p 1=shunt resistance (?), p 2=shunt capacitance (F)
A transmission line: parameter vector p must contain: p 0=2 (type), p 5=characteristic impedance (a constant real value in ohms), p 6=velocity of propagation (m/s). Argument p 7 represents the length of the line (m).
(c) 2003 Howard Johnson and Signal
Consulting, Inc. Permission granted for personal use only
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3.6.6 Lossless Transmission Line.mcd
The function G, implemented in MathCad notation
4.6 Useful Fourier Transform Pairs
Pulse of width b
As you progress in your use of MathCad you will eventually to make a function like this that takes b as an argument and returns a frequency-domain vector. The definition of parameterized functions that return vectors requires some MathCad trickery. The idea is to first define a general function in the freqeuncy domain using continuous arguments. Then define a new function that evaluates the general function at a vector of specific frequencies, given by vector ω.
In MathCad syntax, the arrow over the function PulB_continuous() indicates that it is to be evaluated at every point in the vector ω and the result returned shall be formatted as a vector.
PulB b ()PulB_continuous ωb ,()
→ ????????:=Pulse of width a ·(N ·?T )
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PulN a ()PulN_continuous ωa ,()
→ ????????:=Clock waveform with M complete cycles
ClkN M ()ClkN_continuous ωM ,()
→ ????????:=The delay operator shifts right by amount τ. This function looks simpler than the others because it doesn't require an "if-then" clause, so I can directly apply the vector-evaluation overbar operator on the continuous-freqeuncy definition of the function. The functions that contain "if-then" clauses are not
directly compatible with the overbar operator, which is why they are implemented in two steps.
Dly τ()e
j ?ω?τ?→ ??:=Linear-ramp LPF with 10-90% rise/fall time r
Linr r ()Linr_continuous ωr ,
()
→ ????????:=Quad r ()Quad_continuous ωr ,()
→ ????????:=(c) 2003 Howard Johnson and Signal
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Gaussian LPF with 10-90% rise/fall time r
Gaus r ()Gaus_continuous ωr ,
()
→
????????:=Table 4.2 Example Code (continued)
Excitation rise/fall time (s)
r 10010
12
??:=Excitation is a combination of the pulse function with a waveshaping filter, in this case a gaussian rising edge.
NOTE: The overbar operator indicates that the multiplication is taken point-by-point for each element of PulN() and Gaus()Create vector representing the overall system gain myG G psource pline ,pload ,():=Output signal defined in the
frequency domain
X0Excitation myG ?()→
??????:=Inverse transformation of
freqeuncy-domain vector X0 to produce time-domain vector x0 (see Figure 4.4)
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3.6.6 Lossless Transmission Line.mcd
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