Impedance of Linear, Time-Invariant, Lumped-Element Circuits

The following relations are well worth committing to memory, as I will use them many times over during the course of this text. If they are not already familiar to you, see [1] , or any other general electrical engineering reference.

The impedance magnitude of a linear, time-invariant lumped-element inductor , as measured with a sinusoidal input at frequency f , is

Equation 1.1

graphics/01equ01.gif

 

where

X L is the impedance magnitude (ohms),

 

f is the measurement frequency (Hz),

 

L is the inductance (Henries).

The impedance magnitude of a linear, time-invariant, lumped-element capacitor , as measured with a sinusoidal input at frequency f , is

Equation 1.2

graphics/01equ02.gif

 

where

X C is the impedance magnitude (ohms),

 

f is the measurement frequency (Hz),

 

C is the capacitance (Farads).

When dealing with step waveforms (like digital signals), an appropriate frequency of analysis for estimating the effective impedance of a parasitic element, over a time scale corresponding to a rising (or falling) edge, is

Equation 1.3

graphics/01equ03.gif

 

where

f knee is the assumed frequency for sinusoidal analysis (Hz),

 

t 10 “90% is the 10% to 90% rise (or fall) time of the circuit (seconds).

The knee frequency , f knee , is a crude estimate of the highest frequency content within a particular digital signal (Figure 1.1). Signaling channels whose parasitic impedances are not significant at all frequencies up to and including the knee frequency tend to pass digital signals undistorted.

Figure 1.1. The power spectral density of random data signal has nulls at multiples of the data rates and a maximum effective bandwidth related to its rise and fall time.

graphics/01fig01.jpg

At the knee frequency the average spectral power density of a random digital signal with Gaussian rising and falling edges lies 6.8 dB below the straight slope shown in Figure 1.1. Obviously, there is no crisply defined knee at this point. Some authors define the bandwidth of a digital signal as 0.35/ t 10 “90% , which happens to lie at the “3 dB point in Figure 1.1, closer to the meat of the spectrum associated with a rising edge. For some calculations this may produce a more accurate result, however, I still prefer to use [1.3], thinking of it as a conservative over -estimate of bandwidth. If the parasitic elements in my circuit remain insignificant up to [1.3] then I know they can safely be ignored. If, on the other hand, the parasitic elements begin to give difficulty at frequency [1.3] I immediately turn to more accurate techniques (such as Spice, Fourier-Transforms, or Laplace Transforms) to ferret out the exact circuit response.

POINTS TO REMEMBER

  • The knee frequency , graphics/002equ01.gif Hz, is a crude estimate of the highest frequency content within a particular digital signal.
  • The frequency 0.35/ t 10 “90% Hz may better approximate the meat of the spectrum associated with a rising edge.


Fundamentals

Transmission Line Parameters

Performance Regions

Frequency-Domain Modeling

Pcb (printed-circuit board) Traces

Differential Signaling

Generic Building-Cabling Standards

100-Ohm Balanced Twisted-Pair Cabling

150-Ohm STP-A Cabling

Coaxial Cabling

Fiber-Optic Cabling

Clock Distribution

Time-Domain Simulation Tools and Methods

Points to Remember

Appendix A. Building a Signal Integrity Department

Appendix B. Calculation of Loss Slope

Appendix C. Two-Port Analysis

Appendix D. Accuracy of Pi Model

Appendix E. erf( )

Notes



High-Speed Signal Propagation[c] Advanced Black Magic
High-Speed Signal Propagation[c] Advanced Black Magic
ISBN: 013084408X
EAN: N/A
Year: 2005
Pages: 163

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