Atoms resonate. Oceans resonate. Your vocal chords resonate. The Tacoma Narrows bridge (Figure 1.2) used to resonate in high winds. Vibrating, periodic phenomena abound in nature, especially within highspeed digital electronics.
Figure 1.2. The Tacoma Narrows bridge in Washington State, known locally as "Galloping Gertie," heaving under the influence of heavy winds.
The technical requirements for resonance are simple. First, there must be two or more reservoirs for the storage of some amount of energy E along with a relatively lossfree mechanism for exchanging the energy between them. Such a system almost always harbors one or more modes of oscillatory behavior whereby the total stored system energy can slosh back and forth among the various storage modes.
Second, there must be a source of external power coupled into (at least) one of the reservoirs. The coupling mechanism must be capable of delivering power P into the system without unduly disturbing the resonant behavior.
Third, the external stimulation must excite the periodic resonant behavior you wish to excite. Otherwise, the system won't start resonating. In technical terms, this means that for linear systems the spectral power density of the source must overlap with the natural resonant frequency (or frequencies) of the system. For nonlinear phenomena, this spectral condition is usually helpful but not always required (see [2] ).
Resonant systems may be characterized by their quality factor, Q . The Q measures the reluctance of a system to give up its naturally stored energy. A system with high Q tends to resonate, or ring, for many cycles. A system with low Q damps quickly.
The value of Q is technically defined as the ratio of energy stored to the average energy dissipated per radian of oscillation. The range of Q values typically encountered in digital electronics ranges from less than unity (no resonance) to more than 10 6 (a highly resonant crystal oscillator).
Let's do a specific example. Consider the simple RLC resonator depicted in Figure 1.3. It is representative of a short, unterminated pcb connection with a heavy capacitive load at the receiver. The resistor R SERIES represents the output impedance of the driving gate. In such a circuit, for a single resistance in series with the resonant elements, the Q equals
Equation 1.11
Figure 1.3. A short, unterminated pcb trace with a heavy capacitive load resembles this simple LC seriesresonant circuit.
We'll assume in this example the Q is greater than one. Drive this circuit with a periodic waveform, and let the sine wave frequency match the natural frequency of oscillation. Assume the driver rise/fall times are just fast enough to perform this function, so the driving signal looks pretty sinusoidal. From these conditions we may derive the output amplitude of the voltage across the capacitive load. It may surprise you to find that the output amplitude can grow quite large.
To begin the analysis, break down the driving signal into two components : a steadystate DC offset and a sinusoid at frequency f . The amplitude of the DC offset is V CC /2 (that's the average value of the output). The amplitude of the fundamental frequency is also V CC /2. When added to the DC offset, this brings the peak excursions just up to V CC and down to zero. We will do the DC and AC analyses separately.
The DC analysis for this circuit is easy. Since input and output are connected with an inductor , the average value of the output must equal the average value of the input. Therefore, the DC component of the output waveform is precisely V CC /2.
The AC analysis for this circuit is not too hard either. It may be derived using an energybalance equation. This equation will balance the energy delivered to the circuit during one radian of oscillation with the energy dissipated within the circuit over the same period.
Once the circuit has reached steadystate operation, the power delivered to the circuit per radian of oscillation is P /2 p f . The amount of energy dissipated per radian of oscillation (from the definition of Q ) is E/Q . These two quantities must balance:
Equation 1.12
where 
P is the power delivered to the circuit, 
E is the total energy stored within the circuit, and 

Q is the quality factor for the resonator, equal to . 
You are driving the circuit at its resonant frequency, and you know that the input impedance of a perfect series LC resonator operated at its resonant frequency is zero. Therefore, the AC voltage at point (B) in the circuit must be zero. That implies that the full AC driving voltage of ± V CC /2 must appear across resistor R SERIES . The power P delivered to the circuit must therefore equal ½( V CC /2) 2 / R SERIES .
We also know that the total energy E sloshes back and forth between the inductor and the capacitor. At the peak of the output voltage, all the energy is retained within the capacitor , and none in the inductor (the current through L at this moment is zero). If the AC component of the output swings from + V OUT to  V OUT , the energy E stored within the capacitor at the peaks must equal .
Substituting all into equation [1.12], we can now write an energybalance equation in terms of the AC input amplitude V CC /2 and the AC output amplitude V OUT .
Equation 1.13
Rearranging terms,
Equation 1.14
Recognizing that ,
Equation 1.15
Combining the C terms on the righthand side,
Equation 1.16
Finally, recognizing for a seriesresonant circuit the definition of ,
Equation 1.17
And taking a square root of both sides,
Equation 1.18
We see that the amplification factor for this circuit, that is, the ratio of the AC output amplitude to the amplitude of the AC fundamental in the driving signal, equals the circuit Q . Therein lies the importance of Q . It defines ( roughly ) the amount of amplification that may occur within a resonant circuit. Note that the complete output waveform is a superposition of the DC response V CC /2 and the amplified AC signal. The preceding analysis applies only when Q >> 1.
Now let's move on to a different problem. Figure 1.4 provides some useful approximations for the maximum degree of overshoot expected when a resonant circuit is driven by a step input .
Figure 1.4. The higher the Q, the greater the overshoot.
As illustrated in Figure 1.4, the higher the Q , the greater the overshoot. For Q of 1/2 or below the circuit is overdamped and no visible overshoot occurs. Values of Q greater than 1/2 create overshoot. These approximations apply to any resonant circuit that has unit response at DC and is possessed of only one main resonance, at a frequency well below the knee frequency of the driving step source, with a Q greater than one.
An excessive value of R SERIES pushes the Q below 1/2, giving the circuit a sluggish , overdamped response. The overdamped response in such a circuit is dominated by the action of the resistor R SERIES charging the capacitor. Given particular values of L and C , the best choice of R SERIES for digital applications is usually whatever value forces Q closest to unity. That gives the best risetime without undue amounts of overshoot. In other applications, such as the feedback filter for a control system, one might strive for a Q somewhat below 1/2.
Occasionally you will encounter an LC tank circuit driven by a current source and loaded by some parallel resistance. For a single resistance in parallel with the resonant elements, the Q equals
Equation 1.19
For any LC resonator that includes series and parallel elements, the inverses of the Q 's add
Equation 1.20
Distributed transmission lines also harbor resonant modes. As in all resonance problems, they are caused by multiple reservoirs of energy coupled together with a lowloss mechanism.
To understand the energy reservoir formations along a transmission line, imagine a line driven with a specific risetime t r . Every point p n along the line is surrounded by a quantity of dielectric medium that serves as a storage repository for electric field energy. It is also surrounded by a magnetically permeable medium that serves as a storage repository for magnetic field energy. In circuit terms, the voltage and current at point p n form two independent circuit variables , each of which can serve as an energy storage reservoir. Therefore, every transmission line has at least two energy storage reservoirs.
Furthermore, if two points p n and p m along the line are separated by a delay greater than t r , then over a scale of time commensurate with one rising edge, they each represent distinct, uncoupled, independent energy storage reservoirs. The longer the line, the more independent reservoirs exist and the more possibilities develop for resonant behavior.
Common strategies for combating resonance in transmission lines are threefold. Combinations of the three techniques are often employed.
POINT TO REMEMBER
Fundamentals
Transmission Line Parameters
Performance Regions
FrequencyDomain Modeling
Pcb (printedcircuit board) Traces
Differential Signaling
Generic BuildingCabling Standards
100Ohm Balanced TwistedPair Cabling
150Ohm STPA Cabling
Coaxial Cabling
FiberOptic Cabling
Clock Distribution
TimeDomain Simulation Tools and Methods
Points to Remember
Appendix A. Building a Signal Integrity Department
Appendix B. Calculation of Loss Slope
Appendix C. TwoPort Analysis
Appendix D. Accuracy of Pi Model
Appendix E. erf( )
Notes