PoleZero Analysis and Transient Analysis in High Speed Design
Overshoot and ringing in transient analysis. Image credit.
As part of high speed design, transmission line effects in traces and impedance matching normally receive a lot of attention. This is understandable, as switching speed in a digital IC is the primary factor that determines whether termination needs to be considered in a transmission line. However, any RLC circuit can exhibit a transient response, which appears as overshoot or undershoot and a subsequent damped oscillation during switching. This can even propagate into the power bus in a PCB. This is shown in the above graph, where the output from a switching IC is compared with the voltage on a power bus.
When Transients Become Important
The transient response in a circuit becomes more problematic as the switching speed increases; the voltage induced in an RLC network or in a transmission line due to parasitic capacitance/inductance is larger when the switching speed is faster. Impedance matching in a system with transmission lines is important for suppressing reflections, but the transient response can still appear as an underdamped oscillation in the voltage/current seen at the receiver.
The behavior of transients becomes important in any circuit that contains an equivalent RLC portion or is itself an equivalent RLC network (this is essentially every circuit thanks to parasitics). This includes impedance matching networks, circuits with capacitive and/or inductive loads, and many other circuits. In lowlevel circuits with narrow noise margin, or on clock pulse trains, this can produce unintended switching in a downstream IC. If a circuit is overterminated at the source (i.e., the transient response is overdamped), the current takes a longer amount of time to rise to full scale, effectively increasing the propagation delay.
When designing an impedance matching network or decoupling network, you should try to aim for critical damping, although this is not always possible. As part of transient analysis, polezero analysis is a great tool you can use to quickly understand the transient behavior of your system. This allows you to identify whether a given circuit or model is overdamped/underdamped, and you can use the results to reach critical damping using series termination.
PoleZero Analysis in Transient Analysis
If you are working with a lumped transmission line model or with simpler circuits, it is rather easy to determine the damping constant in your circuit using Kirchoff’s laws and/or the Telegrapher's equations. With more complicated circuits, it can be more difficult to determine the damping constant in your circuit. This is where polezero analysis can aid transient analysis of your circuit. This allows you to identify the damping constant and the oscillation frequency in a circuit when it is driven with digital or analog pulses.
If you are familiar with transfer functions and Laplace transforms, then you are already familiar with the idea of poles and zeroes in a circuit’s response. Pole analysis is based on calculating the damping constant and oscillation frequency in a circuit, effectively showing you maxima in the transfer function. As most circuits are purely involve first order or second order derivatives of the charge in the circuit, the output from a polezero simulation will generally reveal two possible poles in your circuit. Higher order circuits may have many more poles and/or zeroes (3 or more). Calculating these values by hand directly from the transfer function for very complex circuit can be difficult as it may require solving a third degree or higher polynomial, and the problem can become intractable.
Polezero analysis automates this process for you. The example below shows the output from polezero analysis. If we look at the graph, we see there are two poles and one zero. Note that the real parts of these values are negative. The two poles are complex conjugates of each other (as they should be), and the zero lies along the real axis.
Example output from polezero analysis as part of transient analysis
What Does This Mean in Transient Analysis?
The location of the poles tells us two things in transient analysis. First, the real part of the pole is the damping constant in the circuit. In the above plot, the real part of the poles is negative (about 315 rad/sec), meaning the transient response decays over time. The imaginary part is the frequency at which the transient response will oscillate (about 1 kHz). In this case, the transient response will produce an underdamped oscillation. Note that, if the poles were located on the right half of the graph (i.e., the real part of the poles was positive), then this system would be unstable, and the transient response would grow over time.
The zeroes of a transfer function refer to specific frequencies that produce a zero output in the circuit. In the above example, the zero is located at the origin, meaning that a DC driver will not pass a current into the circuit. If the zero were located elsewhere along the imaginary axis, then the value on the imaginary axis corresponds to a particular frequency that will not produce a current in the circuit.
Polezero analysis is just one of many tools you can use for transient analysis in that it tells you two aspects of a given circuit:

Stability: this is important in circuits with feedback and it tells you the limits of the driving frequency you should use in your circuit.

Damping: you can immediately determine whether the response is overdamped, underdamped, or critically damped.
Note that this does not tell you the resonance frequency of the circuit; this is best done by looking at the transfer function for your circuit, or you can calculate it manually from the real and imaginary parts of a pole, just as you would for any oscillator. Note that some circuits with multiple complex conjugate pair poles will have multiple resonances.
If you complete polezero analysis and you find that your circuit exhibits an undesired response (i.e., an underdamped response in an impedance matching network), you can iterate through different component values in your circuit to determine the component values that produce the desired response. This allows you to critically damp the response in your circuit such that you can eliminate overshoot/undershoot.
You can examine multiple transients when optimizing an impedance matching network
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