Extracting Ringing Parameters from Underdamped Second-Order Systems
Underdamped second-order systems can exhibit transient oscillations when excited with a step function stimulus.
The underdamped oscillation in a complex system, such as a complex multiband filter/impedance matching network, can have a transient response that is intractable to determine analytically.
Some simple numerical techniques can be used with circuit simulation data to extract the two basic parameters that describe underdamped oscillations in second-order systems.
Before being rebuilt following a collapse, the Tacoma Narrows bridge was an underdamped second-order system.
On November 7, 1940, the Tacoma Narrows bridge began undulating under gale winds and eventually collapsed. The oscillation seen across the bridge was so severe that it caused major deformation, leading to structural failure. This was due to periodic forcing under gale winds, which excited a resonance in the bridge and caused large displacements from equilibrium. One can see, just from the fact that a resonance was excited, that this system was underdamped.
Structural engineers have learned from this incident, and now place dampers on structures to prevent these resonances and large oscillations. These underdamped second-order systems are still used as resonators, filters, matching networks, and much more in electronics. These circuits can get quite complex and involve multiple stages, and extracting the specific resonance and damping parameters can be complicated or intractable for complex circuit networks.
When you have access to a circuit simulator, you can use transient analysis data from your circuits to extract the relevant parameters describing the behavior of your system. This is a much easier process than deriving the transfer function, poles and zeros, or the differential equation that describes your system.
Analyzing Underdamped Second-Order Systems
What makes a second-order system, and why do we worry about this behavior? When we refer to a system as “second-order,” we’re generally referring to the highest time derivative in its governing differential equation. Second-order systems have a second-order time derivative, as well as at least a zeroth order derivative. The standard theoretical techniques used to analyze ordinary differential equations apply in these systems.
The equation below shows the general ordinary differential equation for a second-order system. The response in the system u(t) depends on two parameters and the forcing function f(t).
General differential equation for a second-order linear system with forcing function f(t).
Note that this second-order system could be derived using separation of variables for a system with spatial and temporal dependence. The classic example in electronics is any electrically long circuit element, such as a transmission line.
The first point to understand when analyzing underdamped second-order systems is to understand the physical meaning of the parameters in the above equation. The two parameters that govern system behavior, specifically an underdamped transient response, are not always time-invariant. Any second-order system could have constant or time-dependent parameters that govern the system’s response under continuous forcing or the transient impulse response. One example of such a second-order system with time-dependent parameters is a parametric amplifier.
What makes a system underdamped is the value of the damping constant ꞵ compared to the value of the natural frequency ω0. If ω0 < ꞵ, then we have an underdamped second-order system and the transient response will be an underdamped oscillation. The response would appear like that shown below. Note that this looks like a form of ringing; any circuit with resonance has the potential to exhibit this type of behavior when the damping constant is low enough.
Extracting System Parameters from Transient Analysis Data
If you have a complex circuit with multiple networks, it can be quite difficult to extract the damping constant and natural frequency analytically. When multiple RLC networks are cascaded, a differential equation such as that shown above can be very difficult to derive using Kirchoff’s Laws and Ohm’s Law. Similarly, determining ω0 and ꞵ from a transfer function may reveal that there are multiple possible values, and finding each value may require solving a third degree or higher polynomial.
However, if you have a SPICE simulator built into your circuit design application, you can get time-domain data showing the transient response. An example is shown below; the orange curve shows the time-domain transient response and the blue dashed curve shows the decay envelope for the system. This curve was taken using a unit step function as the driving stimulus, and the current in the circuit was measured in the simulation.
Transient response for underdamped second-order systems.
The damped oscillation frequency can be seen immediately in the orange curve; it’s just 2π divided by the time between peaks in the transient response. You can then calculate the natural frequency from this value.
The damping constant can be found by plotting the peak value above the final equilibrium current on a logarithmic scale. The damping constant will be the negative slope of this line, as shown in the plot below. Here, the simulated damping constant was 2, and the extracted damping constant is 2.0157.
Extracting the damping constant for underdamped second-order systems.
More Advanced Techniques
What about frequency-domain calculations? This is where you need to look at poles and zeros in the transfer function. Properly interpreting the poles in these systems is very important, as it helps explain stability, instability, transient behavior, and resonance in a single plot. This reveals some implicit assumptions in the above examination:
The system is assumed to be stable without limit cycles. This may not be true in a system with feedback (e.g., a comparator circuit).
The system is assumed to be uncoupled from any other system. This is related to stability and to the fact that there is only a single complex conjugate pair of poles governing the system’s transient response.
The transient behavior in the frequency domain can also be examined using a circuit simulator built into your circuit design tools. When you have complex systems that require transient modeling, you can gather time-domain transient responses or determine the poles from a transfer function for underdamped second-order systems.