- Describing Signal-Integrity Solutions in Terms of Impedance
- What Is Impedance?
- Real vs. Ideal Circuit Elements
- Impedance of an Ideal Resistor in the Time Domain
- Impedance of an Ideal Capacitor in the Time Domain
- Impedance of an Ideal Inductor in the Time Domain
- Impedance in the Frequency Domain
- Equivalent Electrical Circuit Models
- Circuit Theory and SPICE
- Introduction to Modeling
- The Bottom Line

## 3.8 Equivalent Electrical Circuit Models

The impedance behavior of real interconnects can be closely approximated by combinations of these ideal elements. A combination of ideal circuit elements is called an equivalent electrical circuit model, or typically, just a model. The drawing of the circuit model is often referred to as a schematic.

An equivalent circuit model has two features: it identifies how the circuit elements are connected together (called the circuit topology) and it identifies the value of each circuit element (referred to as the parameter values or parasitic values).

Chip designers, who like to think they produce drivers with perfect, pristine waveforms, view all interconnects as parasitics in that they can only screw up their wonderful waveforms. To the chip designer, the process of determining the parameter values of the interconnects is really parasitic extraction, and the term has stuck in general use.

It is important to keep in mind that whenever we draw circuit elements, they are always ideal circuit elements. We will have to use combinations of ideal elements to approximate the actual performance of real interconnects.

There will always be a limit to how well we can predict the actual impedance behavior of real interconnects, using an ideal equivalent circuit model. This limit can often be found only by measuring the actual impedance of an interconnect and comparing it to the predictions based on the simulations of circuits containing these ideal circuit elements.

There are always two important questions to ask of every model: how good is it and what is its bandwidth? Remember, its bandwidth is the highest sine-wave frequency at which we get good agreement between the measured impedance and the predicted impedance. As a general rule, the closer we would like the predictions of a circuit model to be to the actual measured performance, the more complex the model may have to be.

It is good practice to always start the process of modeling with the simplest model possible and grow in complexity from there.

Take, for example, a real decoupling capacitor and its impedance as measured from one of the capacitor pads, through a via and a plane below it, coming back up to the start of the capacitor. This is the example shown previously in Figure 3-3. We might expect that this real device could be modeled as a simple ideal capacitor. But, at how high a frequency will the real capacitor still behave like an ideal capacitor? The measured impedance of this real device, from 10 MHz to 5 GHz, is shown in Figure 3-8, with the impedance predicted for an ideal capacitor superimposed.

**Figure 3-8. Comparison of the measured impedance of a real decoupling capacitor and the predicted impedance of a simple first-order model
using a single C element and a second-order model using an RLC circuit model. Measured with a GigaTest Labs Probe Station.**

It is clear that this simple model works really well at low frequency. This simple model of an ideal capacitor with a value of 0.67 nF is a very good model. It's just that it gives good agreement only up to about 70 MHz. Its bandwidth is 70 MHz.

If we expend a little more effort, we can create a more accurate circuit model with a higher bandwidth. A more accurate model for a real decoupling capacitor is an ideal capacitor, inductor, and resistor in series. Choosing the best parameter values, we see in Figure 3-8 that the agreement between the predicted impedance of this model and the measured impedance of the real device is excellent, all the way up to the bandwidth of the measurement, 5 GHz in this case.

We often refer to the simplest model we create as a first-order model, as it is the first starting place. As we increase the complexity, and hopefully, better agreement with the real device, we refer to each successive model as the second-order model, third-order model, and so on.

Using the second-order model for a real capacitor would let us accurately predict every important electrical feature of this real capacitor as it would behave in a system with application bandwidths at least up to 5 GHz.

It is remarkable that the relatively complex behavior of real components can be very accurately approximated, to very high bandwidths, by combinations of ideal circuit elements.