- 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

## 3.10 Introduction to Modeling

As pointed out in Chapter 1, equivalent circuit models for interconnects and passive devices can be created based either on measurements or on calculations. In either case, the starting place is always some assumed topology for the circuit model. How do we pick the right topology? How do we know what is the best circuit schematic with which to start?

The strategy for building models of interconnects or other structures is to follow the principle that Albert Einstein articulated when he said, “Everything should be made as simple as possible, but not simpler.” Always start with the simplest model first, and build in complexity from there.

Building models is a constant balancing act between the accuracy and bandwidth of the model required and the amount of time and effort we are willing to expend in getting the result. In general, the more accuracy required, the more expensive the cost in time, effort, and dollars. This is illustrated in Figure 3-12.

**Figure 3-12. Fundamental trade-off between the accuracy of a model and how much effort is required to achieve it. This is a fundamental
relationship for most issues in general.**

When constructing models for interconnects, it is always important to keep in mind that sometimes an OK answer, *now*, is better than a more accurate answer late. This is why Einstein's advice should be followed: start with the simplest model
first and build in complexity from there.

When the interconnect structure is electrically short, the simplest circuit model to start with is one composed of lumped circuit elements. When it is uniform and electrically long, the best circuit model to start with is an ideal transmission line model. This property of electrical length is described in a later chapter.

The simplest lumped circuit model is just a single R, L, or C circuit element. The next simplest are combinations of two of them, and then three of them, and so on. The key factor that determines when we need to increase the complexity of a model is the bandwidth of the model required. As a general trend, the higher the bandwidth, the more complex the model. However, every high-bandwidth model must still give good agreement at low frequency; otherwise it will not be accurate for transient simulations that can have low-frequency components in the signals.

For discrete passive devices, such as surface-mount technology (*SMT*) terminating resistors, decoupling capacitors, and filter inductors, the low-bandwidth and high-bandwidth ideal circuit model
topologies are illustrated in Figure 3-13. As we saw earlier for the case of decoupling capacitors, the single-element circuit model worked very well at low frequency.
The higher-bandwidth model for a real decoupling capacitor worked even up to 5 GHz for the specific component measured. The
bandwidth of a circuit model for a real component is not easy to estimate except from a measurement.

**Figure 3-13. Simplest starting models for real components or interconnect elements, at low frequency and for higher bandwidth.**

For many interconnects that are electrically short, simple circuit models can also be used. The simplest starting place for a printed-circuit trace over a return plane in the board, which might be used to connect one driver to another, is a single capacitor. Figure 3-14 is an example of the measured impedance of a one-inch interconnect and the simulated impedance of a first-order model consisting of a single C element model. In this case, the agreement is excellent up to about 1 GHz. If the application bandwidth was less than 1 GHz, a simple ideal capacitor could be used to accurately model this one-inch-long interconnect.

**Figure 3-14. Measured impedance of a one-inch-long microstrip trace and the simulated impedance of a first- and second-order model. The
first-order model is a single C element and has a bandwidth of about 1 GHz. The second-order model uses a series LC circuit
and has a bandwidth of about 2 GHz.**

For a higher bandwidth model, a second-order model consisting of an inductor in series with the capacitor can be used. The agreement of this higher-bandwidth model is about 2 GHz.

As we show in a later chapter, the best model for an electrically long, uniform interconnect is an ideal transmission line model. This T element works at low frequency and at high frequency. Figure 3-15 illustrates the excellent agreement between the measured impedance and the simulated impedance of an ideal T element across the entire bandwidth of the measurement.

**Figure 3-15. Measured impedance of a one-inch-long microstrip trace and the simulated impedance of an ideal T element model. The agreement
is excellent up to the full bandwidth of the measurement. Agreement is also excellent at low frequency.**

An ideal resistor-circuit element can model the actual behavior of real resistor devices up to surprisingly high bandwidth.
There are three general technologies for resistor components, such as those used for terminating resistors: axial lead, *SMT*, and integrated passive devices (IPDs). The measured impedance of a representative of each technology is shown in Figure 3-16.

**Figure 3-16. Measured impedance of three different resistor components, axial lead, surface-mount (SMT), and integrated passive device
(IPD). An ideal resistor element has an impedance constant with frequency. This simple model matches each real resistor at
low frequency but has limited bandwidth depending on the resistor technology.**

An ideal resistor will have an impedance that is constant with frequency. As can be seen, the *IPD* resistors match the ideal resistor-element behavior up to the full-measurement bandwidth of 5 GHz. *SMT* resistors are well approximated by an ideal resistor up to about 2 GHz, depending on the mounting geometry and board stack-up,
and axial-lead resistors can be approximated to about 500 MHz by an ideal resistor. In general, the primary effect that arises
at higher frequency is the impact from the inductive properties of the real resistors. A higher-bandwidth model would have
to include inductor elements and maybe also capacitor elements.

Having the circuit-model topology is only half of the solution. The other half is to extract the parameter values, either from a measurement or with a calculation. Starting with the circuit topology, we can use rules of thumb, analytic approximations, and numerical-simulation tools to calculate the parameter values from the geometry and material properties for each of the circuit elements. This is detailed in the next chapters.