Electrical Properties of Interconnects
4.1 Ideal vs Real Circuit Elements
Be careful when you use the term ideal to describe an interconnect.
An ideal interconnect does not mean it is transparent. It does not mean that an ideal interconnect behaves like a wire element, as though it did not have any resistance or capacitance or inductance.
The term ideal refers to the circuit elements we use to describe the electrical properties. Each R, L, and C element in a schematic is ideal in that their properties are precisely defined in the circuit simulator.
Usually, the simulator defines each ideal circuit element in terms of how the element treats the voltage across it, V, and the current through it, I.
The three common ideal circuit elements are defined by:
V I V
R C L
dV dI
I
dt dt
= = =
The value of the parameter: the resistance, R, the capacitance, C, and inductance, L, of each ideal circuit element, is each perfectly constant over frequency. These terms are the parameters that define each ideal circuit element, also referred to as their figures of merit.
A figure of merit is a powerful concept we will use over and over again in many engineering applications. It is one number that
characterizes a circuit element or a behavior. It is usually a term that is intrinsic to the device and does not vary under external conditions.
If we know the figure of merit of a component, we know an important property of how it will behave.
For example, each parameter associated with an ideal circuit element is a figure of merit. The resistance of a resistor is one number that characterizes the behavior of that resistor. It is a figure of merit. The voltage of a battery is a figure of merit that describes the battery. A mutual inductance between two signal-return path loops is a figure of merit that describes the amount of noise that might be created.
Every ideal electrical circuit is composed of two parts: the various circuit elements contained in the circuit and how they are connected together, and the parameter values of each circuit element. We refer to how the various circuit elements are connected together as the circuit topology.
Our ability to accurately predict the voltage and current waveforms in a real circuit is limited by how accurate a model we can build using ideal circuit elements including their parameter values and the circuit topology.
We can only measure real components. We can only simulate or calculate with ideal circuit elements.
The general process we will follow to translate a real circuit we can measure into an ideal circuit we can calculate is to obey the guideline proposed by Einstein when he said, “Everything should be made as simple as possible, but not simpler.”
Always start with the simplest model and grow in complexity only as needed.
It is truly remarkable that relatively complex real circuits can be accurately approximated with combinations of simple, ideal circuit
4.1 Ideal vs Real Circuit Elements 89
elements. The process of taking a complex system and describing it with combinations of ideal circuit elements is called strategic simplification.
Every experienced engineer keeps these two worlds completely separate. There is the real world of real physical components and interconnects with some geometrical dimensions made from some combination of materials with material properties, and the ideal world composed ONLY of the ideal circuit elements.
Train your engineer’s mind’s eye to see the equivalent circuit composed of ideal circuit elements when you look at the real components of an electronic product. When you look at the wires, see ideal resistors, inductors, and capacitors.
One of the most valuable skills of any experienced engineer is to be able to visualize the abstract world of ideal circuit
elements whenever they look at real components.
These two separate and distinct worlds are illustrated in Figure 4.1.
Figure 4.1 There are two separate and distinct world views, the real world and the ideal world. When you see the real components, think of the equivalent ideal circuit
elements.
We can only build real prototypes and measure real voltages and currents using components from the real world. We can only build
virtual prototypes and perform a calculation or simulation of predicted voltages or currents using ideal circuit elements.
A natural confusion arises when we use the terms resistor, capacitor, and inductor to describe both a real, physical component we
assemble into a solderless breadboard or circuit board, and the ideal circuit elements used in a SPICE simulator. While we use the same names, these are not the same components. This is why using the preface real or ideal is so important to remove the ambiguity.
A real capacitor, for example, is a physical component with conductors in some shape and separation with some dielectric between them. The conductors don’t even have to have a uniform shape, but can be a convoluted, 3D structure.
An ideal capacitor is very precisely defined. The only quality that defines any ideal capacitor element is one figure of merit, its capacitance. For an ideal capacitor, its capacitance is absolutely constant and never changes no matter what the rise time or
frequency components of the voltages imposed on it. This is why the capacitance of an ideal capacitor is such a great figure of merit.
Likewise, a real resistor has some shape and material properties. An ideal resistor has just one parameter, its resistance. A real inductor is some conductor with some shape to it. An ideal inductor has just one parameter associated with it, its inductance. These distinctions between real components and ideal components are illustrated in Figure 4.2.
4.1 Ideal vs Real Circuit Elements 91
Figure 4.2 The distinction between real and ideal components.
The electrical properties of a real capacitor, such as its measured impedance, are approximated to first order, by an ideal capacitor.
This is a perfectly fine ideal model to use in some cases. This is why an ideal capacitor model is so useful.
However, at higher frequency, the measured impedance of a real capacitor does not match the simulated, or predicted impedance, of an ideal capacitor. At this point, to achieve a better approximation, it is necessary to grow the complexity of the ideal model by adding more components. It is still an ideal model, it is just more complex.
To distinguish the two ideal models that differ in complexity but also in their accuracy in approximating the real component, we use the terms first-order model and second-order model or even third-order model.
For example, a first-order model of a real capacitor is a simple ideal C element. A second-order ideal circuit model for a real capacitor is a series RLC circuit. Each ideal element has an R, L, or C value constant with frequency. Yet their combined circuit impedance varies with frequency, and this is what matches the real measured impedance of a real capacitor.
When the simulated impedance of an RLC circuit matches the measured impedance of a real capacitor, the values of the R, L, and C elements become figures of merit to describe the real capacitor.
An example of the measured impedance of a real capacitor and the simulated impedance of an ideal C circuit and an ideal RLC circuit is shown in Figure 4.3.
Figure 4.3 Top: Measured impedance of a real 0603 MLCC ceramic capacitor, shown in the inset, as the red circles. The black line is the simulated impedance of an ideal capacitor. Bottom: The same measured data with the simulated impedance of an ideal
series circuit of ideal R, L, and C elements. The match is so good it is hard to see the simulated line.
The two different models to approximate the measured impedance of a real capacitor, the single ideal C circuit model, and the ideal RLC circuit model, are both ideal models. One is a first-order
approximation, and the other is a second-order approximation of the real component.
EVERY model we use to approximate a real component is an ideal model. Just referring to an ideal model of a component is ambiguous.
4.1 Ideal vs Real Circuit Elements 93
We need to further clarify what circuit topology and parameter values we are using in the ideal model.
It is remarkable that combinations of these simple ideal circuit elements predict impedances, voltages, or currents that match the actual measured values in real circuits incredibly well. This is the value of ideal circuit models.
With accurate models and a circuit simulator such as SPICE, we can predict the voltage and current waveforms on any node of a circuit in the time or frequency domain that we would measure with a scope, for example.
Keep in mind that in a schematic, the real physical interconnects are always modeled as ideal wires. The schematic assumes the
interconnects are transparent. This approximation may or may not predict the actual performance. This approximation may be too simple, in which case, the circuit’s performance will depend on how we engineer the interconnects.
We may choose to approximate the properties of a real interconnect as an ideal wire element, which predicts there is no voltage drop across the interconnect no matter the current through it. An ideal wire circuit element is an ideal model of the interconnect, which also happens to be a transparent model.
We may choose to approximate the properties of a real interconnect using ideal R and L circuit elements, assuming the impact on circuit performance from an ideal capacitance is negligible. This is an ideal interconnect model.
We may choose to approximate the properties of a real interconnect in terms of just an ideal resistor element. This is an ideal
interconnect model.
In order to assign a parameter value to each ideal circuit element, we need to know how the geometry and material properties of the real structures are translated into the parameter values. This process is covered in this chapter.