When using a model to help in the design process, it is important to use the most appropriate type of model, as doing otherwise can waste computing power and time, and either provide too little detail or far too much.
In a pure black box model the internal workings of a device are not described, and the model simply solves a numerical problem without reference to any underlying physics. This solution usually takes the form of a set of transfer parameters or empirical rules that relate the output of the model to a set of inputs.
For example, a standard current-gain transistor model is a form of black box model where the internal workings of the transistor are ‘hidden’. In such a model, the output current is related to the input current at the transistor base by a linear current gain.
In a grey box model, some or all of the mechanisms describing the behaviour of a device are known, but all are not fully represented in the model. In a grey box model, certain elements within the model can be approximated by rules.
If we continue our transistor model analogy, then a grey box model of a transistor would be more complex, and would model some of the internal transistor operation. For example, the Ebers-Moll transistor model is more ‘grey’ than the basic current-gain model because it begins to describe and break down the operation of the transistor into several elements. However, there are still some approximations in this type of model.
A white box model contains as much detail as the simulation model can provide and no approximations are made using bulk parameters. Such detail in a model is only used in situations where the simulation results must closely match those produced in reality and often consume large amounts of computing power.
A pure white box model cannot exist as it is essentially a copy of reality! For a transistor, although a pure white box model does not exist, ‘whiter’ models than Ebers-Moll have been devised. For example, some transistor models actually model the movement of electrons and holes through the p- and n-type silicon materials that make up the transistor.
The choice between these three types of model will depend upon the level of detail we require in the model. This is ultimately dependent upon how closely we wish the model’s behaviour to match that of the real system.
Categorise the following model descriptions in terms of black, grey and white box. Explain your choice in each case.
1. A model of an operational amplifier relating the output to the two inputs by a linear relationship.
2. A model of the heat transfer from the surface of a PCB to the ambient air where we model the movement of the air over the surface.
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A black box model is used where the response of a system is not broken down into its underlying mechanisms, and is represented by an empirical description or set of transfer parameters that do not describe any internal physics. Compared with grey and white box models, black box models have advantages and disadvantages:
Advantages of black box models:
Disadvantages of black box models.
Good examples of black box models are any form of parametric model, where transfer parameters are used to relate model outputs to model inputs.
A resistor’s final steady state temperature is governed by the equation:
. . . . . Eqn.1
where P is the power applied to the resistor, and A is the resistor’s area and T0 is the ambient temperature.
This is a type of black box model, with three inputs, P, A and T0, and one output, T. We see that a model based on Equation 1 has the following properties.
Other general examples of black box models include:
None of these types of model attempt to model by low-level application of the physics of the problem, they just try to get the right answer by using a high-level process.
Black box models are useful when an answer to a specific problem is required and the flexibility to change aspects of a model and see the effect is not. The required flexibility of a model depends upon its long-term objectives as part of the design process. If the purpose of the model is only to provide quick, approximate answers, based on a pre-determined set of input parameters, then a black box model is appropriate. In such a case, flexibility is not required, as the overall design has already been fixed. For example, the maximum power handling capability of a component could be related to its dimensions, material thermal properties and electrical properties. A rule-based black box model could be set up to determine a component’s power handling based on these parameters. However, such a model is not flexible, and if a different answer is required, or new parameters are added, it usually means re-designing the model from scratch rather than just modifying the existing rule.
The majority of simulation models are grey box models. A grey box model provides a physical representation, but some of the physics is approximated. For example, in a PCB thermal model, convection at the board surfaces is often represented by a convective heat transfer coefficient. Though this represents the amount of power escaping from the board in terms of physical parameters, it hides the underlying process of air flow across the PCB surface, laminar or turbulent. It is a high-level model of a low-level process. This is typical of a grey box model.
What do you think the advantages and disadvantages of a grey box model are when compared to a black box model?
Other typical approximations that are made in a grey box thermal model:
In cases where flexibility is required, we need a more generic model, that can be adapted to model variations in a design. A grey box model provides more flexibility, and enables us to use modelling to optimise a design, rather than just tell us answers based on a fixed design. This includes determining the effects of variations in device geometry, right through to a change in material parameters. Grey box models can be used for design sensitivity analysis, where we want to see how sensitive a design is to a particular design aspect, such as how component temperature is related to the position of components on a PCB. The flexibility of a grey box model also allows us to extract rules that describe the behaviour of a device. In this sense, a grey box model can be used to form a black box model that could be used as a design tool itself. For example, we could use a grey box thermal model of a PCB to obtain a relationship between the area of a resistor and its temperature rise for the same component power dissipation. Figure 1 shows a curve that could have been generated from a grey box model.
The results shown in Figure 1 could then be extracted and used in the form of a table lookup as part of a black box rule-based model.
A white box model is the most detailed type of model, and is as close as possible to a full description of the real device. The physical processes are described at as low a level as possible, with no approximations or bulk parameters used. So the PCB thermal simulation would model the actual laminar flow of air across the PCB surface, instead of using a heat transfer coefficient to describe the heat loss through convection. In this sense, it would no longer be a pure thermal model, as we would also require an air-flow model as well.
Advantages of white box models:
Disadvantages of white box models:
A white box model is only really necessary where we need the flexibility of a grey box model plus a high level of detail in the physical processes. If we need to determine the sensitivity of a design to the smallest of details, then we may need a white box model. In every other sense, a white box model can be used for the same applications as a grey box model, but provides greater realism.
Categorise the following in terms of black, grey and white box models. Explain your answer in each case.
If we wanted to make a ‘whiter’ model than the one in part 2, what aspect of the model might we change?
The question of which model type to use for a specific modelling task can be answered by considering the requirements of a model.
A CPU is mounted to a PCB. To provide the necessary cooling, a heat-sink and fan arrangement is mounted to the CPU. Firstly, list any parameters that might affect the temperature of the CPU and then discuss a set of successive approximations that may need to be made to model such an arrangement using a grey box approach.