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In mathematics, physics, and engineering, abstract concepts are an indispensable foundation for the study and comprehension of concrete models. As concepts within these fields become increasingly detached from physical entities and more associated with mental events, thinking shifts from analytical to conceptual-abstract. Fundamental topics taken from the abstract algebra (aka: modern algebra) are unquestionably abstract. Historically, fundamental concepts taught from the abstract algebra are detached from physical reality with one exception: Boolean operations. Even so, many abstract algebra texts present Boolean operations from a purely mathematical operator perspective that is detached from physical entities. Some texts on the abstract algebra introduce logic gate circuits, but treat them as perceptual symbols. For majors of pure or applied mathematics, detachments from physical entities is not relevant. For students of Computer and Electrical Engineering (CpE/EE), mental associations of Boolean operations are essential, and one might argue that studying pure Boolean axioms are unnecessary mental abstractions. But by its nature, the CpE/EE field tends to be more mentally abstract than the other engineering disciplines. The depth of the mathematical abstractions that we teach to upper-division CpE/EE majors is certainly up for questioning.

In traditional computer and electrical engineering curricula, juniors must complete a course sequence in micro- electronic devices and circuits (often referred to as microelectronics or electronic circuits). Depending on the academic institution, this sequence is typically two semesters or three trimesters in duration. Classic textbooks may include [

One of the successions of topics covered in this sequence is digital circuits, which includes a meticulous study of the dynamic switching characteristics of a basic metal-oxide semiconductor field-effect transistor (MOSFET) inverter. Prior to this juncture, students have to spend several weeks learning about the comprehensive physical structure and current-voltage characteristics of these devices, followed by how MOSFETs are biased to operate as linear amplifiers. Preceding the microelectronics sequence, CpE/EE majors traditionally complete a course in digital logic design. This is where the student learns about fundamental binary and hexadecimal arithmetic opera- tions, the basic logic gates as mathematical operators, the various transistor-transistor logic (TTL) gate IC fami- lies, their combinational and sequential switching architectures, hardware reduction techniques, and hardware description language (e.g., VHDL, [

When CpE/EE juniors are ready to study digital circuit configurations in microelectronics, they first should learn how an inverter circuit is formed using discrete components. This is initially introduced as resistor-transis- tor logic (RTL), followed by the complementary MOS (CMOS) configuration. It’s here where students’ know- ledge of 1) transistor i-v characteristic curves; 2) TTL logic gates, gets unified. Students learn that a MOSFET must be operated in either the cutoff or triode regions if it’s to function as a logic inverter. Rather than using theoretical “1 s” and “0 s” to represent logic states, students acknowledge that these binary representations are distinctive [_{OL}) or some “full” amount (e.g., V_{OH}), and that these voltages represent MOSFET operation in the cutoff and triode regions, respec- tively. What’s more, the student learns that the switching transition from V_{OL} to V_{OH} (and vice versa) is not instantaneous, and that there are tangible energy consequences for such an ideality. It’s at this critical juncture that the student experiences sensorimotor reenactment [

Simultaneously, students learn that familiar MOSFET parameters (e.g., electron mobility µ_{n} and SiO_{2} capa- citance C_{ox}) factor into the modeling of lumped drain-source dynamic resistance and inverter switching speed. Hence, it’s here where the CpE/EE begins to understand the very real switching characteristics of a discrete transis- tor and particularly, switching time and power dissipation consequences. These distinctives [

If the CpE/EE student chooses to pursue more advanced courses in the areas of digital electronic circuits or digital logic operations, they have a variety of advanced-undergraduate courses to choose from [

Hence, the CpE/EE intent on establishing a rigorous, mathematical description of switched electronic systems might consider courses in Switching Systems and Automata, Switching Theory and Logic Design, or just Switch- ing Theory. Texts examined on these topics tend to be presented from a rigorous, algebraic treatment [

Regardless of the presentation, the texts examined are traditionally presented to advanced graduate-level stu- dents in computer engineering, computer science, pure, or applied mathematics. It’s not characteristic for un- dergraduate CpE/EE’s to learn how to take pure or abstract mathematical concepts and conceptualize these ab- stractions into something tangible (e.g., a physical digital circuit). Consequently, undergraduate CpE/EE’s are not likely to cross-correlate Boolean switching abstractions with the physical properties of a transistor. The au- thor sees the benefit of such a unified exposure at a level that is appropriate for junior CpE/EE’s.

Conceptual-abstract thinking consists of one’s ability to effectively find connections or patterns between abstract ideas. The subsequent task is to then piece ideas together to form a coherent picture [

From a classical cognitive viewpoint [

The MOSFET has many very specific defining features such as enhancement of a current flow channel via the application of an electric field E, a threshold voltage V_{t}, an oxide capacitance C_{ox}, and an aspect ratio W/L. These features are essential for describing the concept of “MOSFET” and distinguishing it as a unique three-terminal device. The feature of a new abstract stimulus [

What about a case where there are no defining features of a device abstraction but characteristic features only? For instance, a truth table might have the characteristic features of a three-input NAND gate. But in the absence of defining features, how can one make a concrete determination that it’s data from a 74LVC1G10 [

An exercise is proposed to promote the association of the physical characteristics of dynamic switching in a CMOS inverter with concepts taken from the abstract algebra. The objective is to present the CpE/EE junior with a basic algebraic abstraction and through a series of fundamental steps, engage the students’ abstract and inductive thinking into realizing the abstraction into the physical switching characteristics of a CMOS inverter. CMOS technology was chosen because inverter switching-speeds and steady-state response times approximate a hypothetical switch.

Consider the algebraic symbol x as being a number of the set S, such that x Î S. The task is to manipulate the symbol x of the set S using a binary operation.

DEFINITION I: A binary operation on a set S is a rule that associates with each ordered pair of elements in S a unique element of S. It’s a mapping which sends elements of the Cartesian product S ´ S → S. Because the re- sult of performing the operation on a pair of elements of S is again an element of S, the operation is called a closed binary operation on S.

DEFINITION II: Given an element x in S, if there exists and element y Î S such that x Ú y = 1 and x Ù y = 0, then y is called a complement of x.

From these definitions perform the following tasks:

a) Arrange the following objects into a representation of the operation f: S →

b) Construct a complete truth table for the operation in a.

c) Use the following Boolean theorems to prove complementation:

Theorem 1: 0 Ù x = 0; 1 Ú x = 1

Theorem 2: x Ù (x Ú y) = x = x Ú (x Ù y)

As a CpE/EE major you’re aware that real-life electronic circuits can perform complementation (e.g., a TTL 7404 NOT gate). Likewise, you’re aware that TTL circuitry doesn’t act alone; the output of one logic gate ulti- mately feeds into the input of another. Consequently, there are parasitic capacitances that realistically affect the speed of the complementation operation. As a result of this capacitance, V_{in} is not complemented instantaneously but involves a propagation delay t_{p} as illustrated in _{pHL}, where the subscript “pHL” represents the propagation time it takes a complementation circuit to switch (map) a 1 (HIGH) to a 0 (LOW).

Consider now the complementation model (

where R_{P}_{(ON)} is the drain-to-source resistance of Q_{P} during its on state, and C represents the lumped-loading parasitic capacitance.

e) Discuss under what condition this seemingly continuous-time solution in d. approximates the Boolean ab- straction S→

where V_{t} is the threshold voltage of transistor Q_{P}. Discuss in detail how hole mobility µ_{p} (cm^{2}/V∙s) and the SiO_{2} capacitance C_{ox} (Farads/meter^{2} or Amp-second/Volt∙meter^{2}) affect the speed of complementation and therefore, how these two parameters limit the instantaneous complementation of the abstraction S →

An exercise has been proposed for the purpose of engaging students’ thinking into realizing an algebraic ab- straction into the physical switching characteristics of a basic CMOS inverter circuit. This exercise is meant to be used in a clinical setting with advanced-junior CpE/EE majors.

The intellectual merit of this exercise is that it promotes conceptual-abstract and deductive thinking in the CpE/EE junior via a non-traditional presentation of complementation, where mental abstractions that are nor- mally presented in a pure and theoretical manner are represented by a set of defining features congruent with a CpE/EE’s assumed knowledge level in microelectronics.

The broader impact is that of advancing the clinician’s understanding of how a small and unique group of un- dergraduates practice abstract and deductive thinking in the process of making familiar mental associations (i.e., structured representations) when presented with a pure mathematical abstraction. This work bridges cognitive theories of object concepts [

The subsequent research ambitions of the author are to establish the validity and reliability of this exercise as an abstract reasoning assessment in a clinical setting. The objective is to dispense this exercise between the pre- and post-test administration of an established abstract and/or intelligence test battery, e.g., [