79,283 research outputs found
Finding the Minimum Input Impedance of a Second-Order Unity-Gain Sallen-Key Low-Pass Filter without Calculus
We derive an expression for the input complex impedance of a Sallen-Key second-order low-pass filter of unity gain as a function of the natural frequency , o quality factor Q and the ratio of the resistors of the filter. From this expression, it is shown that the filter behaves like a single capacitor for low frequencies and as a single resistor at high frequencies. Furthermore, the minimum input impedance magnitude is found without using calculus. We discovered that the minimum input impedance magnitude is inversely proportional to Q and can be substantially less than its highfrequency value. Approximations to the minimum input impedance and the frequency at which it occurs are also provided. Additionally, PSpice simulations are presented which verify the theoretical derivations
Finding the Minimum Input Impedance of a Second-Order Twofold-Gain Sallen-Key Low-Pass Filter Without Calculus
We derive an expression for the input complex impedance of a Sallen-Key second-order low-pass filter of twofold gain as a function of the natural frequency ωo and the quality factor Q. From this expression, it is shown that the filter behaves like a Frequency Dependent Negative Resistance (FDNR) element for low frequencies and as a single resistor at high frequencies. Furthermore, the minimum input impedance magnitude is found without using calculus. We discovered that the minimum input impedance magnitude is inversely proportional to Q and can be substantially less than its high-frequency value. Approximations to the minimum input impedance and the frequency at which it occurs are also provided. Additionally, PSpice simulations are presented which verify the theoretical derivations
Effect of Scaffolding on Helping Introductory Physics Students Solve Quantitative Problems Involving Strong Alternative Conceptions
It is well-known that introductory physics students often have alternative
conceptions that are inconsistent with established physical principles and
concepts. Invoking alternative conceptions in quantitative problem-solving
process can derail the entire process. In order to help students solve
quantitative problems involving strong alternative conceptions correctly,
appropriate scaffolding support can be helpful. The goal of this study is to
examine how different scaffolding supports involving analogical problem solving
influence introductory physics students' performance on a target quantitative
problem in a situation where many students' solution process is derailed due to
alternative conceptions. Three different scaffolding supports were designed and
implemented in calculus-based and algebra-based introductory physics courses to
evaluate the level of scaffolding needed to help students learn from an
analogical problem that is similar in the underlying principles but for which
the problem solving process is not derailed by alternative conceptions. We
found that for the quantitative problem involving strong alternative
conceptions, simply guiding students to work through the solution of the
analogical problem first was not enough to help most students discern the
similarity between the two problems. However, if additional scaffolding
supports that directly helped students examine and repair their knowledge
elements involving alternative conceptions were provided, students were more
likely to discern the underlying similarities between the problems and avoid
getting derailed by alternative conceptions when solving the targeted problem.
We also found that some scaffolding supports were more effective in the
calculus-based course than in the algebra-based course. This finding emphasizes
the fact that appropriate scaffolding support must be determined via research
in order to be effective
On the convergence of Regge calculus to general relativity
Motivated by a recent study casting doubt on the correspondence between Regge
calculus and general relativity in the continuum limit, we explore a mechanism
by which the simplicial solutions can converge whilst the residual of the Regge
equations evaluated on the continuum solutions does not. By directly
constructing simplicial solutions for the Kasner cosmology we show that the
oscillatory behaviour of the discrepancy between the Einstein and Regge
solutions reconciles the apparent conflict between the results of Brewin and
those of previous studies. We conclude that solutions of Regge calculus are, in
general, expected to be second order accurate approximations to the
corresponding continuum solutions.Comment: Updated to match published version. Details of numerical calculations
added, several sections rewritten. 9 pages, 4 EPS figure
Extending du Bois-Reymond's Infinitesimal and Infinitary Calculus Theory
The discovery of the infinite integer leads to a partition between finite and
infinite numbers. Construction of an infinitesimal and infinitary number
system, the Gossamer numbers. Du Bois-Reymond's much-greater-than relations and
little-o/big-O defined with the Gossamer number system, and the relations
algebra is explored. A comparison of function algebra is developed. A transfer
principle more general than Non-Standard-Analysis is developed, hence a
two-tier system of calculus is described. Non-reversible arithmetic is proved,
and found to be the key to this calculus and other theory. Finally sequences
are partitioned between finite and infinite intervals.Comment: Resubmission of 6 other submissions. 99 page
Order-of-Magnitude Influence Diagrams
In this paper, we develop a qualitative theory of influence diagrams that can
be used to model and solve sequential decision making tasks when only
qualitative (or imprecise) information is available. Our approach is based on
an order-of-magnitude approximation of both probabilities and utilities and
allows for specifying partially ordered preferences via sets of utility values.
We also propose a dedicated variable elimination algorithm that can be applied
for solving order-of-magnitude influence diagrams
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