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Clay U.
Student at University of Florida, Tutor for two years
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MATLAB
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Question:

Matlab is a relatively simple programming environment with a vast expanse of functions to utilize. In order to utilize all of Matlab's functionality, programming knowledge is required. In some cases where very large matrices are being reiterated every run through a loop, the performance of a program may suffer.

Clay U.

In order to be able to work with very large matrices, it is very important to pre-allocate the size of arrays that are continually changing. If a program is running too slowly, a major speed increase can be found by simply declaring an initial matrix using the $$zeros(m,n)$$ function, where m is the number of rows and n is the number of columns of the matrix.

Calculus
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Question:

An interesting topic that bridges both electrical engineering and calculus is the derivation of Euler's Formula. $(e^{iwx} = cos(w*x) + isin(w*x)$)

Clay U.

The proof for this equation lies in the series expansion for all elements involved. The expansion for $$e^x$$ is $$\sum^{\infty}_{n=0} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$$ The expansion for $$sin(x)$$ is $$\sum^{\infty}_{n=0} \frac{(-1)^n}{(2n+1)!} x^{2n+1}\quad = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots \\$$ and the expansion for $$sin(x)$$ is $$\sum^{\infty}_{n=0} \frac{(-1)^n}{(2n)!} x^{2n}\quad = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$$ It can be seen from these expansions, that if the exponent of $$e^x$$ was changed to $$e^{ix}$$, the resulting series would equal exactly the sum of the expansion of $$cos(x) + i*sin(x)$$ Another interesting way to prove Euler's Formula comes from the derivation of both sides. If it is true that $$e^{iwx} = cos(w*x) + isin(w*x)$$, then the derivatives of both sides should be equal. Doing a quick derivation proves that $(\frac{d}{dx}e^{iwx} = \frac{d}{dx}cos(w*x) + \frac{d}{dx}isin(w*x)$) $(iwe^{iwx} = -wsin(w*x) +iwcos(w*x)$) which can be rewritten as $((iw)*e^{iwx} = (iw)*(cos(w*x) +isin(w*x))$)

Electrical Engineering
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Question:

In electrical engineering, a common problem that arises when dealing with digital and analog signals is signal bounce. It can be a cause for miscounting an event, triggering an unwanted alarm, or causing irreparable damage to sensitive electronic equipment.

Clay U.

Engineers have come up with a variety of ways to deal with bounce, and I will list three. The simplest way to ignore the effects of a signal bounce is to slow down your process, ignoring all values while the bounce occurs. A typical example of this is having a delay in a microcontroller program, waiting long enough until all bounces die out. A slightly more complicated way to solve this problem involves using an SR latch. If a switch that has bad signal bounce is connected between two different NAND gates, and the circuit is set up properly, the negative effects of bounce will be eliminated. This is accomplished by connecting the inputs of each NAND gate to each other's outputs. The final, and most complicated way to solve signal bounce is by using the characteristic impedances of the transmission lines and the load in order balance the line and the load. In order to accomplish this, capacitors, inductors, or transmission lines terminated in open or short circuits can be placed in series or parallel to the load. The values of these elements can be calculated using equations derived from Maxwell's Equations, or through using a Smith Chart. Once all net reflections are eliminated, signals will pass more smoothly, and sensitive electronic equipment will not be jeopardized by incoming power spikes.

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