Chapter 7: Q6P (page 374)

In Problems 4 to 10, the sketches show several practical examples of electrical signals (voltages or currents). In each case we want to know the harmonic content of the signal, that is, what frequencies it contains and in what proportions. To find this, expand each function in an appropriate Fourier series. Assume in each case that the part of the graph shown is repeated sixty times per second.
6. Triangular wave; the graph consists of two straight lines whose equations you must write! The maximum voltage of occurs at the middle of the cycle.

Short Answer

Expert verified

The voltage function is equal to $V (t) =50- \frac{400}{π^{2}} \sum_{nodd}^{\infty} \frac{cos (120πnt)}{n^{2}}$

Step by step solution

Given Information

The given curve consists of two straight lines whose equations is equal to

$V (t) = {\begin{cases} \in [0, \frac{τ}{2}] \\ \in [\frac{τ}{2},τ] \end{cases}$ where $τ= \frac{1}{60},A=100, \dot{A = 12000}$

Definition of Fourier series

A Fourier series is that a sum that be a periodic function as a sum of sine and cosine waves. It can be written as

$f (x) = \frac{a_{0}}{2} + \sum_{n=1}^{\infty} [a_{n} cos \frac{nπx}{L} {+b}_{n} sin \frac{nπx}{L}]$

The corresponding Fourier coefficients are

$\begin{array}{l} a_{0} = \frac{1}{L} \int_{2L} f (x) dx \\ a_{n} = \frac{1}{L} \int_{2L} f (x) cos \frac{nπx}{L} dx \\ b_{n} = \frac{1}{L} \int_{2L} f (x) sin \frac{nπx}{L} dx \end{array}$

Evaluate the Fourier Coefficients an

The mean value is $\frac{a_{0}}{2} =50$

Solve for the value of a_n

$\begin{array}{l} a_{n} =2 \frac{2}{τ} [\int_{0}^{\frac{τ}{2}} \frac{2πnt}{τ} dt] \\ = \frac{}{τ} [\int_{0}^{\frac{τ}{2}} (tcos \frac{2πnt}{τ}) dt] \\ = \frac{}{τ} {[\frac{τt}{2πn} sin \frac{2πnt}{τ} + {(\frac{τ}{2πn})}^{2} cos \frac{2πnt}{τ}]}_{0}^{\frac{τ}{2}} \end{array}$

Solving, further

$\begin{array}{l} = \frac{^{2}}{{τπ}^{2} n^{2}} ({(-1)}^{1+n} -1) \\ = \frac{}{π^{2} n^{2}} \\ = \frac{2×12000× \frac{1}{60}}{π^{2} n^{2}} = \frac{400}{π^{2} n^{2}} \end{array}$

Where is odd.

Evaluate the Fourier Coefficients bn

The function is even about $t= \frac{τ}{2}$ which means that $b_{n} =0$

Thus the voltage function is equal to $V (t) =50- \frac{400}{π^{2}} \sum_{nodd}^{\infty} \frac{cos (120πnt)}{n^{2}}$

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Short Answer

Step by step solution

Given Information

Definition of Fourier series

Evaluate the Fourier Coefficients an

Evaluate the Fourier Coefficients bn

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