Phys 312 - Assignment 8 - Due 26 Mar 98

1.

Fundamental constants from LED data. In class, we measured the threshold voltage to get appreciable light from various LED's (red, yellow, green, blue). Use these data to obtain a rough estimate of h/e, assuming that all the energy eV gained by an electron passing through the LED is transferred to a single photon. Plot (by hand) the data to check this simple assumption.

2.

Circuit parameters. A general circuit is characterized by the three quantities R, C, L.

(a)

What are the dimensions of these quantities in the SI? Optional: what are their dimensions in the gaussian system (they are a lot simpler and more intuitive, but in this course we have agreed to use SI).

(b)

What are the dimensions of the product LC? What is its physical meaning?

(c)

What are the dimensions of the product RC? What is its physical meaning?

(d)

What are the dimensions of R/L? What is its physical meaning?

(e)

Form a dimensionless combination of R, C, L. What is its physical meaning?

3.

Passive low pass filter. As I mentioned in class, one stage of an archaic AM receiver consists of a low pass filter which removes the RF (radio frequency) carrier wave, leaving the AF (audio frequency) signal. The simplest passive filter (i.e., a filter which does not involve active elements such as transistors), consists of a resistor and a capacitor.

(a)

Sketch this filter

(b)

Show that

$$\left\vert {\frac{V_{{\rm out}}}{V_{{\rm in}}}}\right\vert ^{2}={\frac{1}{1+(\omega RC)^{2}}}.$$ (1)

Plot this transfer function and show that it has the right properties to function as a low pass filter.

(c)

Choose appropriate values of R and C for an AM receiver.

(d)

Our simple filter has a very slow "roll-off"; ideally, one would like a "brick wall" transfer function which is 1 up to some frequency $\omega _{0}$ and zero otherwise. Some improvement is obtained by incorporating an inductor into the filter, as discussed in class. Sketch this filter.

(e)

Find the transfer function for this filter. Show that the response at low frequencies is the flattest when $R=\sqrt{2L/C}$. This can be done graphically by plotting the transfer function as a function of a dimensionless frequency for several values of a dimensionless parameter that is proportional to L. Note that for L=0 we are back to the previous case.

(f)

How does the transfer function behave at high frequencies?

4.

Fourier series.

(a)

Derive the Fourier series for the square wave of period $2\pi ,$ defined by the periodic repetition of

$$f(x)=\left\{ \begin{array} {lll} 1 & {\rm if} & 0<x\leq \pi \\ -1 & {\rm if} & -\pi <x<0 \end{array} \right.$$

Note that f(x) is discontinuous at x=0 and at $x=\pi$. In this case, the Fourier series reduces to a series of sines or cosines (which?); write down both the complex exponential form and the sine or cosine form of the series.

(b)

Examine the convergence of the Fourier series to the square wave by plotting the sum of the first few terms. How many Fourier amplitudes are needed to produce an accurate square wave? Look carefully at what happens near x=0.

(c)

What is the bandwidth for this signal?

5.

Power spectrum. Here we combine the results of problems 3 and 4.

(a)

What is the normalized power spectrum of the square wave $V_{{\rm in}}f(\omega t)$?

(b)

If the square wave is passed through the filter of problem 3a, what is the output signal V_out? Make an accurate plot of $V_{{\rm out}}(t)/V_{{\rm in}}$ as a function of t/T for a representative value of $RC\omega .$

(c)

Same question for the filter of problem 3d, when it is optimized for low-pass behavior.

(d)

What is the normalized power spectrum of the filtered signal in the two cases ? Formulas are sufficient.

(e)

What is the bandwidth of the filtered signal in the two cases?