E11 Lab #2
Background Information
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## The Diode

The diode is a circuit element that (ideally) will pass current in only one direction.  It is analogous to a one way valve that will only allow water (or air) to flow in one direction.  The circuit diagram for a diode is shown below, along with an image of two real diodes, and an equation that gives the theoretical relationship between voltage and current for a diode.

 Circuit diagram Real Diodes Current-Voltage Relationship

Note that the diode has two terminals, labeled anode and cathode.  On a real diode the cathode is marked by a stripe on one end (on the black epoxy diode, it is a silver stripe at the right; on the glass diode it is a black stripe, also on the right).   The current and voltage are related by the equation shown.  In this equation Is is a quanitity called the saturation current (and is typically less than 1 mA), q is the charge on an electron, n is a number between 1 and 2 called the emission coefficient, k is Boltzmann's constant, and T is the absolute temperature.  At room temperature the quantity kT/q is about 25 mV.  We can then rewrite the equation as

Note that from this equation you can see that if the voltage, v, across the diode is positive, large currents result.  But if the voltage is negative, the exponential term drops to zero, and a very small (-Is) current results.  This brings us to the ideal diode which is one that will conduct current from the anode to the cathode with no voltage drop across it (i.e., it behaves as a wire, or short-circuit), but will not conduct current in the opposite direction (it behaves as an open-circuit when the cathode is at a higher potential than the anode).  In practice we usually use a model of the diode that doesn't conduct until the voltage across the diode reaches some threshold (usually in the range of about 0.5-0.7 volts for silicon diodes).  This model diode can be realized by an ideal diode in series with a voltage source.

The current-voltage characteristic for the three diode models is shown below.

In this lab we will only use the latter two models -- we will not use the exponential model.

## Amplitude Modulation

Modulation is the process of altering a signal so it can be transmitted more easily.  A common form of modulation is amplitude modulation in which the amplitude of a carrier signal is altered to transfer information about a signal.  Consider the image below.

The signal of interest is shown in yellow.  It is a sine wave at about 1 kHz.  Typically this would be a more complicated wave such as a audio signal, but we will deal with sine waves for simplicity.  If we want to transmit this audio signal over radio, we need to convert it to a radio frequency signal.  To do this we perform a simple multiplication:

where is the frequency of our signal and is the frequency of the carrier signal (typically in the radio frequency range 500-2000 kHz for amplitude modulation).  In the example above the signal is at about 1 kHz and the carrier is only about 20 kHz so you can see the relationship between them.  The signals are shown in more detail below.

The modulated signal (magenta) is made up of a carrier (with period Tc) and our signal (with period Ts) which is typically an audio signal.  The audio signal is also called the envelope and is shown in yellow on the image.  If the carrier is sufficiently high frequency (in the radio frequency, or RF, range) we could transmit this signal through the air.

After the modulated signal is transmitted, we need to demodulate it to get the envelope (our audio signal) back.  Before going over the demodulation process, let us consider a simple RC circuit (which will be used in the demodulator).

## An RC Circuit

There are many RC circuits that can be built, but the only one we will need to deal with is the one below.

A capacitor is a device that can store charge.  The amount of charge is proportional to the capacitance and the voltage across it, .  It takes charge being deposited on a capacitor to generate a voltage across.  Since it takes time to deposit, the voltage across the capacitor can not change instantaneously.  The current is the time derivative of the charge, so the relationship be current and voltage for a capacitor is,   If the voltage across the capacitor at t=0 is vc0, then the voltage as a function of time is

That's all you need to know (for this lab).

## Amplitude Demodulation

An AM modulated signal and the rectified signal (only the voltages greater than zero) are shown below.  In the lab you will learn to do this.

We want to recover the envelope of the signal (the yellow line on the image above of the modulated signal).

We can do this (approximately) by putting using a resistor and a capacitor to smooth out the variations in the signal.  The graph below shows two peaks from the rectified signal, isolated.  If we pick the RC time constant to be too long we get a curve like the blue one and we can miss many of the peaks, if we pick it to short we get one like the red line and we don't sufficiently smooth out the peaks, if we pick it "just right" we get the green line which just joins the peaks.  In the lab you will build a circuit that will charge up the capacitor at each peak and let it discharge through a resistor.

The problem is obviously one of choosing the right capacitor.  How do we do this?

Choosing the time constant for a demodulator

Choosing the appropriate time constant is tricky because there is no single "best" value.  Near the peaks and valleys of the envelope we'd like a long time constant because the voltage changes very little between successive peaks of the carrier.  But a circuit with a long time constant would miss many of the peaks when the envelope is charging more rapidly.

We will pick the time constant just short enough to ensure that we don't miss any of the peaks of the carrier when the signal is changing most rapidly.  In other words we want the drop of the demodulated signal between two peaks to be exactly equal to the maximum drop of the modulated signal.

During one period of the carrier, Tc, the maximum change in the heights of successive peaks of the modulated signal is approximately equal to the maximum derivative of the signal multiplied by the time between peaks.  In other words

The maximum decrease occurs at the point labeled "t=0" below.

Consider a modulated signal

Assume that t=0 is when the the modulation signal is at A volts and decreasing, as shown above, the demodulated signal is decreasing as.

So the maximum decrease in the demodulated signal is simply the difference between the value at t=0 and the value at t=Tc.

For the RC decay to just hit successive peaks we set the two maximum decreases to be equal to each other.  To wit,

if we assume then we can perform a Taylor series expansion (keeping only the first term).

Any value of the time constant near this value will suffice for our purposes.  It is not necessary to know "A" or "B" or precisely.  You should use this equation when choosing a time constant for the circuit you will build in lab.