Engineering 11 – Electrical Circuit Analysis

 

Lab #2

9/28/03

 

 

 

Diodes and Demodulation

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Aron Dobos    &   Adem Kader

Abstract

            In this experiment, the electrical characteristics of diodes and capacitors were observed and applied in practical contexts.  Applications of the aforementioned components for transforming alternating current to direct current as well as in signal demodulation areas were explored.  The measurements obtained conclusively verified the correct functioning of the diode, and confirmed its utility in everyday circuitry

 

Introduction

            Diodes are circuit elements that allow current to flow through them in only one direction.  They have two terminals called the anode and cathode, and are represented schematically by the following symbol:

            Current flows only from the anode to the cathode with ideally no voltage drop.  In practice, however, diodes generally create approximately a 1 volt potential drop across the terminals.  Equations describing the electrical characteristics of diodes will be discussed later.  In terms of practical applications, diodes are frequently used to transform a sinusoidal alternating current (AC) signal into direct current (DC).  Consider the following circuit:

            This circuit is called a half-wave rectifier.  If a sinusoidal alternating current is applied to Vin, the diode will only allow currents flowing in the positive direction to pass through.  Therefore, the current measured at Vout will always be positive, but it will be 0 when the input current is negative.

            To smooth out the choppy output current, a capacitor can be used.  A capacitor is a circuit element that can store a certain amount of electrical charge and release it at a later point.  Suppose we wired a capacitor in parallel with the 10kΩ resistor:

            When the diode allows current to flow, the capacitor accumulates a certain amount of charge.  When the current changes polarity, the capacitor releases its stored charge over a short period of time, keeping the current at Vout from reaching 0 quickly.  Depending on the choice of capacitor, very smooth DC output currents can be achieved.

            Another useful application of diodes and capacitors can be found in AM radios.  AM stands for amplitude modulation, and it is a technique by which a relatively low frequency electrical signal (for example, a signal representing audible sounds) can be modulated with a high frequency "carrier" signal that can travel long distances.  The trick is to decode the modulated signal at a receiving station.  A demodulator circuit that looks exactly like the one depicted above can accomplish the translation given appropriate choices for the resistor and capacitor values.  Consider the rectified signal below: 

            By carefully choosing a capacitor that will smooth out the signal between peeks, the original input signal can be accurately recreated.  However, if the capacitance chosen is too small, the transformed signal will be fuzzy, and if too big, significant signal detail will be lost as well.

 

Theory

            The electrical characteristics of real-world diodes can be represented by the current-voltage relationship indicated below

i = Is * ( e^( 40 * v / n ) - 1 )

            where v is the voltage across the diode.  Note that when v is positive, the current through the diode is large.  However, when v becomes negative, the exponential term goes to 0, leaving i = Ia - 1 .  For analysis purposes, we will only use the diode model represented in the second two graphs included below.

            Capacitors can store an amount of charge that is directly proportional to the capacitance and the voltage across the terminals: Q = C*V.    It takes time for a capacitor to accumulate charge and to release it, so the current is the derivative with respect to time of the charge.  It follows then that

ic = C * (dv / dt)

            With regards to amplitude modulation and demodulation, suppose the input signal is of the form

Vi(t) = A + B * sin( ws * t )

            Since this signal Vi must be modulated to a high frequency, we can multiply it with some high frequency carrier signal

Vc(t) = sin( wc * t )

            to get the final modulated signal

Vm(t) = (A + B * sin( ws * t )) * ( sin( wc * t ) ) .

            To calculate the appropriate capacitance required for the demodulator, we can use the following equation

R * C = A / ( ws * B )

where A is the value of the input signal voltage when the derivative of the signal is greatest, B is the difference between A and the peak of the input voltage, and ws is the frequency of the input signal.  This equation is arrived at using

which is the equality for the two maximum decreases in the wave.  Using the above equality and assuming Tc<<τ, a Taylor series expansion (keeping the first term yields

R * C = A / ( ws * B )

 

            Using a resistance of 10 kΩ, a capacitor of 0.030 mF was calculated as an acceptable capacitance (the calculation can be seen in the Results section).

 

Procedure

            The attached procedure was followed for this lab. 

Due to some technical problems with the oscilloscope, some of our results are distorted and not clear.

 

Results

Graph 1:

Graph 1 shows the initial regular sine wave and the modulated sine wave.

 

Graph 2:

This graph shows the demodulated signal.  On the graph (as can be seen in the table under the graph) V1(2) and V2(2) show the values of (A + B) and (A – B) respectively.  A and B are the coefficients we will use to find C.

 

A + B = 5.6 V    ß V1(2)

A – B = 1.6 V    ß V2(2)

Thus,

A = 3.6 V

B = 2.0 V

From these values of A and B, C can be calculated using the following equality

 

Plugging in the values,

which yields,

 

Graph 3:

This graph shows how the alternating current is transformed into a one way (not direct) current using a diode.  Since the diode lets current flow in one direction only, the negative current is set to 0 current and the positive current is left as it originally was.

 

Graph 4:

Graph 4 shows the change in the initial sine wave with the addition of a 1kΩ resistor and a 0.1μF capacitor.

 

Graph 5:

This graph shows the change in the initial signal due to the addition of a 10kΩ resistor and a 0.1μF capacitor.

 

Graph 6:

Graph 6 shows us the modification of the signal with a 10 kΩ resistor and a 0.2μF capacitor.

 

Discussion

 

            The diode model does a good job modeling the behavior of a real world diode in our experiment.  It lets no current when the AC is in the negatives and lets the current through when the AC is in the positives.  As shown on Graph 3, the voltage drop between the original signal and the rectified signal when the diode allows current flow is nearly constant, meaning that the constant voltage drop diode model is a good approximation of the real world results.

 

The printout showing Vin and Vout with the 0.1 μF capacitor is attached (see graphs 4 and 5).  When the resistance was decreased, the capacitor discharged much faster because there was a smaller resistor between the capacitor and the ground making it much easier for the capacitor to let the charge free.  When the capacitance was increased, the charge stored in the capacitor increased and this resulted in the capacitor taking more time to let the stored charge free.  Looking at graphs 5 and 6, this difference can be observed.  Graph 6 shows us how a 0.2 μF capacitor (compared to that of graph 5 which is 0.1 μF) took more time, causing a straighter line between the two peaks, since the larger capacitance allowed more charge to be accumulated.

 

The output of the Wavetek generator was much more than the drop in the Exact generator because it had a much larger impedance (600Ω) compared to that of the Exact generator (50Ω).

 

The modulated signal was a high pitch sound because the frequency of the modulated signal was very high, and thus was nearly inaudible.

 

The appropriate value for the capacitor was calculated as shown in the Results section.  The modulated signal and the original input are shown together on graph 1 and the modulated signal and the original signal are shown on graph 2.

 

 

Conclusion

 

Diodes can be used in AC to DC current conversion applications.  As a result, electricity can be sent over long distances from power plants to people’s homes in AC, and then can be rectified into DC currents for use in nearly all electronic devices.  Also, we have shown that a relatively low frequency audio signal can be modulated with a high carrier frequency, making it possible to transmit the signal over long distances.  Furthermore, we have also shown that a modulated (generally high frequency) signal can be demodulated to reproduce the original signal with significant accuracy.  This method can be used to transmit radio signals and get audible sound by demodulation.