E11 Lab #4
Prelab
2008

Read this section before coming to lab.  You should look over the whole document, but if you don't completely understand something move on.  You shouldn't spend more than an hour reading it.

First Order Time Domain Response

This laboratory exercise will familiarize you with the time response of first order circuits.  You will be analyzing a wide variety of circuits, so if you can come up with a common framework for the analysis, your work will be much easier.  To this end, we will make much use of equation 7.59 from the 8th edition of Nilsson and Riedel's "Electric Circuits."

## Before you begin:

Measuring time constants with the digital scopes

To measure the time constant of a signal, you can follow this procedure:

1.) Get an image similar to the one below, which shows the input at the top of the screen, and the output at the bottom.

2.) Activate the voltage cursors (press the "Voltage" button in the "Measure" section of the scope front panel.  Make sure the cursors are set to measure the appropriate channel on the oscilloscope (channel 2 in the image above).  Put voltage cursor #1 (bottom cursor=43.75mV) so that it measures the steady-state voltage of the output, and put cursor #2 so that it measures the initial voltage (top cursor=1.000V).  Make sure you record these values, you will need them in your report.

3.) Note the value of ΔV.    In this image ΔV=956.2mV.  The time constant occurs when the voltage difference is equal to e-1 (or 37%) of this value.  In this case the time constant will occur when this difference decreases to 352mV.

4.) Move voltage cursor #2 until the voltage difference is the value calculated in the previous step.  This is shown by the yellow marking labelled ΔV in the image below.

5.) The time constant is equal to the difference in time between t=0 (when the input changes), and when the voltage difference has been reduced to 37% of its original value (or, said another way - it has gone through 63% of its total change).  Use the time cursors (as shown above) to measure the time constant (1.080 seconds in this image).  Make sure you record this value as well.  Note that I expanded the time scale from the previous image to get a more accurate measurement.  The large you make the image on the screen, the more accurate your measurement will be.

The 555 timer/oscillator

In this lab you will be using an integrated circuit that is new to you, the 555 timer (datasheet) which can use first order circuits to make oscillators.   A block diagram (from the datasheet) is shown below,

but you needn't concern yourself with what is inside.  Here is a connection diagram to make the device work as an oscillator (from the datasheet).

For our purposes we can model the device as shown below.  The pins on the integrated circuit (1 through 8) are shown in dark green.  Voltages important for analysis are shown in red.  Note R1, R2, R3 and the switch are internal to the device.  We add RA, RB, C and the speaker (in addition to Vcc and ground).

As far as we are concerned there are only three facts that are necessary to understand how this circuit works.

1. If the voltage Vc is greater than V2, the switch closes.

2. If the voltage Vc is less than V1, the switch opens.

3. If the voltage Vc is between V1 and V2, the switch does not change states.

If you hook up the circuit as shown, the voltage on pin 3 is low (Ground) when the switch is closed, and high (Vcc) when the switch is open.  Pin 3 (attached to the speaker) and pins 2 & 6 (labeled Vc) look like this (from datasheet):

When the voltage Vc (lower trace) is between V1 (1.67 volts) and V2 (3.33 volts) and the switch is open (pin 3 is high) the capacitor charges with a time constant of (Ra+Rb)C.  When the voltage gets to V2 (3.33 volts) the switch closes (pin 3 goes low) and the voltage drops with a time constant of RbC.  When the voltage drops to V1 (1.67 volts) the switch opens (pin 3 goes high) and the capacitor charges ...

It can be shown the the time that the output is high (Vc rising) is given by:

The time the output is low (Vc falling) is given by:

The frequency of oscillation is given by:

In deriving these equations consider the following:

• to find T1, let t=0 be the time at which the output just goes high.

• What is Vc(0)?

• What would be the forced value of Vc (i.e., Vc(t) as t goes to infinity))?

• What is the time constant.

• T1 is defined by Vc(T1)=Vcc*2/3.

• to find T2, let t=0 be the time at which the output just goes low.

• What is Vc(0)?

• What would be the forced value of Vc (i.e., Vc(t) as t goes to infinity))?

• What is the time constant.

• T2 is defined by Vc(T2)=Vcc*1/3.

The function generator

To analyze some of the circuits you may need to consider the internal impedance of the function generator.  The Thevenin equivalent of the Wavetek Model 20 function generator is an ideal voltage source, Vs, in series with a 50 Ω resistor.  For most of the other function generators in the lab, the resistance is just 50 Ω.  In some of the circuits used in this lab, this resistor may affect your measurements. This is depicted in the diagram below.

Relating back to lab 1 (Extra - not neccessary)

Recall in lab 1 you built a level shifter circuit with three resistors.  The original circuit had two constraints (maximum and minimum A/D voltage).  I added a third constraint - the Thevenin equivalent resistance.  Let's examine why I did so.  The circuit below is from the data sheet for a microcontroller (a small computer) called a "PIC."

The diagram is excessively complicated for our purposes.  I will simplify it for you.

The circuit within the dotted line is your level shifter circuit, the square with an "X" through it represents the pin on the microcontroller, the capacitance is internal to the microcontroller.  The microcontroller pin is the input to an A/D convertor (as in the article describing the level shifter in first lab).  We wish to measure VAIN as accurately as possible (that is, we would like the voltage at the point labelled ANx to be as close to VAIN as possible).  The microcontroller data sheet recommends that Rs be no larger than 10 kΩ.

• What effect does "Rs" have on the static (constant input voltage) operation of the circuit?

• What effect does "Rs" have on the dynamic (input voltage is changing - you may assume a step change) operation of the circuit?

• How does Rs have to the level shifter circuit?

On to the lab!

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