1. Objective: 

Perform laboratory measurements that reinforce the learning of Thevenin's theorem of equivalent circuit and Norton's theorem of equivalent circuit.

Here are related materials: pre-lab:   and app:  
Wiring diagram:

2. Introduction:

Consider the circuit below.

Figure 1. R-2R ladder circuit for 8-bit digital-analog conversion. Note how the currents n various loops flow from sources (bit ON) to sinks (bit OFF). The width of the current arrow represents the current magnitude (the vector length has no significance). The vertical bars with a red and blue end represent the node voltages. The red and blue ends of each bar mark the voltage polarity, and the height of a bar represents the voltage  magnitude.

What is it? Something that appears to be regular and periodic must have some sort of design. Indeed, this circuit a very common circuit, known as an R-2R ladder. It looks like a ladder put on its side. Each ladder step has a 2R resistor and is connected to adjacent steps (left and right) with 1R resistor. It is useful for a modern electronic application: digital-analog conversion, or DAC. If you want to peek ahead a bit more, your next project is actually to build a DAC with this. This is the interesting part of the circuit and how it works: It accepts 8 voltage inputs labeled as V0 - V7, and gives a voltage output, Vout as shown on the upper right corner of the figure. It can be proven - in fact, you will be doing the lab to show it - that the output is:

The expressions in Eq. 2 is known as a binary expression of a number. You might have heard of binary numbers. We have 10 digits and it is convenient for us to express a numerical value with base 10, or decimal expression, like this: For the transistor-transistor logic that our entire computing technology is based on, it is like having only two fingers, and a quantity is expressed in base 2: which is like the circuit above. In fact, what the circuit does is to convert the 8 input voltages if they are binary, either 1 or 0, into an analog voltage representing the value of the 8-bit number as above. The voltage representing bit 1 (ON) can be any value you choose, e. g. 5 V in this lab. For bit 0 (OFF), it is obviously 0 V or the ground. In this lab assignment, you will build, analyze, and test this circuit using Thevenin and Norton equivalent circuit concepts.

3. Description:

How do we analyze this circuit? If we use either node voltage or mesh current method, we will have 8 equations with 8 unknowns (8 node voltages or 8 mesh currents). It is certainly impractical to solve the circuit this way. Imagine what happens if we have 16 bits? However, we can use a method that you learn in ECE 2201, the method of equivalent circuit with Thevenin's or Norton's theorem.

You will conduct measurements that verify Thevenin's equivalent circuit theorem to understand how to solve this R-2R resistor network. The circuit is actually a lot simpler than you might think. The reason is that it is just a repetition of a basic step like a ladder: if you know how to make one, you know how to make them all as shown below.




It can be built one step at a time. Look at the start: it begins as just a simple, equal-segment voltage divider: splitting V0, the bit-0 voltage into two with 2 2R resistors. Then, we connect the bit-1, V1 ladder step to its output (the red node and ground). Then, we connect again with an identical structure ladder with bit-2, V2 voltage, and so on. All ladder steps are structurally identical: a voltage source representing the added bit with two resistors (2R and R) in series. The left side is connected to the existing ladder, and the right side is the output. The exact value of the bit voltage of a ladder is irrelevant to solving the problem. It is just a voltage source.

Consider we have, say a 4-step ladder. What happens when we add the 5th step to it? If the entire 4-step ladder can be represented with a Thevenin's or Norton's equivalent circuit (the illustration above uses Thevenin's EC), then we can solve the problem quite easily with the added 5th step (shown as m+1, m=4)

What happens if the old Thevenin resistance value is also R ?


A very simple result indeed: The new ladder with a new step has exactly the same Thevenin resistance as the old one. In fact, you now can infer that if the first ladder has Thevenin resistance = R, then every ladder after that has exactly the same Thevenin resistance.

And for the voltage, the new Thevenin voltage is just 1/2 of the sum of the previous Thevenin and the new ladder voltage. Let's substitute:



It is equally simple to solve each step with Norton's EQ as shown in the Figure below




 In essence, this lab is designed for you to carry out experimentally the steps we show in the recursive proof: you will show experimentally that the Thevenin resistance is the same for all ladders, and that the Thevenin voltage for each successive ladder you add is 1/2 of the sum of the previous Thevenin voltage and the new ladder voltage.

4. Components and Circuit Layout

You will need:

- 9 resistors of value 2R = 2 kOhm - 7 resistors of value R,= 1 kOhm.
- for testing, you will need 8 LEDs so that you know which bit you turn on and which bit you turn off. To drive the LED, you will need an octal line/LED driver chip, and the one you have is an integrated circuit named 74LS244.
- an 8-bit dip switch so that you can manually flip each bit on or off.

You should use bit-ON voltage 5 V.