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Systems with Loads

Module by: Don Johnson. E-mail the author

Summary: This module discusses actual loads used for measuring the output from a circuit. A load model and examples are provided.

Suppose we want to pass the output signal into a voltage measurement device, such as an oscilloscope or a voltmeter. In system-theory terms, we want to pass our circuit's output to a sink. For most applications, we can represent these measurement devices as a resistor, with the current passing through it driving the measurement device through some type of display.

Figure 1: The simple attenuator circuit shown in this module is attached to an oscilloscope's input. The input-output relation for the above circuit without a load is: v out = R 2 R 1 + R 2 v in v out R 2 R 1 R 2 v in .
Figure 1 (circuit4c.png)

In circuits, a sink is called a load; thus, we describe a system-theoretic sink as a load resistance R L R L . Thus, we have a complete system built from a cascade of three systems: a source, a signal processing system (simple as it is), and a sink.

We must analyze afresh how this revised circuit, shown in Figure 1, works. Rather than defining eight variables and solving for the current in the load resistor, let's take a hint from other analysis (series rules, parallel rules). Resistors R 2 R 2 and R L R L are in a parallel configuration: The voltages across each resistor are the same while the currents are not. Because the voltages are the same, we can find the current through each from their v-i relations: i 2 = v out R 2 i 2 v out R 2 and i L = v out R L i L v out R L . Considering the node where all three resistors join, KCL says that the sum of the three currents must equal zero. Said another way, the current entering the node through R 1 R 1 must equal the sum of the other two currents leaving the node. Therefore, i 1 = i 2 + i L i 1 i 2 i L , which means that i 1 = v out (1 R 2 +1 R 1 ) i 1 v out 1 R 2 1 R 1 .

Let R eq R eq denote the equivalent resistance of the parallel combination of R 2 R 2 and R L R L . Using R 1 R 1 's v-i relation, the voltage across it is v 1 = R 1 v out R eq v 1 R 1 v out R eq . The KVL equation written around the leftmost loop has v in = v 1 + v out v in v 1 v out ; substituting for v 1 v 1 , we find

v in = v out ( R 1 R eq +1) v in v out R 1 R eq 1
(1)
or
v out v in = R eq R 1 + R eq v out v in R eq R 1 R eq
(2)

Thus, we have the input-output relationship for our entire system having the form of voltage divider, but it does not equal the input-output relation of the circuit without the voltage measurement device. We can not measure voltages reliably unless the measurement device has little effect on what we are trying to measure. We should look more carefully to determine if any values for the load resistance would lessen its impact on the circuit. Comparing the input-output relations before and after, what we need is R eq R 2 R eq R 2 . As R eq =1 R 2 +1 R L -1 R eq 1 R 2 1 R L -1 , the approximation would apply if 1 R 2 1 R L 1 R 2 1 R L or R 2 R L R 2 R L . This is the condition we seek:

Voltage measurement:

Voltage measurement devices must have large resistances compared with that of the resistor across which the voltage is to be measured.

Exercise 1

Let's be more precise: How much larger would a load resistance need to be to affect the input-output relation by less than 10%? by less than 1%?

Solution

R eq = R 2 1+ R 2 R L R eq R 2 1 R 2 R L . Thus, a 10% change means that the ratio R 2 R L R 2 R L must be less than 0.1. A 1% change means that R 2 R L <0.01 R 2 R L 0.01 .

Example 1

Figure 2
Figure 2 (circuit15.png)

We want to find the total resistance of the example circuit. To apply the series and parallel combination rules, it is best to first determine the circuit's structure: What is in series with what and what is in parallel with what at both small- and large-scale views. We have R 2 R 2 in parallel with R 3 R 3 ; this combination is in series with R 4 R 4 . This series combination is in parallel with R 1 R 1 . Note that in determining this structure, we started away from the terminals, and worked toward them. In most cases, this approach works well; try it first. The total resistance expression mimics the structure:

R T = R 1 ( R 2 R 3 )+ R 4 R T R 1 R 2 R 3 R 4
(3)
RS R S
(4)
R T = R 1 R 2 R 3 + R 1 R 2 R 4 + R 1 R 3 R 4 R 1 R 2 + R 1 R 3 + R 2 R 3 + R 2 R 4 + R 3 R 4 R T R 1 R 2 R 3 R 1 R 2 R 4 R 1 R 3 R 4 R 1 R 2 R 1 R 3 R 2 R 3 R 2 R 4 R 3 R 4
(5)
Such complicated expressions typify circuit "simplifications". A simple check for accuracy is the units: Each component of the numerator should have the same units (here 3 3 ) as well as in the denominator ( 2 2 ). The entire expression is to have units of resistance; thus, the ratio of the numerator's and denominator's units should be ohms. Checking units does not guarantee accuracy, but can catch many errors.

Another valuable lesson emerges from this example concerning the difference between cascading systems and cascading circuits. In system theory, systems can be cascaded without changing the input-output relation of intermediate systems. In cascading circuits, this ideal is rarely true unless the circuits are so designed. Design is in the hands of the engineer; he or she must recognize what have come to be known as loading effects. In our simple circuit, you might think that making the resistance R L R L large enough would do the trick. Because the resistors R 1 R 1 and R 2 R 2 can have virtually any value, you can never make the resistance of your voltage measurement device big enough. Said another way, a circuit cannot be designed in isolation that will work in cascade with all other circuits. Electrical engineers deal with this situation through the notion of specifications: Under what conditions will the circuit perform as designed? Thus, you will find that oscilloscopes and voltmeters have their internal resistances clearly stated, enabling you to determine whether the voltage you measure closely equals what was present before they were attached to your circuit. Furthermore, since our resistor circuit functions as an attenuator, with the attenuation (a fancy word for gains less than one) depending only on the ratio of the two resistor values R 2 R 1 + R 2 =1+ R 1 R 2 -1 R 2 R 1 R 2 1 R 1 R 2 -1 , we can select any values for the two resistances we want to achieve the desired attenuation. The designer of this circuit must thus specify not only what the attenuation is, but also the resistance values employed so that integrators—people who put systems together from component systems—can combine systems together and have a chance of the combination working.

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