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.
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 devices must have large
resistances compared with that of the resistor across which the
voltage is to be measured.
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%?
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
.
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)
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
Ω3
Ω
3
) as well as in the denominator (
Ω2
Ω
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.
