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Homework Set #1 MAS.863 Sensor Systems for Interactive Environments
Distributed : Thursday February 14
th
, 2008
Due : Thursday February 21
st
, 2008
Homework Set #1
Simple Circuit Analysis
The purpose of this problem set is to familiarize yourself with Kirchos voltage and
current laws, and the operation of simple op-amp circuits. To accomplish this, you will need
to use the following:
Standard Circuit Nomenclature :
Discrete components, named node voltages, and in-
put/output signals are in upper case. For example V
in
, the voltage into a circuit, R
3
is resistor number 3, V
1
is the voltage a node 1.
Properties of named components, such as resistors, are in lower case. For example, a
resistor R
3
would have a voltage of v
3
across it, and a current of i
3
through it. The
standard convention is to always have positive current ow from the positive voltage
terminal to the negative voltage terminal. The current through a circuit may be repre-
sented by an arrow above, or an arrow on the symbol. They are used interchangeably
for clarity.
+ v
1 i
1
=
+ v
1 R
1
i
1
Figure 1: Equivalent circuit element representations.
Voltage +
V
1
Current
I
1
Ground
Figure 2: Voltage and Current circuit source symbols, which supply power to circuits. The
+, , and arrow denote voltage polarity and current direction. The ground symbols is
dened at the point in the circuit where there is 0V .
1 Ohms Law :
The voltage drop across a resistor is proportional to the current owing
through it and is modeled by Ohms Law.
v
1
= i
1
· R
1
(1)
+ v
1 i
1
Figure 3: Resistor notation showing v
1
, the voltage across the resistor, and i
1
, the current
through the resistor.
For example, if 5mA of current is owing through a resistor of 10k, then the voltage
drop across the resistor will be 5mA·10k = 50V olts. It is very important to remember
that the voltage drops in the direction of the current ow, as the electrons move from a
higher potential to a lower potential. To ensure that you do not get your voltages and
currents backwards, you should always label them before performing any calculations.
Kircho s Current Law :
To satisfy the conservation of charge (think ow of electrons),
all currents owing into a node must sum to zero, [1] page 3.
i
1
=
n
m=2
i
m
(2)
+ v
1 R
1
i
1
+
v
2 R
2
i
2
V
A
+
v
3 R
3
i
4
+
v
n R
n
i
n
Figure 4: A series of n resistors connected together at a single node V
A
.
This property is know as Kirchos Current Law. A node is any place in a circuit
where two or more elements touch. Since the number of electrons at any point must
be conserved, the current owing into the node must equal the current owing out of
that node, regardless of the number of circuit elements that meet at that node.
Kircho s Voltage Law :
The sum of all voltages around a loop of circuit elements is zero.
If the voltage is raised by a certain amount due to a circuit element, it must be lowered
by that same amount somewhere else in the circuit to compensate. In many circuit
diagrams, symbols are used to represent certain xed voltages, rather than drawing a
line to complete the loop. When a voltage is referred to in a circuit, it is the voltage
drop from that point to ground, unless the voltage drop across a specic component is
asked for. For example, in the circuit below, the voltage drop across R
1
is v
1
, but the
voltage at the R
1
and R
2
juncture is v
2
. This law applies to loops regardless of other
connected circuits.
2
+ V
s
+
loop
+ v
1 R
1
i
1
+
v
2 R
2
i
2
blackbox
Figure 5: Given a continuous loop of circuit elements, the sum of the voltages must be zero.
In this example V
s
= v
1
+ v
2
.
See [2] section 4.4 or [3] page 35.
Ideal Op-amp Model :
The ideal op-amp can be viewed as a device which
 
draws no current into its input pins v
in+
and v
in
.
 
is able to source an innite amount of current at its output pin (v
out
), i.e. zero
output resistance.
 
has innite gain A
v
.
A slightly more accurate model on an op-amp includes a nite gain factor A
v
. The
op-amp amplies the dierence between its input pins and presents this voltage at the
output v
out
= A
v
· (v
in+
v
in
). This gain factor will vary with frequency in a non-
ideal op-amp. As the frequency of the input signals increase, the gain of the op-amp
decreases. This gain/frequency relationship is known as the gain-bandwidth product
(GBW), since the gain times the frequency is xed. For example, if you have a signal
at 10kHz which is being amplied by an op-amp with a GBW of 2M Hz, the maximum
voltage gain you would be able to achieve would be 2M Hz/10kHz = 200.
v
in+ +
v
in
+V
cc
V
cc
v
out
A
v
Figure 6: Diagram of an ideal Op-amp where v
out
= A
v
· (v
in+
v
in
). The positive and
negative V
cc
s supply the power to the Op-amp.
3 Problem 1
The following two resistor circuit is assembled in a series connection, as one resistor follows
the other in a series. For this circuit, answer the following questions.
V
in
+
v
1 R
1
i
1
V
out
+
v
2 R
2
i
2
Figure 7: Circuit for problem 1. i
1
is the current owing into R
1
and i
2
is the current owing
into R
2
. V
out
is unconnected.
a. What is the relationship between i
1
and i
2
?
b. What is the voltage drop across R
1
in terms of i
1
and R
1
?
c. What is the voltage drop across R
2
in terms of i
2
and R
2
?
d. What is the voltage at V
out
in terms of v
2
?
e. What is the voltage at V
in
in terms of v
2
and v
1
?
f. What is the relationship between V
in
and V
out
in terms of R
1
and R
2
? (Note that this
is the voltage divider equation)
g. What is V
in
/i
1
in terms of R
1
, and R
2
? (Note that this is the formula for summing
resistors in series)
4 Problem 2
The following two resistor circuit is assembled in a parallel connection, as one resistor runs
parallel to the other. For this circuit, answer the following questions.
V I
+
v
1 R
1
i
1
+
v
2 R
2
i
2
Figure 8: Circuit for problem 2.
a. What is the voltage drop across R
1
in terms of i
1
and R
1
?
b. What is the voltage drop across R
2
in terms of i
2
and R
2
?
c. What is the relationship between v
1
, v
2
, and V ?
d. What is the relationship between i
1
, V , and R
1
?
e. What is the relationship between i
2
, V , and R
2
?
f. What is the relationship between I, i
1
, and i
2
?
g. What is
V
I
in terms of R
1
and R
2
? (Note that this is the formula for summing two
resistors in parallel)
5 Problem 3
The following op-amp circuit is an inverting amplier, so called because it inverts and am-
plies an input signal. Assume the ideal op-amp relationships given at the beginning of the
problem set and answer the following questions. +
v
v
1
+
R
1
i
1
V
in
+ v
2 R
2
i
2
V
out
v
+
Figure 9: Circuit for problem 3.
a. What is the relationship between i
1
and i
2
?
b. What is the voltage at (v ) in terms of V
in
, i
1
, and R
1
?
c. What is the voltage at (v ) in terms of V
out
, i
2
, and R
2
?
d. Using solutions to part a, b, and c, What is i
2
in terms of V
out
, V
in
, R
1
, and R
2
?
e. Using the ideal op-amp equation, what is the voltage at V
out
in terms of the op-amp
gain A
v
, (v ), and (v
+
)?
f. What is the voltage at (v
+
)?
g. Using c, e, and f, what is i
2
in terms of V
out
, A
v
, and R
2
?
h. Using d and g, what is V
out
/V
in
in terms of A
v
, R
1
, and R
2
?
i. Assuming A
v
is much greater than one, such that (1 + 1/A
v
)
1, what does your
result from h simplify to? (Note that this is the gain of an inverting amplier under
ideal conditions)
j. If you were using an op-amp with a gain-bandwidth-product of 2M Hz, what would be
the highest frequency at which your answer to i would hold true? (Assume A
v
is high
enough to neglect its eect)
k. Using the equations in a, b, and c, what is V
out
in terms of (v ), V
in
, R
1
, and R
2
?
l. Using the results of i and k, what is the value of (v )? (Note that this gives us our third
ideal op-amp law, that an op-amp with negative feedback (i.e. a connection between
its output and the inverting input terminal) will always work to make (v
+
) = (v ))
6 Problem 4
The following op-amp conguration is called a summing amplier. To analyze this congu-
ration, note that it has negative feedback, and therefore the third op-amp rule of (v
+
) = (v )
will hold. Answer the following questions. +
v
v
1
+
R
1
i
1
V
in1 v
2
+
R
2
i
2
V
in2
+ v
3 R
3
i
3
V
out
v
+
Figure 10: Circuit for problem 4.
a. What is the voltage at (v
+
) and (v )?
b. What is i
1
in terms of v
1
and R
1
?
c. What is i
2
in terms of v
2
and R
2
?
d. What is i
3
in terms of V
out
and R
3
?
e. What is the relationship between i
1
, i
2
, and i
3
?
f. What is V
out
in terms of v
1
, v
2
, R
1
, R
2
, and R
3
? (Note that this performs the mathe-
matical sum of the two inputs with scaling factors inversely proportional to the input
resistor values)
7 Problem 5
The following op-amp circuit is a non-inverting amplier conguration, so called because
it amplies the input signal without inverting it. Again, assume all three ideal op-amp rules,
and answer the following questions. +
v
+
V
in
v +
v
2 R
2
i
2
v
1
+
R
1
i
1
V
out
Figure 11: Circuit for problem 5
a. What is the relationship between (v ), (v
+
) and V
in
?
b. What is the relationship between i
1
and i
2
?
c. Using your answer to Problem One, part f, what is V
out
in terms of (v ), R
1
, and R
2
?
d. Using your result from a and c, what is the gain (V
out
/V
in
) of this circuit? (Note that
this is the gain for an ideal non-inverting amplier)
e. What is the smallest gain value attainable with this conguration?
8 P