For each circuit shown in [link], the current
equals
.
What is the voltage across each element and what is the voltage
in each case?
For the last circuit, are there element values that make the voltage
equal zero for all time?
If so, what element values work?
Again, for the last circuit, if zero voltage were possible, what circuit element could substitute for the capacitor-inductor series combination that would yield the same voltage?
Solving Simple Circuits
Write the set of equations that govern
Circuit A's
behavior.
Solve these equations for
:
In other words, express this current in terms of
element and source values by eliminating non-source
voltages and currents.
For Circuit B, find the value for
that results in a current of 5 A passing through
it.
What is the power dissipated by the load resistor
in this case?
Circuit A
Circuit B
Equivalent Resistance
For
each of the following
circuits, find the equivalent resistance using
series and parallel combination rules.
circuit a
circuit b
circuit c
circuit d
Calculate the conductance seen at the terminals for
circuit (c) in terms of each element's conductance.
Compare this equivalent conductance formula with the
equivalent resistance formula you found for circuit (b).
How is the circuit (c) derived from circuit (b)?
Superposition Principle
One of the most important consequences of circuit laws
is the Superposition Principle: The current
or voltage defined for any element equals the sum of the
currents or voltages produced in the element by the
independent sources. This Principle has important
consequences in simplifying the calculation of ciruit
variables in multiple source circuits.
For the depicted circuit, find the
indicated current using any technique you like (you
should use the simplest).
You should have found that the current
is a linear combination of the two source values:
.
This result means that we can think of the current as
a superposition of two components, each of which is
due to a source. We can find each component by setting
the other sources to zero. Thus, to find the voltage
source component, you can set the current source to
zero (an open circuit) and use the usual tricks. To
find the current source component, you would set the
voltage source to zero (a short circuit) and find the
resulting current. Calculate the total current
using the
Superposition Principle. Is applying the Superposition
Principle easier than the technique you used in part
(1)?
Current and Voltage Divider
Use current or voltage divider rules to calculate the
indicated circuit variables in [link].
circuit a
circuit b
circuit c
Thévenin and Mayer-Norton Equivalents
Find the Thévenin and Mayer-Norton equivalent
circuits for the
following circuits.
circuit a
circuit b
circuit c
Detective Work
In the depicted
circuit, the circuit
has the v-i relation
when
.
Find the Thévenin equivalent circuit for
circuit
.
With
,
determine such that
.
Bridge Circuits
Circuits having the form of [link] are termed bridge circuits.
What resistance does the current source see when nothing is connected to the output terminals?
What resistor values, if any, will result in a zero voltage for
?
Assume
,
,
and
.
Find the current
when the current source
is
.
Express your answer as a sinusoid.
Cartesian to Polar Conversion
Convert the following expressions into polar form. Plot
their location in the complex plane.
The Complex Plane
The complex variable
is related to the real variable
according to
Sketch the contour of values
takes on in the complex plane.
What are the maximum and minimum values attainable by
?
Sketch the contour the rational function
traces in the complex plane.
Cool Curves
In the following expressions, the variable runs from zero to infinity.
What geometric shapes do the following trace in the complex plane?
Trigonometric Identities and Complex Exponentials
Show the following trigonometric identities using complex exponentials.
In many cases, they were derived using this approach.
Transfer Functions
Find the transfer function relating the complex
amplitudes of the indicated variable and the
source shown in [link].
Plot the magnitude and phase of the transfer
function.
circuit a
circuit b
circuit c
circuit d
Using Impedances
Find the differential equation relating the indicated
variable to the source(s) using impedances for each circuit
shown in [link].
circuit a
circuit b
circuit c
circuit d
Measurement Chaos
The following simple circuit was constructed but the signal measurements were made haphazardly.
When the source was
, the current
equaled
and the voltage
.
What is the voltage
?
Find the impedances
and
.
Construct these impedances from elementary circuit elements.
What is the transfer function between the source
and the indicated output current?
If the output current is measured to be
, what was the source?
Circuit Design
Find the transfer function between the input and the
output voltages for the circuits shown in
[link].
At what frequency does the transfer function have a
phase shift of zero? What is the circuit's gain at
this frequency?
Specifications demand that this circuit have an output
impedance (its equivalent impedance) less than 8Ω
for frequencies above 1 kHz, the frequency at which
the transfer function is maximum. Find element values
that satisfy this criterion.
Equivalent Circuits and Power
Suppose we have an arbitrary circuit of resistors that we collapse into an equivalent resistor using the series and parallel rules.
Is the power dissipated by the equivalent resistor equal to the sum of the powers dissipated by the actual resistors comprising the circuit?
Let's start with simple cases and build up to a complete proof.
Suppose resistors
and
are connected in parallel.
Show that the power dissipated by
equals the sum of the powers dissipated by the component resistors.
Now suppose
and
are connected in series.
Show the same result for this combination.
Use these two results to prove the general result we seek.
Power Transmission
The network shown in the figure represents a simple power transmission system.
The generator produces 60 Hz and is modeled by a simple Thévenin equivalent.
The transmission line consists of a long length of copper wire and can be accurately described as a 50Ω resistor.
Determine the load current
and the average power the generator must produce so that the load receives 1,000 watts of average power.
Why does the generator need to generate more than 1,000 watts of average power to meet this requirement?
Suppose the load is changed to that shown in the second figure.
Now how much power must the generator produce to meet the same power requirement?
Why is it more than it had to produce to meet the requirement for the resistive load?
The load can be compensated to have a unity power factor (see exercise) so that the voltage and current are in phase for maximum power efficiency.
The compensation technique is to place a circuit in parallel to the load circuit. What element works and what is its value?
With this compensated circuit, how much power must the generator produce to deliver 1,000 average power to the load?
Simple power transmission system
Modified load circuit
Optimal Power Transmission
The following figure shows a general model for power transmission.
The power generator is represented by a Thévinin equivalent and the load by a simple impedance.
In most applications, the source components are fixed while there is some latitude in choosing the load.
Suppose we wanted the maximize "voltage transmission:"
make the voltage across the load as large as possible.
What choice of load impedance creates the largest load voltage?
What is the largest load voltage?
If we wanted the maximum current to pass through the load, what would we choose the load impedance to be?
What is this largest current?
What choice for the load impedance maximizes the average power dissipated in the load?
What is most power the generator can deliver?
One way to maximize a function of a complex variable is to write the expression in terms of the variable's real and imaginary parts, evaluate derivatives with respect to each, set both derivatives to zero and solve the two equations simultaneously.
Big is Beautiful
Sammy wants to choose speakers that produce very loud music.
He has an amplifier and notices that the speaker terminals are labeled
"
source."
What does this mean in terms of the amplifier's equivalent circuit?
Any speaker Sammy attaches to the terminals can be well-modeled as a resistor.
Choosing a speaker amounts to choosing the values for the resistor.
What choice would maximize the voltage across the speakers?
Sammy decides that maximizing the power delivered to the speaker might be a better choice.
What values for the speaker resistor should be chosen to maximize the power delivered to the speaker?
Sharing a Channel
Two transmitter-receiver pairs want to share the same
digital communications channel. The transmitter signals
will be added together by the channel. Receiver design
is greatly simplified if first we remove the unwanted
transmission (as much as possible). Each transmitter
signal has the form
where the amplitude is either zero or
and each transmitter uses its own frequency
.
Each frequency is harmonically related to the bit
interval duration
,
where the transmitter 1 uses the the frequency
.
The datarate is 10Mbps.
Draw a block diagram that expresses this
communication scenario.
Find circuits that the receivers could employ to
separate unwanted transmissions. Assume the received
signal is a voltage and the output is to be a
voltage as well.
Find the second transmitter's frequency so that the
receivers can suppress the unwanted transmission by
at least a factor of ten.
Circuit Detective Work
In the lab, the open-circuit voltage measured across an
unknown circuit's terminals equals
.
When a 1Ω resistor is place across the terminals, a
voltage of
appears.
What is the Thévenin equivalent circuit?
What voltage will appear if we place a 1F capacitor
across the terminals?
Mystery Circuit
We want to determine as much as we can about the circuit lurking in the impenetrable box shown in [link].
A voltage source
V
has been attached to the left-hand terminals, leaving the right terminals for tests and measurements.
Sammy measures
V when a 1 Ω resistor is attached to the terminals.
Samantha says he is wrong.
Who is correct and why?
When nothing is attached to the right-hand terminals, a voltage of
V is measured.
What circuit could produce this output?
When a current source is attached so that
amp, the voltage
is now 3 V.
What resistor circuit would be consistent with this and the previous part?
More Circuit Detective Work
The left terminal pair of a two
terminal-pair circuit is attached to a testing
circuit. The test source
equals
([link]).
We make the following measurements.
With nothing attached to the terminals on the
right, the voltage
equals
.
When a wire is placed across the terminals on
the right, the current
was
.
What is the impedance “seen” from the terminals
on the right?
Find the voltage
if a current source is attached to the
terminals on the right so that
.
Linear, Time-Invariant Systems
For a system to be completely characterized by a
transfer function, it needs not only be linear, but also
to be time-invariant. A system is said to be
time-invariant if delaying the input delays the output
by the same amount. Mathematically, if
,
meaning
is the output of a system
when
is the input,
is the time-invariant if
for all delays
and all inputs
.
Note that both linear and nonlinear systems have this
property. For example, a system that squares its input
is time-invariant.
Show that if a circuit has fixed circuit elements
(their values don't change over time), its
input-output relationship is time-invariant.
Hint: Consider the differential
equation that describes a circuit's input-output
relationship. What is its general form? Examine the
derivative(s) of delayed signals.
Show that impedances cannot characterize
time-varying circuit elements (R, L, and C).
Consequently, show that linear, time-varying systems
do not have a transfer function.
Determine the linearity and time-invariance of the
following. Find the transfer function of the linear,
time-invariant (LTI) one(s).
diode
Long and Sleepless Nights
Sammy went to lab after a long, sleepless night, and
constructed the circuit shown in [link].
He cannot remember what the circuit, represented by the
impedance , was. Clearly,
this forgotten circuit is important as the output is the
current passing through it.
What is the Thévenin equivalent circuit seen by
the impedance?
In searching his notes, Sammy finds that the circuit
is to realize the transfer function
Find the impedance
as well as values for the other circuit elements.
What is the impedance
at the frequency of
the source?
Black-Box Circuit
You are given a circuit that has two terminals for
attaching circuit elements.
When you attach a voltage source equaling
to the terminals, the current through the source equals
.
When no source is attached (open-circuited terminals),
the voltage across the terminals has the form
.
What will the terminal current be when you replace the
source by a short circuit?
If you were to build a circuit that was identical
(from the viewpoint of the terminals) to the given
one, what would your circuit be?
For your circuit, what are and
?
Solving a Mystery Circuit
Sammy must determine as much as he can about a mystery
circuit by attaching elements to the terminal and
measuring the resulting voltage. When he attaches a 1Ω
resistor to the circuit's terminals, he measures
the voltage across the terminals to be
.
When he attaches a 1F capacitor across the terminals,
the voltage is now
.
What voltage should he measure when he attaches
nothing to the mystery circuit?
What voltage should Sammy measure if he doubled the
size of the capacitor to 2 F and attached it to the
circuit?
Find the Load Impedance
The depicted circuit
has a transfer function between the
output voltage and the source equal to
.
Sketch the magnitude and phase of the transfer
function.
At what frequency does the phase equal
?
Find a circuit that corresponds to this load
impedance. Is your answer unique? If so, show it to be
so; if not, give another example.
Analog “Hum” Rejection
“Hum” refers to corruption from wall socket
power that frequently sneaks into
circuits. “Hum” gets its name because it
sounds like a persistent humming sound. We want to find
a circuit that will remove hum from any signal. A Rice
engineer suggests using a simple voltage divider circuit
consisting of two series impedances.
The impedance
is a resistor. The Rice engineer must decide between
two circuits for the impedance
.
Which of these will work?
Picking one circuit that works, choose circuit element
values that will remove hum.
Sketch the magnitude of the resulting frequency
response.
An Interesting Circuit
For the circuit shown in [link],
find the transfer function.
What is the output voltage when the input has the form
?
What is the transfer function between the source and
the output voltage?
What will the voltage be when the source equals
?
Many function generators produce a constant offset in
addition to a sinusoid. If the source equals
,
what is the output voltage?
An Interesting and Useful Circuit
The depicted circuit
has interesting properties, which
are exploited in high-performance oscilloscopes.
The portion of the circuit labeled "Oscilloscope"
represents the scope's input impedance.
and
(note the label under the channel 1 input in the lab's
oscilloscopes). A probe is a device
to attach an oscilloscope to a circuit, and it has the
indicated circuit inside it.
Suppose for a moment that the probe is merely a wire
and that the oscilloscope is attached to a circuit
that has a resistive Thévenin equivalent
impedance. What would be the effect of the
oscilloscope's input impedance on measured voltages?
Using the node method, find the transfer function
relating the indicated voltage to the source when
the probe is used.
Plot the magnitude and phase of this transfer
function when
and
.
For a particular relationship among the element
values, the transfer function is quite simple. Find
that relationship and describe what is so special
about it.
The arrow through
indicates that its value can be varied. Select the
value for this capacitor to make the special
relationship valid. What is the impedance seen by
the circuit being measured for this special value?
Find the differential equation relating the output
voltage to the source.
What is the impedance “seen” by the capacitor?
Analog Computers
Because the differential equations
arising in circuits resemble those that describe
mechanical motion, we can use circuit models to describe
mechanical systems. An ELEC 241 student wants to
understand the suspension system on his car. Without a
suspension, the car's body moves in concert with the bumps
in the raod. A well-designed suspension system will smooth
out bumpy roads, reducing the car's vertical motion. If
the bumps are very gradual (think of a hill as a large but
very gradual bump), the car's vertical motion should
follow that of the road. The student wants to find a
simple circuit that will model the car's motion. He is
trying to decide between two circuit models ([link]).
Here, road and car displacements are represented by the voltages
and
, respectively.
Which circuit would you pick? Why?
For the circuit you picked, what will be the
amplitude of the car's motion if the road has a
displacement given by
?
Sketch the magnitude and phase of your transfer
function. Label important frequency, amplitude and
phase values.
Find
when
.
Fun in the Lab
You are given an unopenable box that has two terminals
sticking out. You assume the box contains a circuit. You
measure the voltage
across the terminals when nothing is connected to them
and the current
when you place a wire across the terminals.
Find a circuit that has these characteristics.
You attach a 1 H inductor across the terminals. What
voltage do you measure?
What is the transfer function relating the complex amplitude of the output signal, the current
, to the complex amplitude of the input, the voltage
?
What equivalent circuit does the load resistor
see?
Find the output current when
.
Why Op-Amps are Useful
The
circuit of a
cascade of op-amp circuits illustrate the reason why
op-amp realizations of transfer functions are so useful.
Find the transfer function relating the complex
amplitude of the voltage
to the source. Show that this transfer function equals
the product of each stage's transfer function.
What is the load impedance appearing across the first
op-amp's output?
[link] illustrates that sometimes
“designs” can go wrong. Find the transfer
function for this op-amp circuit, and then show that
it can't work! Why can't it?
Find the transfer function relating the voltage
to the source.
In particular,
,
,
,
,
and
.
Characterize the resulting transfer function and
determine what use this circuit might have.
Designing a Bandpass Filter
We want to design a bandpass filter that has transfer
the function
Here,
is the cutoff frequency of the low-frequency edge of the
passband and
is the cutoff frequency of the high-frequency edge. We
want
and
.
Plot the magnitude and phase of this frequency
response. Label important amplitude and phase values
and the frequencies at which they occur.
Design a bandpass filter that meets these
specifications. Specify component values.
Pre-emphasis or De-emphasis?
In audio applications, prior to
analog-to-digital conversion signals are passed through
what is known as a pre-emphasis circuit that
leaves the low frequencies alone but provides increasing
gain at increasingly higher frequencies beyond some
frequency
. De-emphasis
circuits do the opposite and are applied
after digital-to-analog conversion. After
pre-emphasis, digitization, conversion back to
analog and de-emphasis, the signal's spectrum
should be what it was.
The op-amp circuit here has been
designed for pre-emphasis or de-emphasis (Samantha can't
recall which).
Is this a pre-emphasis or de-emphasis circuit?
Find the frequency
that defines the transition from low to high
frequencies.
What is the circuit's output when the input voltage is
, with
?
What circuit could perform the opposite function
to your answer for the first part?
What is this circuit's transfer function? Plot the
magnitude and phase.
If the input signal is the sinusoid
,
what will the output be when
is larger than the filter's “cutoff frequency”?
Optical Receivers
In your optical telephone, the receiver circuit had the form
shown.
This circuit served as a transducer, converting light
energy into a voltage
. The photodiode acts as a current source,
producing a current proportional to the light intesity
falling upon it. As is often the case in this crucial
stage, the signals are small and noise can be a
problem. Thus, the op-amp stage serves to boost the
signal and to filter out-of-band noise.
Find the transfer function relating light intensity
to
.
What should the circuit realizing the feedback
impedance
be so that the transducer acts as a 5 kHz lowpass
filter?
A clever engineer suggests an alternative
circuit to accomplish the same
task. Determine whether the idea works or not. If it
does, find the impedance
that accomplishes the lowpass filtering task. If
not, show why it does not work.
Reverse Engineering
The depicted
circuit has been developed by the TBBG
Electronics design group. They are trying to keep its
use secret; we, representing RU Electronics, have
discovered the schematic and want to figure out the
intended application. Assume the diode is ideal.
Assuming the diode is a short-circuit (it has been
removed from the circuit), what is the circuit's
transfer function?
With the diode in place, what is the circuit's output
when the input voltage is
?