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hello welcome to lecture 1 of electrical circuits 1
In this lecture first we're gonna provide a brief course overview
The course subject matter is overall about system modeling analysis and design.
I must spend the first part of this lecture talking about what these things
are and introducing you to some of the basics of the philosophy of where
we're going to from this point forward for the rest of the semester
after that I'll introduce some basic circuit parameters
voltage current charge power. we will need those
throughout the rest of the semester when we're dealing with our circuit analysis
toward the end to the lecture we will introduce the passive sign convention
the passive sign convention is very important
if you don't use it properly before you start your analyses
your analyses will not make any sense
Luckily just because it's extremely important doesn't mean that it's difficult it's
really really easy to do you just have to make sure that you do it right and
you pay attention to it
the related educational module here his section 1.1
there some suggested information that
you should know before starting the class there's also some suggested
additional classes that you might want to be taking at the same time you take
this one
so it is suggested that
as a prerequisite before starting this class you should have at least
a basic exposure to electricity and magnetism
in general that would be presented in a second semester college level physics
class
a high school physics class may be appropriate
may give you the information you need if it was a good one
You should also have probably a couple of semesters of calculus under your belt before you
start this class
we will be doing integrations in this class we will be
finding maximum and minimum of functions we will need some calculus related tools
Strongly recommend that has a core requisite to this class you take a
differential equations class. We will be writing and solving differential
equations during this class
okay If you're taking a differential equations class at the same time
they will give you a lot of theoretical background that will help you understand
what we're doing in this class and why we're doing it
now
if pressed I will admit that both the sets the requirements are a little bit
weak
I've had students make it through the class without having some
or even any of these. However
you need to be extremely motivated if you're going to do that
so when I introduce a topic that require some prerequisite or corequisite:
knowledge
I will explain it briefly at the appropriate point in this class but I
won't get into a lot of
background information about it. Our primary goals for this class is to
provide an introduction to modeling and analysis and design
of electrical circuits. Now quite often we will use
what is called a systems level approach to the electrical circuit, the
sign analysis and modeling that simply means that we will
represent our physical device as some kind of system
The system will have some input and some
output denoted by U of T and Y(t)
u(t) is provided to the system by the outside environment
y(t) is provided by the system
back to the outside environment. The system itself can often be considered to
be
a black box. The modeling aspect of this class
is generally determine the functional relationship between the output
y (t) and the input u(t). The system is going to be performing some process on u(t)
to produce y(t). That functional relationship consists of the model of
the system
In this class most generally the functional relationship between y(t) and
u(t) will be an ordinary differential equation
Although for the first part of this class quite often these functional
relationships will be algebraic
now one important thing to note is that once we've got this functional
relationship we will often start
thinking about the system's in terms of these functional relationships. Rather
than the original physical device
that has its good side and its bad side
it's good side is that it gives you an ability to handle a wide range of
systems
We will be emphasizing electrical circuits but a lot of the techniques provided in
this class are appropriate to mechanical systems
thermal systems, fluidic systems. Okay the functional relationship is really
what's important
the downside of thinking in terms of these functional relationships
is that sometimes it easy to lose track of the above
the fact that this was originally a physical device
there will be limitations on how accurate this mathematical
representation is.
Okay, if you have forgotten that this was originally a physical device
you have lost the ability to assess whether your mathematical relationship
is actually reflecting the physics of what's going on in here
I now want to define what I mean by modeling analysis and design
when we model the system what we're doing is determining the functional
relationship between the input and the output that I talked about on the
previous slide
we're taking the physical device were analyzing that physical device in order
to determine the mathematical relationship between the input and the
output in general
system analysis means that given this functional relationship and some
specific
u(t) we determined some y(t)
For example we have some voltage applied to our circuit were representing
our circuit is some
mathematical expression we may want to determine
what the power in some resistor is for that
specific input. So we're determining
some specific output for some given specific input.
System design is maybe the most complex
of the three categories and we'll just getting to design a little bit in
this class
Design means that what we're doing is creating a system
which provide some desired relationship between the input and the output. We know
what we want the system to do
quite often that's represented a some mathematical functional relationship
between input and output
we want to do what is called realizing and that mathematical relationship by
creating a physical device that will perform that process
now I wanna provide a graphical representation of what I mean by
modeling analysis and design and their relationship to the physical world
we really wanna firmly fixed on our minds that there is a physical
entity happening we are using mathematics to represent that entity
to some extent. We have to kind of divorce those two things in order to
keep track of both of them
So we have some real-world, which consists of a physical
system. Some electrical circuit in general for us
we will also be representing this physical world
mathematically. Now we're gonna always start out with some physical system for
us this will be an electrical circuit for a mechanical engineer it may be a
thermal system
or some kind of fluidic system we're gonna take this physical system
and write some mathematical relationships that we feel govern the
physics of what's going on in here we've now transferred ourselves from the real
world
into mathematics land once were in mathematics land we can
analyze what we think this physical system will do
so depending on the complexity of this mathematical model we may do some hand
calculations to determine the desired parameters the
the specific output that we expect this system to give us
more realistically for any reasonably complicated system will probably do our
analysis using a computer will simulate the response of the system using a
computer
now we really need to be concerned about whether
this analysis results actually match
the real world so in the real world
generally we're gonna take this system and test it
generally under the same conditions that were doing the analysis for
or we can do the analysis specifically for the conditions that we're
testing
then we'll take the test data and compare that to the analysis results
and see whether they agree to a satisfactory level for us
if they don't agree satisfactorily that means our original mathematical model
left out something important about the physics of this system
we may have to go back and revise our mathematical model and we do this
analysis section and continue to compare
our analyses with the test data now once we have revised our mathematical model
so that it reflects reality
consistent to what we're testing we may use the result in these analyses to
modify our system
or redesign it so we may try to get some desired response out of this physical
system by redesigning part to the physical system
now leaving this in mathematics land is generally not appropriate for an
engineer were always
wanting to build something that will do something in the real world
so what's we created this design will go back and
implement that design build the actual circuit or system
and test it now at this stage you also still
run the risk that your analysis that created this design may not be
reflecting reality accurately enough
your design may not actually meet the requirements
that may require you to go back and revise your original mathematical model
so one really important thing that I want to get across
with this slide is that design especially is a very
iterative process okay you end up trying something maybe it doesn't work out you
may have to go back and revise things
and try again
now as I mentioned earlier your model may or may not reflect
the physical behavior of the system that you want to reflect
there are actually a number of different approaches towards modeling systems
so regardless of the fact that in this class we will stick with the most simple
approaches for system modeling
I'm gonna give a quick overview of kind of the broad categories of modeling
approaches so you can sort of see where we stand in the overall
world hey
a lumped parameters model is appropriate
if system parameters change slowly relative to the component response time
Okay the component whatever it is at electrical mechanical thermal
fluidic has a lot of time to respond to whatever's happening
if you have that condition met
in general the typeof model you'll end up with consists up ordinary
differential equations
now they can be algebraic but for the most part I will be emphasizing
time-varying stuff which implies differential equations
here is an example of an lumped parameter system
it's a simple slinky okay if I hang the slinky from one hand
and move my hand up and down very slowly
the bottom of the slinky moves up and down
now this looks like a spring with a mass at the bottom of it
at any given time the spring
is either compressing or expanding all parts of the springer doing the same
thing at any given time
the mass which consists of several coils of the the spring at the bottom
is not really stretching
I can't conceptualize this system as the following
I may have some support at the upper
end with a spring with spring rate k
upon which is hanging a mass m
My input might be the motion of the top with the spring
and my output might be the motion of the mass
now this spring i can assume is massless
the mass I may assume has no spring rate
those are approximations this spring has mass
just for the purposes of my model I'm going to assume
that it doesn't. What I'm gonna hope is that the mass of the spring is not
important
Likewise I'm going to hope that the relative motion of those little coils at the
bottom of the spring
is not going to contribute significantly to y(t)
and i can assume that this has no spring rate what I've done
is lumped the system parameters i've lumped all the spring rate at one point
I've lumped all the mass
at another point as long as I don't move
this spring very rapidly that's probably going to be okay
now conversely if parameters are changing rapidly relative to the
component response time
we'll have to use what is called distributed parameter system
not all parts of the component will be responding the same way because the
information can't get through the component
fast enough if you're doing a distributed parameters model
in general your modeling approach will consist of partial differential
equations
as you might guess we'd much rather do this
than this
here's my example the distributed parameter systems
it's the same slinky that I used before
please notice that although the system hasn't changed
the system may be doing something that requires me to model
the system differently because different conditions apply
so now I'm gonna take this slinky and stretch it tight
if I change the displacement of one end of this link you very rapidly just by
releasing one coil
what I end up doing is propagating a wave down the slinky it bounces off
my hand at the other end propagates back as it
goes back and forth it turns into kind of a jittery motion of the spring
Okay
not all parts of the spring are moving the same way at the same time
i cant lump the mass and the spring rate
because it any given time part of the spring is acting like a mass part is
acting like the spring rate
it has to be distributed throughout the entire system
it's governed by a partial differential equation now
linear systems are those which
have as the name implies linear relationships between the dependent
variables in the system
these are very nice they tend to be governed by linear differential
equations or as a special case linear algebraic equations
the really nice thing about this is linear
equations have a unique solution
if you can write these equations and solve them you will always end up with only
one solution
here's my example linear system it's still my slinky
if I stretch an ideal spring
Hooke's law tells me that the displacement that I get is proportional
to the force that I apply
so if I stretch it to some position increase the force
incrementally I will increase the displacement incrementally
the force and the displacement will be related
by a linear relationship with some constant of proportionality which is the spring
rate
graphically that looks like this
if I plot the force versus displacement were x is displacement
and f is force. Hooke's law tells me that F is equal to k times x
Where k is the spring rate of the spring . That is a linear relationship.
If I plot those
two things I get a straight line
The slope of this line is the spring rate k.
So, for a linear system all the governing relationships between
dependent variables have to be linear if I plot one dependent variable against
another I would end up with a straight line
if the relationships between dependent variables in a system or nonlinear
that is if I plot two dependent variables against one another in the end
up with something that is not
a straight line the system itself is said to be nonlinear
if the system is nonlinear it still governed by differential equations but
the differential equations themselves are nonlinear
that has a couple of major drawbacks over linear differential equations
first they're just more difficult to solve flat-out
second if you do get them solve they may actually have more than one solution
more than one solution implies that the system can operate at more than one
point so you may design this system to operate at some point
if its nonlinear it may decide to go ahead and operate at some other point
which you may not be very happy about we'd much rather design linear systems
and nonlinear systems
these are tough to deal with. Problem
is that any real system is ultimately going to be nonlinear to some extent
what we tend to try to do is to take a non linear system
an approximate it as a linear system designed to that
and hope that everything works out
my slinky can also have a nonlinear behavior
Hooke's law is a good way to model this system if the displacement is small
if however I displace this a lot
the more I stretch it the harder and harder it gets to stretch the spring
in fact in the extreme after a stretch this far enough so that its uncoil
the coils and a straight wire
I would have to plastically deform the wire in order to continue to stretch in
anymore
that would require a lot a force just to get a little bit
displacement likewise if I try to compress this spring
once I compressed it to the point where there is no air gap between the coils
I have to re apply a huge amount of force if I wanna try to compress this
anymore
the middle actually has to be form in order to compress it.
if I plot this stuff graphically
this plot the same parameters that I used before displacement
and force for relatively small displacements or small forces
yeah Hooke's law is gonna hold the force
and the displacement are gonna be approximately linearly related
as the displacement gets larger and larger though
it may cause more and more force to be applied in order to get
the same relative displacement. Right.
in them limit as this spring entirely on Cornell's it takes
a lot love forced to get a
tiny bit of displacement likewise if I compress the spring
once the air gap between the coils is gone it's going to require a lot of
forced to get a very small amount of displacement
this is now a nonlinear relationship
whether we can approximate it as a linear
relationship or whether we have to take into account this nonlinearity
is a decision that we have to make one more modeling this system
finally there what are called time-varying systems
in a time-varying system the physical parameters in the system may vary as a
function of time
for example a light bulb the resistance of the filament changes as the light
bulb heats up
it may appear that the physical parameters on the system are changing
with time
if that happens the system is governed by
differential equations with time-varying coefficients
we can also consider our slinky to potentially be a time-varying system
if I'm operating the slinky and bouncing it around and I decide to change for
example the temperature
on a regular schedule. The physical parameters of this slinky
like the Young's modulus which contributes to the spring rate of the
slinky
those may change as I change the temperature I'd have to change the
temperature
quite a lot granted but I could make them change
if the temperature is changing as a function of time it may look like the
slinky is
parameters are changing as a function of time
the resulting governing equations would now be
differential equations with time-varying coefficients
in this class in electrical circuits one we will
restrict the category of models that we will be creating
we're only gonna worry about lumped parameters models of linear
time invariant systems so
the governing equations that we create will be either algebraic equations with
constant coefficients
or linear differential equations with constant coefficients
were doing the easiest possible case in this class
there are a few basic electrical parameters that will be dealing with
throughout the course of the semester I'd like to introduce those now
the most fundamental electric circuit parameter is
charge it is abbreviated as lower case q
the units of charge are Coulombs capital C
One Coulomb is defined as about 6.24 times ten to the 18th electrons the
fundamental unit
of charges 11 electron charge has to occur in integer numbers
of electrons this minus sign is because
when this these analyses approaches were originally being generated people did
not think in terms of moving electrons around they were thinking in terms of
motion a positive charge it wasn't until later
that they realize that charge motion is done by moving electrons
this negative sign accounts for that difference
in outlook now although charge is extremely fundamental to any electrical
analysis we won't be using it directly very often in this class
what we are interested in is not charge but the motion of charge we're
interested in transferring energy we want to get
energy from one place to another in our electrical circuit in order to perform
some
useful task so this parameter won't get used a lot
however we have a couple of others that will allow us to
quantify charge motion the first of these
is current its abbreviated as lower case I
it is the rate of change of charge with time
so I is dq dt in mathematical form
so what we're doing is sitting at some point in our circuit and watching
electrons go by us
the rate at which they're going by gives us the current. The units of current or
amperes abbreviated as capital A. Units of
charge are coulombs. Units of time is seconds so
amperes are actually Coulombs per second in more fundamental units
our next circuit parameter that's gonna be a great direct importance to us in
this class is voltage
Usually abbreviated as lower case v
voltage quantifies the change in energy
of a unit charge at two different physical locations
so we're taking a charge or we're looking at the same charge
in two different places two different points in our circuit
there's gonna be a voltage difference but characterizes the difference in
energy of those two charges or that one charge it two different locations
so voltage is DW DQ or rate of change
energy with change in charge
units of voltage are V
capital V volts
Units of energy is Joules, units of charge is Coulombs
Volts are Joules per Coulomb
the final parameter of direct concern to us at the moment is gonna be power
power is consistent with your physics class
the rate of change of energy with time
so power is dW/dt where W is
energy now I can use the chain rule to expand this into dW/dq
times dq/dt and I can think of each of them canceling themselves
for a multiplicative process
dW/dq was previously defined as voltage
dq/dt was previously defined as current
so power in the electrical world is voltage times current
the units a power Watts consistent with what you've used previously to represent
power
there are two main categories of electrical circuit elements that will be
dealing with throughout this class
the first is passive circuit elements
passive circuit elements are defined by the fact that the total energy overall
time
that is delivered to the circuit element by the rest to the circuit
has to be non-negative the circuit the passive circuit element can either store
energy or dissipate energy it can't create
energy okay so you can store energy in a passive circuit element or you can
dissipate energy in a passive circuit element
but you can't create energy any energy in that circuit elements has to have
been provided
to the element by the circuit at some point in time
Contrary to this active circuit elements
can actually supply energy to the circuit that energy is coming from some
external source
so from the circuit standpoint this
element is creating energy in providing it to the circuit
a battery for example is performing a chemical reaction which is converted to
electrical energy that electrical
energy can be provided to an electrical circuit to perform some useful task
as far as the circuit is concerned that energy was just created by that battery
now I want to introduce the passive sign convention this is extremely important
it's very easy but it's extremely important
if you don't get the passive sign convention right then none of
analyses will be meaningful you cannot rely on any of your results
essentially in the passive sign convention we're going to assume
the sign of the current relative to the sign
of the voltag. So we're gonna assume a current direction
relative to a voltage polarity individually we can make the current
direction
any direction we want it to be but once we've chosen that current direction we
must be consistent with our
assumed voltage polarity. Likewise we could assume
a voltage polarity once we've done that though
our current direction must agree with that voltage polarity
the assumption that we must make is that positive current
enters the node which is assumed to be at the higher voltage
for example for this particular circuit element
if I assume this voltage polarity the plus/minus on the voltage gives me the
polarity it indicates that the voltage at this node is assumed to be higher
than the voltage it this node
so if I assume this voltage polarity I have to assume that current is
entering the upper voltage node now that's not the only assumption I could
make
it would be perfectly legitimate to assume that the voltage at the lower
node is higher than the voltage at the upper node
however if I do that I must assume that the current is
entering the bottom node now. Current has to enter
the node which is at the higher voltage
for active circuit elements we don't need and to make any assumptions
the current directions are given for current sources the voltage polarities
are given for voltage sources
so reiterating this. You can assume arbitrarily
either the voltage polarity or the current direction
however once you've chosen one or the other of those
that dictates the assumed direction on the
other parameter
the reason that individually they don't matter is that what you're doing is
assigning
reference directions for these you have a reference voltage polarity you have a
reference current direction it tells you
what you think the positive direction is.
Okay. your numbers can turn out to be negative
okay if you do your analysis if you get a negative result that just means that
you assume the wrong voltage polarity if your voltage turns out to be negative in
you need to swap the polarity
if your current turns out to be a negative number
you simply assume the wrong current direction these guys are just telling
you
how to interpret a positive relative to a negative result
of your analyses
now I want to do a few examples of assigning the passive sign conventions
the first example I have a circuit
which has a 9 volt source notice that the voltage polarity is given on this
active circuit element
it also has a current source which is 0.5 A
the current direction is given to us there we don't need to make
any sign assumptions on these active elements
what we do need to do is assign our passive sign conventions
to the four passive elements
represented by gray boxes now what I've done is given you
what I i and going to dictate is the positive direction for either voltage or
current
on each of those you need to assign
the positive value for the other parameter on
each of these passive elements now the way this is going to work best as if you
stop the video at this point
take a shot at doing the problem after you've tried it yourself
come back restart the video watch me do the problem
so go ahead positive video and give it a try
okay for this circuit element I've told you that
the current i want is assumed to be
positive going from right to left
that's my assumed direction of positive current that dictates the voltage
polarity for element 1
current has to be entering the positive voltage node
this has to be my assumed voltage polarity for element 1
in element 2 I've assume that this lower note is that a higher voltage than
the
upper node that dictates the assumption of
the direction of assumed positive current for this element
so for element 2, i2 has to be going into the positive voltage node i2
if it's positive is going from bottom to top
element 3 the assumption is that positive voltage
is at the upper node the voltage polarity here is given
that tells me that I need to have positive current for element three as
going from top to bottom
element 4 the positive current direction is given as being
going from top to bottom current has to enter the positive voltage node
that dictates the assumed voltage polarity for element 4
a few hints relative to the passive sign convention
now students recognize that this is extremely important to get this right
sometimes that means that they spend a lot of time trying to get it right
in some sense just because it's important
doesn't necessarily mean that it's difficult generally
it's not productive to try to figure out what the correct positive directions are
before you start analyzing the circuit
just blindly assign assumed voltage polarities and current directions
before you start analyzing
Than, go ahead and analyze the circuit negative values will tell you that you
just assume the wrong direction don't try to
pick the correct direction before you do the analysis the analysis has to tell
you what the correct direction is.
so when you're approaching a problem
arbitrarily choose either the voltage polarity or the current direction for each
passive circuit element I don't care which one you choose. Pick one of
them. Than, label the other parameter for each one of those circuit elements
so that they agree with the passive sign convention
Okay. Later on when we talk about doing analysis methods
you will do your analysis using those
assumed signs if you end up with a negative value it's no big deal it just
means that the original assumption was incorrect
here's another example
I want to assign reference voltage polarity some current directions for
each
of the passive circuit elements in this circuit
I have a 2 A source: its current direction
is positive downward now in this particular example I haven't given you
either the voltage polarity or the current direction you're going to have
to determine both of them
so please give that a try before you come back and watch me do the problem
okay
now i can assume
either the current direction or the voltage polarity arbitrarily for each of
these elements
so for this element let's call it element 1 I'm going to assume that the
voltage polarity
is positive at this node relative to this note
that assumption
individually doesn't matter however once I've made that assumption I have to
assume that
i1 is going positive into this node
now if you made an alternate assumption
I'm perfectly happy if you assume that the voltage
is positive at this lower node relative to the upper node
that's fine now just make sure that you assume
that i1 is going in to the positive voltage node
for this element let's assume that it's element 2 and than i2 is positive in
this direction
once I've assumed a positive current direction
that tells me what the voltage polarity
assumed direction is. the positive voltage must be at the node which the
current is
entering. Element 3 say I assume that
voltage is positive at the upper node relative to the lower node
that then tells me that I have to assume
that i3 is positive downward positive current enters the positive
voltage node
again that assumption is not unique for example
say I wanted to assume that the positive current and element 3 is
i3 is positive going upwards
that's fine as long as you assume
that the polarity of the voltage is such that the voltage at the lower node
is higher than at the upper node
then later when I analyze the circuit I just know
how to interpret positive versus negative results
if v3 turns out to be a negative number it just means that
the voltage is higher at the upper node than it is at the lower node
to do another example the same thing
in this particular circuit I have a slightly different configuration of
three passive circuit elements and I have now a voltage source of three volts
with a given polarity rather than the current source that I have before
please notice that the source that I give you
doesn't affect your choice on the passive sign convention I don't need to
use this information to try to decide which direction are positive voltage in
and which directions are positive current
I'd choose them arbitrarily for each individual passive circuit element
so for circuit element one i can assume either a voltage
polarity or current direction let's assume that the current
I one is positive from left to right that means the voltage
v1 has to be assumed to be higher
at the leftmost node then it is at the right
if I assume for element 2 to that the voltage clarity
v2 is such that the higher voltages at the upper node
that's fine as long as you assume that the current
i2 is entering the positive voltage node
element 3 I can assume
either the voltage polarity or the current direction let's assume
that I assume that
the current is going from the bottom to the top i3 is going upwards
positive
that means that v3 it's assumed higher voltage has to be at the lower
node
Okay, the assumption of the current directions in the voltage polarities of
this element
aren't affected by my other assumptions I picked them
each independently as long as they agree for each passive element I'm in good
shape when I do my analysis
One more example relative to the passive sign convention
I have this particular circuit here it has
two passive circuit elements and a 4V voltage source
I'm gonna claim that I have picked my passive sign convention for each of
these two elements
in this way I'm assuming that v1 is higher
at this node than it is at this node and i1 is going positive
right to left so i1 is entering the positive
voltage node of this element second element
I've assume that v2 is higher at this node than it is at this node. Thus I need to
assume that I2
is positive into the positive voltage node
now I'm in a claim that I've secretly gone off and done some analysis
after I've analyzed the circuit I determined that i1 turns out to be
negative three milli amps
I2 is positive three milliamps, V1 is a negative 1.5 volts
and v2 is 2.5 volts positive
I want you to redraw the circuit showing
the voltages and currents and their actual directions rather than their
assumed directions
so give that a shot come back watch me do it
okay my voltage source hasn't changed
and I still have
to passive circuit elements in this
configuration.
V1 is assumed to be positive here relative to here.
v1 is -1.5 volts that simply means that the
actual voltage polarity is the opposite of
the assumed voltage polarity so that v1
1.5 volts is a higher voltage at the left node then it is that the right node
the opposite of what I assume. I1 is also negative that means that this
current
is also in the opposite direction to what I assumed.
So i1 is actually in this direction and it is positive three milliamps in
that direction
For element two
I2 is a positive three milliamps that means that this direction is
correct
So we have three milliamp's positive
Likewise voltage v2 is a positive number
so this is the actual correct
voltage polarity I don't need to change that
that conclude lecture 1
next time in lecture 2 will come back and will briefly review the passive sign
convention we really need to make sure that that's done right before we try to
do anything else
also in lecture 2 we will start talking about power generation and power
absorption
those two concepts whether an element
is absorbing are generating power will hinge
very heavily upon understanding the passive sign convention
after we do that we'll start talking about a particular type of circuit
element resistors
and the voltage current relationships
for resistors which is called ohms law