Damped Sinusoids

1
Damped Sinusoids
Were studying basic waveforms because they can
be used as building blocks to construct more
complicated waveforms. Weve already seen how a
unit step function can be combined (by
multiplication) with another waveform (such as a
DC level, or a ramp) to serve as an on/off
switch. A step can also be used with a sinusoid
as an on/off switch. The sinusoid function is
like a function generator, and the step is its
switch. The next slide demonstrates this.
2
Damped Sinusoids
The red plot is a decaying exponential, with t
1 second. The blue plot is a sinusoid with f 5
Hz., f 0, and amplitude equal to the
instantaneous value of the decaying exponential.
Notice how the exponential causes the sinusoids
amplitude to decrease or damp down as t
increases.
3
Shifted Functions
This heading means shifted in time. Shifting the
damped sinusoid shown below to the left means
advancing it in time, shifting it to the
right means delaying it. Lets see how to delay
the waveform by one second. Mathematically, this
damped sinusoid is given by
We know how to delay the unit step factor.
4
Shifted Functions
Delaying the unit step factor by one second
yields the function
This function is shown here. Notice that we
didnt delay the damped sinusoid, we just delayed
the point at which it is turned on. What we have
to do is delay the sinusoid and the exponential
which damps it, as well as the step which turns
it on.
5
Shifted Functions
We can delay the exponential and sinusoidal
factors the same way we delayed the unit step
by changing the argument. We substitute t - 1
for t, which gives us the result on the next
slide
6
Shifted Functions
The delayed waveform is given by
which is plotted here. Notice that its shape is
exactly that of the original damped sinusoid, but
it is shifted to the right by one second.
7
General Waveforms
Were studying basic waveforms such as steps,
ramps, exponentials, and sinusoids because they
can be used as building blocks for more general
waveforms. Weve just seen how a unit step, an
exponential, and a sinusoid can be used to
construct a damped sinusoid. Weve also seen how
those three building blocks can be delayed (or
shifted) to construct a delayed damped sinusoid.
Lets look at a couple of additional examples
of the use of shifted and scaled basic waveforms
to construct more complex waveforms.
8
Shifted Functions
Heres a waveform which is not any single basic
waveform. Lets see how to construct it by
summing several shifted and scaled step functions.
First notice that the waveform is equal to a unit
step for t lt 0.5, so well start constructing
the function with u(t).
9
Shifted Functions
Heres a waveform which we have under
construction along with the first building block,
u(t). Weve actually built the waveform for t lt
0.5.
At t 0.5, the waveform takes another step up
from a value of 1 to a value of 2. Well add
another step, delayed until t 0.5
u(t-0.5)
10
Shifted Functions
Now weve built the waveform for t lt 1. At t
1, the waveform takes a large step down, from
f(t) 2 to f(t) -2. The size of this step is
-4, so
well add a step of size -4, delayed until t 1
u(t-1)
11
Shifted Functions
Were almost done. The waveform under
construction takes one more step up, with a size
of 2, at t 1.5. We add 2u(t-1.5).
12
Shifted Functions
Were done. Weve built the waveform f(t) out of
delayed and scaled step functions
13
Shifted Functions
We can also construct complex waveforms using
ramps as building blocks. Heres one
For t lt 0.5, a ramp with a slope of 2 matches the
waveform were trying to build. The ramp may be
written as 2tu(t)
14
Shifted Functions
Heres the waveform under construction, along
with the ramp that matches it for t lt 0.5. Next,
we need a ramp with a slope of -1 to cancel
the positive slope of the first ramp. The new
ramp must start at t 1, so its delayed by 1
second. The second ramp is tu(t-1)
15
Shifted Functions
We finish constructing the waveform by adding a
third ramp, delayed until t 1 and with a slope
of -1 The waveform weve built is the function
16
Impulse Function
Now well examine the impuluse function, the last
of the basic waveforms were going to study. We
will create an impulse function by starting with
a rectangular pulse, as shown below. The width
of the pulse is t1, and the area of the rectangle
is 1, so the height of the pulse is 1/ t1. The
pulse can be constructed from two step functions
f(t)
Weve assigned a value to the pulses area (its
1), but not to its width t1. t1 can be large or
small. If its small, the pulse height must be
large but if t1 is large, the pulse height must
be small.
t
t1
0
17
Impulse Function
Now, lets start making t1 smaller and smaller.
The pulse gets shorter and shorter, but its
height gets larger and larger. However, the area
under the pulse remains equal to 1. If we let T
approach zero, the amplitude of the pulse
approaches infinity. The pulse approaches
something with infinite amplitude and zero
duration, but its integral or area is still 1.
f(t)
f(t)
f(t)
t
t
t
t1
t1
t1
0
0
0
18
Impulse Function
Heres another way of putting it
The impulse function is represented by d(t).
Its also called the delta function, or the dirac
delta function. One of its properties is that
its integral is equal to 1
d(t)
Its also important to note that d(t) 0
everywhere, except at t 0.
1
t
0
19
Impulse Function
The impulse is usually drawn as an arrow pointing
straight up. Remember, the height of the arrow
doesnt represent the height of the impulse,
which is infinity. The height of the arrow can
and usually does represent the area of the
impulse. The unit impulse d(t) has an area of 1
by definition, but it can be multiplied by a
constant to get a larger
or smaller impulse
Kd(t)
K
If the unit impulse d(t) is multiplied by K, the
area of Kd(t) is K.
t
0
20
Impulse Function
An impulse can also be advanced or delayed in
time, like all the other waveforms weve studied.
To delay an impulse by an amount t1, change the
argument from t to t t1 as shown. Obviously,
the impulse function is an idealization which
cant exist in the real world. No circuit can
generate a pulse which has zero duration and
infinite height. What use is the impulse?
Kd(t-t1)
K
t
t1
0
21
Impulse Function
In network analysis, the impulse function is
arguably the most important of the basic
waveforms. Even though an impulse can never be
generated in the real world, well find that we
can calculate the response of any network to an
impulse stimulus. This impulse response can then
be used to find the response of the network to
ANY stimulus.
Kd(t-t1)
The impulse is also useful as an analytical
modelling tool. We can use an impulse to set up
a networks initial conditions by instantaneously
charging energy storage elements (capacitors and
inductors). This is done to an analytic model of
the circuit, not the actual circuit.
K
t
t1
0
22
Impulse Function
The impulse can also be used analytically to
approximate very short pulses, which do occur in
the real world. The impulse may (will) turn up
in some of the mathematical manipulations and
derivations well be doing later in the course.
In short, although the impulse function doesnt
exist in the real world, it is very useful as an
analytic tool.
Kd(t-t1)
K
t
t1
0
23
Relationship of Step and Impulse
Now well examine the relationship between the
step function and the impulse function. You
should recall that the derivative of the ramp
function is the step function
It turns out that the derivative of the step
function is the impulse function
Kd(t-t1)
K
t
t1
0
24
Relationship of Step and Impulse
It also turns out that the integral from - to t
is a step
The definition of the unit impulse, d(t), has two
parts
d(t)
So its obvious that
1
t
0
25
Relationship of Step and Impulse
And its also obvious that
Now, lets define t 0- as the largest value of
t for which t lt 0, and define t 0 as the
smallest value of t for which t gt 0. Now, for
any t such that t gt 0,
d(t)
1
t
0
26
Relationship of Step and Impulse
And this can be extended to include an impulse
with area K
And we can summarize what weve found
d(t)
1
t
0
27
Relationship of Step and Impulse
We can summarize what weve found
In other words,
d(t)
1
t
0
28
Differentiation and Integration of Waveforms
Its often necessary to differentiate or
integrate a known waveform. If we have a plot of
the waveform from which slopes and areas can be
calculated, graphical differentiation or
integration may be most convenient.
29
Differentiation and Integration of Waveforms
Consider this waveform (which weve examined
elsewhere). It may be divided into four regions
t lt 0, 0 t lt 0.5, 0.5 t lt 1, and t gt 1.
30
Differentiation and Integration of Waveforms
Consider this waveform (which weve examined
elsewhere). It may be divided into four regions
t lt 0, 0 t lt 0.5, 0.5 t lt 1, and t gt 1.
The slope within each region can easily be found
by inspection. Its plotted here, and is the
same as the derivative waveform. Now all we need
to do is apply the techniques we learned earlier
to construct the derivative waveform out of step
functions.
31
Differentiation and Integration of Waveforms
And here it is
Now let try differentiating f(t) analytically.
32
Differentiation and Integration of Waveforms
We found earlier that the expression for f(t) is
Each term in this expression is of the form
Differential calculus tells us that the
derivative of g(t)h(t) is
33
Differentiation and Integration of Waveforms
Differential calculus tells us that the
derivative of g(t)h(t) is
Which we will now apply to each of the three
terms in
First,
The second part of this, 2td(t), reduces to 0
because d(t) is nonzero only for t 0.
Therefore,
34
Differentiation and Integration of Waveforms
The second term of f(t) is
Taking its derivative,
35
Differentiation and Integration of Waveforms
The third term of f(t) is
Taking its derivative,
Summing the derivatives of the three terms gives
us f(t)
This is the same expression we found by graphical
means.
36
Differentiation and Integration of Waveforms
Now lets try integrating the waveform shown
below. This waveform can be divided into five
regions t lt 0, 0 t lt 0.5, 0.5 t lt 1, 1 t
lt 1.5,
and t ³ 1.5.
37
Differentiation and Integration of Waveforms
Integrating the first region,
And the second
38
Differentiation and Integration of Waveforms
The third region
39
Differentiation and Integration of Waveforms
The fourth region
40
Differentiation and Integration of Waveforms
The fifth region
41
Differentiation and Integration of Waveforms
So the expression for the integral of this
waveform is the sum of four ramps
42
Differentiation and Integration of Waveforms
Here is f(t), along with its integral (in red)
43
Average and Effective (rms) Values of Periodic
Waveforms
Many interesting waveforms are periodic. A
periodic waveform repeats at intervals of T,
where T is the period of the waveform. In other
words, if f(t) is a periodic waveform and n is
any integer,
Quite often, we need to find the average value of
a waveform f(t). In general, the average value
is given by
Obviously this may be hard to evaluate
44
Average and Effective (rms) Values of Periodic
Waveforms
Fortunately, for periodic waveforms its a little
easier. We dont have to average the waveform
over all time, just over one period
Lets look at an example.
45
Differentiation and Integration of Waveforms
Examination of this waveform reveals that its
period is T 2 sec.
46
Differentiation and Integration of Waveforms
Well write an expression for this waveform
47
Average and Effective (rms) Values of Periodic
Waveforms
Now, well find the average
48
Average and Effective (rms) Values of Periodic
Waveforms
The waveform for which we just found the average
may be a current or Voltage waveform. If it
represents the Voltage across or the current
through a resistor, the resistor dissipates power
which is proportional to the square of the
Voltage or current
If the Voltage or current is a function of time,
so is power
49
Average and Effective (rms) Values of Periodic
Waveforms
Quite often, the average power is of interest.
If the Voltage or current waveform is periodic,
the power waveform is also periodic, so
The average power is equal to the average (or
mean) of the square of the Voltage waveform (also
called the mean square Voltage), divided by
resistance.
50
Average and Effective (rms) Values of Periodic
Waveforms
For a current waveform,
The average power is equal to the average (or
mean) of the square of the current waveform (also
called the mean square current), multiplied by
resistance.
51
Average and Effective (rms) Values of Periodic
Waveforms
Now well define a new quantity, called the
root-mean-square (or rms) value of Voltage or
current. This is simply the square root of the
mean-square Voltage or current, and is often
called the effective value of the Voltage or
current.
You can think of the rms Voltage or current as
the DC Voltage or current which would result in
the same average power being dissipated by a
resistor as the Voltage or current waveform for
which the rms value was found.
52
Average and Effective (rms) Values of Periodic
Waveforms
Here we see the same waveform for which we
recently calculated the average value. Its a
Voltage waveform, v(t), and v2(t) is also shown.
The period is 2, so we graphically integrate
v2(t) over the range 0 t lt2 and and divide
by 2, yielding

53
Average and Effective (rms) Values of Periodic
Waveforms
The rms value of a sinusoid with amplitude A,
is given by
It should be obvious that this relationship
between amplitude and rms value is only true for
a sinusoid and is not true for any other
waveform. A DVM set to measure AC Voltage or
current will read the correct rms value only for
a sinusoidal Voltage or current. For any other
waveform (e.g., sawtooth or triangle) the correct
rms value is only obtained if a true rms
instrument is used.