Circuit Analysis Using Fourier Series ⭐ RL Circuit Response - Nonsinusoidal Waveform ⭐ Example 1
Summary
TLDRThis video introduces a series on circuit analysis, focusing on the application of Fourier series to analyze non-sinusoidal input signals. The first example explores an RL circuit with a given input voltage expressed as a Fourier series. The video explains how to determine the output signal across the inductor, considering both DC and AC components. It also discusses the circuit's behavior at low and high frequencies and uses the voltage divider rule to find the output voltage. The analysis includes calculating amplitude and phase shifts, transforming the frequency domain solution into the time domain, and plotting the amplitude and phase spectra for a clear understanding of the output signal.
Takeaways
- 🎓 The video series focuses on circuit analysis, specifically discussing the application of Fourier series.
- 🔍 The Fourier series is introduced as a tool for analyzing non-sinusoidal input signals by breaking them into smaller parts.
- 📚 Example 1 involves an RL circuit with a given input voltage represented by a Fourier series expression.
- 🔧 The circuit's response to the input voltage is analyzed, with the output voltage across the inductor being the focus.
- ⚡ The input voltage's Fourier series includes a DC term and frequency-dependent terms (harmonics), with odd values for n.
- 🌉 At very low frequencies, the circuit acts as a short circuit, and at very high frequencies, it acts as an open circuit, making it a high-pass filter.
- 📉 The output voltage is calculated using the voltage divider rule, considering the reactance of the inductor.
- 🔄 The DC component of the output voltage is zero because inductors short out DC.
- 📊 The amplitude and phase of the AC terms are analyzed, with the amplitude decreasing and the phase approaching -90 degrees as frequency increases.
- 📈 The output signal's amplitude and phase spectra are plotted to visualize the signal's behavior across different frequencies.
- 🔗 The video concludes with a summary of the output voltage determination and the plotted spectra, emphasizing the rapid decrease in amplitude for higher harmonics.
Q & A
What is the main topic of the video series?
-The main topic of the video series is circuit analysis, with a focus on discussing the Fourier series in the context of analyzing circuits.
Why is the Fourier series useful in circuit analysis?
-The Fourier series is useful in circuit analysis when dealing with non-sinusoidal input signals, as it allows for breaking down complex signals into smaller, more manageable parts for analysis.
What type of circuit is being analyzed in the first example?
-In the first example, an RL (Resistor-Inductor) circuit is being analyzed, where the input voltage is given by a Fourier series expression.
What are the values of R and L in the given RL circuit?
-The values of R (resistor) and L (inductor) in the given RL circuit are 10 ohms and 4 Henries, respectively.
What is the significance of the DC term in the Fourier series expression of the input voltage?
-The DC term in the Fourier series expression of the input voltage represents the frequency-independent term, which is a constant value separate from the frequency-dependent harmonics.
How does the value of 'k' in the Fourier series expression affect the harmonics?
-The value of 'k' in the Fourier series expression determines the frequency of the harmonics. As 'k' increases, the frequency of the harmonics (n = 2k - 1) also increases, resulting in odd harmonic values such as 1, 3, 5, 7, etc.
What is the fundamental frequency (ω0) in the context of this video?
-The fundamental frequency (ω0) in this context is given by π radians per second, which is the frequency of the first harmonic in the Fourier series.
Why does the output voltage across the inductor approach zero for very low frequencies?
-For very low frequencies, the inductor behaves like a short circuit, which means the output voltage across it approaches zero because the current flows through the path of least impedance, bypassing the inductor.
How is the output voltage of the circuit calculated?
-The output voltage of the circuit is calculated using the voltage divider rule, which states that the output voltage is the input voltage multiplied by the ratio of the reactance of the inductor to the total impedance of the circuit.
What is the significance of the amplitude and phase of the input voltage in the analysis?
-The amplitude and phase of the input voltage are significant because they determine the magnitude and timing of the voltage waveform. In the analysis, these properties are used to calculate the response of the circuit to each harmonic component of the input signal.
How does the video script describe the transition from frequency domain to time domain for the output voltage?
-The script describes the transition from frequency domain to time domain by expressing the output voltage as a sum of cosine terms, each with its amplitude and phase shift, which are derived from the analysis in the frequency domain.
What is the purpose of plotting the amplitude and phase spectrum of the output voltage?
-Plotting the amplitude and phase spectrum of the output voltage helps visualize the contribution of each harmonic to the total output signal and understand how the circuit responds to different frequencies.
Why are even terms not present in the output for this specific RL circuit?
-Even terms are not present in the output for this specific RL circuit because the input signal, as represented by its Fourier series, only contains odd harmonics, and the circuit's response does not generate even harmonics.
Outlines
📚 Introduction to Circuit Analysis and Fourier Series
The video begins by introducing a new series on circuit analysis, focusing on the application of Fourier series. The presenter outlines that the series will cover various circuits and analyze their responses to specific input voltages using Fourier series. The utility of Fourier series is highlighted for analyzing non-sinusoidal input signals by breaking them into smaller parts. The presenter then introduces the first example involving an RL circuit with a given input voltage expressed as a Fourier series. The series includes a DC term and frequency-dependent terms (harmonics). The importance of understanding the circuit's behavior at low and high frequencies is discussed, identifying the circuit as a high-pass filter. The fundamental frequency and the reactance of the circuit components are also introduced, setting the stage for a detailed analysis of the output voltage.
🔍 Analyzing the Output Voltage Using Fourier Series
The second paragraph delves into the calculation of the output voltage across the inductor in the RL circuit. The presenter explains the use of the voltage divider rule to relate the output voltage to the input voltage and the circuit's impedance. The focus is on calculating the reactance of the inductor and how it affects the output voltage. The DC component of the input voltage is analyzed, showing that the output voltage is zero at DC due to the inductor's short-circuit behavior at low frequencies. The presenter then moves on to analyze the AC components, discussing the amplitude and phase of the input voltage's harmonics. The process of converting the sine form of the harmonics to a cosine form for easier analysis is explained. The amplitude and phase of the output voltage are derived from the input voltage expression, providing a complex expression for the output voltage in terms of the circuit parameters and frequency.
📊 Examining the Output Signal's Frequency Components
In this section, the presenter examines the output signal's frequency components by calculating the output voltage for the first few harmonics. The process involves substituting specific values of 'n' into the derived formula to obtain the amplitude and phase for each harmonic. The results show a decreasing amplitude and a phase that approaches -90 degrees as the frequency increases. The presenter then discusses the significance of plotting the amplitude and phase spectra, which provide a visual representation of the output signal's behavior across different frequencies. The absence of even harmonics in the output signal is noted, and the rapid decrease in amplitude for higher frequencies is highlighted. The analysis concludes with a discussion on the accuracy of the results and the potential for including more terms to refine the analysis.
🎓 Conclusion and Preview of Next Example
The video concludes with a summary of the first example, where the presenter has successfully determined the output voltage using Fourier series and plotted the amplitude and phase spectra. The presenter expresses gratitude for the viewers' attention and provides a preview of the next example, promising a continuation of the series with a different circuit and input signal. The summary emphasizes the step-by-step approach taken in the analysis and encourages viewers to look forward to the next installment of the series.
Mindmap
Keywords
💡Fourier Series
💡Circuit Analysis
💡Input Voltage
💡Reactance
💡Resistor (R)
💡Inductor (L)
💡Output Signal
💡Frequency Domain
💡Amplitude
💡Phase
💡Harmonics
Highlights
Introduction to a new series on circuit analysis focusing on Fourier series.
Explanation of the utility of Fourier series in analyzing non-sinusoidal input signals.
Discussion on breaking down complex problems into smaller parts using Fourier series.
Introduction to the first example involving an RL circuit with a given input voltage.
Description of the input voltage as a Fourier series expression.
Explanation of the DC term and frequency-dependent terms (harmonics) in the Fourier series.
Identification of the odd values of n in the harmonics.
Analysis of the circuit's behavior at very low and high frequencies.
Prediction of the output voltage across the inductor without detailed analysis.
Introduction to the fundamental frequency and its calculation.
Application of circuit analysis tools to derive the output voltage expression.
Explanation of the voltage divider rule in the context of the given circuit.
Calculation of the input voltage's phasor and its impact on the output voltage.
Discussion on the DC component of the input voltage and its effect on the output.
Analysis of the AC part of the input voltage and its response in the circuit.
Conversion of the input voltage expression to cosine form for easier analysis.
Derivation of the amplitude and phase of the AC input voltage using phasors.
Substitution of the AC input voltage into the output voltage expression.
Calculation of the output voltage's amplitude and phase shift.
Transformation of the output voltage from frequency domain to time domain.
Explanation of the decreasing amplitude and approaching phase shift with increasing frequency.
Conclusion of the first example and a preview of the next example.
Emphasis on the importance of plotting the amplitude and phase spectrum for analysis.
Observation that even terms are not present in the output for this circuit.
Discussion on the rapid decrease in amplitude and the approach to a -90 degrees phase shift.
Summary of the exercise, including determining the output voltage and plotting the spectrum.
Encouragement for viewers to continue to the next example for further learning.
Transcripts
hi everyone and welcome to a new series
about the circuit analysis and we will
discuss the fourier series in this
analysis
and this is our example number one what
i will do in this video series of
circuit analysis during 3a series we
will discuss several circuit like
circuits and we will apply an input
voltage specific input voltage and we
will look at the response and for that
we will use the fourier series
why we use a fourier series is very
handy if you have a non-sinusoidal input
signal like this
and if you want to break up the problem
in very small parts then the fourier
series is a very
handy tool to work out the analysis we
will see that shortly in more detail
when we go through the example and this
is for example number one and we will
work out another example and example
number two and we will see different
problems so let's jump to our first
example
what do we have we have the following
circuits we have input vs which is our
input voltage which is given by this
fourier
series expression i will go in more
detail shortly
we have a resistor and we have the
inductor in series so we have actually
rl circuit and that's connected to the
output input voltage and the output is
across the inductor
what we want and we have the r and l
already given 10 ohms and four areas for
the inductor
we would like to determine the output
signal v out if the input is given by
this expression so this
is the fourier series of a expression of
a
signal which is of course given in this
mathematical form what you see in this
expression is that the dc term is just
this separate constant term which is the
frequency independent term and these are
the frequency dependent terms that also
call the harmonics
i can see that the harmonics are
dependent on the value of k k will start
at one and will all go to
infinite t so it'll actually mean that
there are infinite terms here so dc term
plus a lot of ac terms
the n in this case which is shown in the
expression for the harmonics is given by
2k minus one for this specific signal
doesn't have to be all the time
so if k is one this will be two times
one minus one will be one if the k is
two this will be two times two minus one
will be three in case three that will be
two times three minus one it would be
five so that means actually the n is
actually odd value so it will be one
three
five seven et cetera
so we will see what kind of response we
have just looking at this input voltage
and also
the resistor and inductor value for this
specific circuit
before we move on we can also say what
kind of circuit is this if i look at the
circuit i see for very
low frequencies for dc
this is actually short so it is actually
that due to the reactance of the
inductor
and if this is for the very high
frequencies it will be an open circuit
so actually for very low frequencies
when i have a shortage means actually
the output will be a zero that means the
very low frequency will be attenuated so
that's actually a high pass voltage just
looking at the circuit without doing any
detailed x analysis so let's look at the
solution because we want to of course
work towards the output signal
so the radio frequency let's first have
that one that is actually very important
it's also called the fundamental
frequency in this case
is given by m pi which is also given
here so we have the omega 0 which is
called the
fundamental frequency it is just pi and
n is what we just discussed is the value
will change depending on this case the k
so we will use then the
circuit analysis tools to work out
towards the output voltage expression
using the input voltage and also the
circuit itself so using vaser we can
write down for the output voltage of the
following using voltage divided rule so
the output has a reactance divided by
the total impedance and times the input
voltage is equal to the output voltage
which is this is just very
straightforward
voltage divider rule and if i just work
out the reactance i would have to j
omega n
l and also for the example exact same
expression in the denominator
if i write down this in more detail just
using the given values i have this
expression this is just a complex
expression we have seen the output now
is dependent on the input using this
dynamics of the circuit
now we will look first again to dc means
actually there is no
ac term involved that means actually
there's just what i have for the input
so n is zero that means also omega n is
zero because the omega n is related to n
so if n is zero that will be omega n is
zero then we have the following input
in phasor is just a half
but the output is zero and we have
discussed this shortly why if the
frequency is zero hertz or zero radians
per second doesn't matter then the
inductor will be short so inductors are
shorted
so just a
wire here and if you
want to calculate or measure the voltage
across a wire that will be of course
zero that is actually why we have a v
out of zero so this actually for just
the dc
if i now move on with ac part so the end
harmonics for this what is the response
of these terms that will be the
following we will look at the vs for the
input voltage actually and just looking
at this expression now we have to do
something
uh very
structured because we will need to work
it out in phasors that means we need to
look at the amplitude and also the phase
of this expression
the amplitude is given by the 2 divided
by pi and also the n also in the
denominator that's actually what you see
here
the face
is minus 0 degrees
why because for the calculations of
the signals the cosine is used as a
reference signal it is
of course not really mandatory but it is
handy if you convert your signals first
to cosine so that's actually what i have
done so to rewrite this
in the cosine form i really need to
place a minus 90 degrees in the cosine
so i will replace this by cosine n by t
minus 90 degrees that will result in
exact same amplitude shown here because
i will place just n here outside the
summation and i have then the phase of
minus 90 degrees due to that
change from the sine to cosine
expression
now if i use
this phase and i will use
the
rectangular form
formula we have discussed this in a
separate video about vases and complex
numbers i have this expression so the
sine of minus nine degrees will be minus
one this will be just 0 and i have this
expression for my
input voltages looking only at the
ac terms so the harmonics so you can see
if n changes this will change
now i need of course this expression for
my ac values because i know the dc but i
don't know the ac yet the v out was just
this expression and it will depend on
the vs which is now given by this
expression so what i will do is next i
will use this expression and i will
substitute the vs which is shown here in
here so i have done this here
shown here
now if i work this out by the
denominator the numerator first i can
also see the following maybe it's better
to discuss this first
and
pi in the numerator and also n pi here
that will cancel each other out
i have a minus j 2 and i have a j4 it
will result in a plus 8 just real value
i have only this in the denominator
shown here and i have only eight in the
numerator
of course i want to have an amplitude
and a phase expression for this for this
complex expression so this will be eight
over the
length of this expression and the phase
shift will be given by this expression
minus arctangent of 4 and pi divided by
10. if you work it out a little bit in
more detail and also write that down as
100 and also 16 n
squared pi squared i have this
expression and this will be of course a
little bit simplified in this form
i will now have this amplitude and also
this phase now i will now
go from the frequency domain to the time
domain so in time domain the v out will
be
this
why because i look at the amplitude and
also at the face only and i will just
place in a template which is just a
cosine of this radian frequency and this
will be just the face which is shown
here
and this is just the amplitude here and
i will place it in front of this cosine
expression that's actually what you need
to do and of course it's dependent on
the n and that is related to k
if i want to rewrite this in a more
detail form
then this will be our expression for the
v out which we'll see in a summation
form
now of course we can work out here many
many terms which is which also can see
from the uh summation term because it
goes from k to equal to one all the way
to infinity so we can work out maybe
10 20 maybe 40 terms maybe 100 terms it
really depends on actually what your
accuracy will be but if i look at the
first four terms you will see what kind
of
sequence we will get so i will look at
case one two three four that means
actually in n
one three five seven and these are the
harmonics
so let's look at the
details case one will mean n is one the
v out which will be given now in the
first term as a green one we'll be just
using this formula
and we'll just substitute n s1
n is one and n is one that is actually
the result and if you work it out you
will get approximately 0.5 cosine pi t
and minus 52 degrees that's just the
first term for k is one
so that's actually the first harmonic
now the second harmonic that means
actually for nf3 specifically the out
the second one will be again just
substitute for ns3 and three and for
this also three you will get this
expression and you will have
approximately 0.21 and a cosine of 3 pi
t minus 57 i mean 75 degrees
so if you of course continue with your
next one k3 and s5 you will have exact
the same procedure
and for the next one and these are the
four
terms i have actually now determined
using just the formula which is shown
here so i have now in total five terms
four from the harmonics and one from the
dc
term which is from the
analysis here what do i do next i will
just collect those five terms four plus
one and i'll i will have this because
this is just zero i will of course just
skip that and i have just v out one all
the way to v out four of course it will
continue all the way to the out infinity
and i have
i don't want i don't want to write them
down all of them because i see already
in the amplitude fashion because it's
sort of 0.5 approximately then it goes
down to 0.21 and then goes down to 0.13
approximately then it goes to 0.09 you
can see that this is really decreasing
very rapidly
you can also see if the ink frequency
just increases the phase is also
approaching minus 90 degrees so you can
also see that from this analysis
so what you see now is in the v out
expression total is this expression this
is just the first term second term
third term and the fourth term of course
i might have the fifth and the sixth
term if you want more accuracy in your
analysis but i already see this is quite
accurate for the first four terms
already
so let's look at the next analysis and
we will discuss in more detail what we
have this is actually the output signal
spectrum which is very helpful for this
specific signal
because if i look at the signal we have
just determined we can plot the
amplitude spectrum and also the phase
spectrum so this is the amplitude
spectrum
this is the phase spectrum of the v out
output voltage and that's just the
amplitude of that
output signal what you see is that at 0
which is a dc there is no output that
means just zero and the phase shift was
also zero degrees so you can see that
clearly in this case
the
at frequency pi radius per second was a
0.498
volts as shown here and the associated
phase was minus 50 degrees you can also
see that here
so if you if you don't know that
anymore you can see that in the output
expression
if you go to the next expression which
is the next turn which is the zero point
205 which is the amplitude and the
associated phase phase is
minus
75 degrees
etc you can then look at the blue one
and also the red one you can follow
actually the amplitude and also the face
in this spectrum
what you also see is that we don't have
the even terms in this circuit so the
even terms will be
not present
at the output for this circuit
and you also see again it will be
much easier to see in this graphical
form but the amplitude will decrease
very rapidly so if i go to for example 9
pi 11 pi
13 pi then you will see that this
amplitude will be even more even smaller
so that will mean that actually the
contribution will be very insignificant
and the facial will approach actually an
isotopic phase
minus 90 degrees for this so that's
actually what we have for this
exercise so i've determined my v out
i've plotted also the spectrum and it
will be giving of course some more
detail and if you want to more accuracy
again you can add the fifth the sixth
term or even more terms if you want to
have it more detailed and also more
accuracy
okay we have now concluded our first
example about circuit analysis using
fourier series i will continue in the
next example example number two and use
a different circuit and we will look at
a different input signal and we will
work it out again step by step looking
how to work it out for the output
response
thanks again for your attention and see
you next time and take care
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