Circuit Analysis Using Fourier Series ⭐ RL Circuit Response - Nonsinusoidal Waveform ⭐ Example 1

Summary

This 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

00:00

📚 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.

05:02

🔍 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.

10:02

📊 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.

15:04

🎓 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.

Keywords

💡Fourier Series

The Fourier Series is a mathematical tool used to represent a function as a sum of sine and cosine functions. It's central to the video's theme as it's applied to analyze non-sinusoidal input signals in circuit analysis. The script mentions using the Fourier Series to break up complex signals into simpler, analyzable parts, exemplified by the input voltage signal being expressed as a Fourier series.

💡Circuit Analysis

Circuit Analysis refers to the study of electrical circuits to comprehend their behavior under various conditions. The video series is dedicated to this concept, focusing on how different circuits respond to input signals. The script discusses analyzing RL circuits with a given input voltage to determine the output voltage.

💡Input Voltage

Input Voltage is the electrical potential difference supplied to a circuit, which drives the current. In the context of the video, the input voltage is described by a Fourier series expression, indicating a non-sinusoidal waveform. It's the starting point for analyzing the circuit's response.

💡Reactance

Reactance is the opposition to the flow of an alternating current, caused by capacitors and inductors in a circuit. The script explains that at very low frequencies, the inductor's reactance is high, effectively 'shorting' the circuit, leading to zero output voltage for low frequencies.

💡Resistor (R)

A Resistor is a passive component in a circuit that opposes the flow of electric current, dissipating electrical energy as heat. In the video, an RL circuit is analyzed, where 'R' is one of the components. The script specifies a resistor value of 10 ohms.

💡Inductor (L)

An Inductor is a component that stores energy in a magnetic field when current flows through it. The video discusses an RL circuit where 'L' is the inductor. The inductor's value is given as four henries, and its behavior is key to understanding the circuit's response at different frequencies.

💡Output Signal

The Output Signal is the voltage or current measured at the output of a circuit in response to an input signal. The script's main goal is to determine this output signal across the inductor for a given input voltage using Fourier series.

💡Frequency Domain

The Frequency Domain refers to the analysis of signals in terms of frequency rather than time. The video uses the frequency domain to analyze the circuit's response to different frequency components of the input signal. The script transitions from time domain to frequency domain to simplify the analysis.

💡Amplitude

Amplitude is the magnitude of a signal's oscillation, which is crucial in understanding the strength of the signal. The script calculates the amplitude of the output signal for different harmonics, showing how it decreases with increasing frequency.

💡Phase

Phase refers to the position of a point in a wave cycle, relative to a reference point. The script discusses phase shifts in the output signal, which are crucial for understanding how the output signal aligns with the input signal over time.

💡Harmonics

Harmonics are integer multiples of the fundamental frequency of a wave. The script mentions that the input signal's harmonics are dependent on the value of 'k', and these are analyzed to determine the output signal's behavior at different frequencies.

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