Question

Explain series resonance and derive resonant frequency

Answer 1

Thus far we have analysed the behaviour of a series RLC circuit whose source voltage is a fixed frequency steady state sinusoidal supply. We have also seen in our tutorial about series RLC circuits that two or more sinusoidal signals can be combined using phasors providing that they have the same frequency supply.

But what would happen to the characteristics of the circuit if a supply voltage of fixed amplitude but of different frequencies was applied to the circuit. Also what would the circuits “frequency response” behaviour be upon the two reactive components due to this varying frequency.

In a series RLC circuit there becomes a frequency point were the inductive reactance of the inductor becomes equal in value to the capacitive reactance of the capacitor. In other words, XL = XC. The point at which this occurs is called the Resonant Frequency point, ( ƒr ) of the circuit, and as we are analysing a series RLC circuit this resonance frequency produces a Series Resonance.

Series Resonance circuits are one of the most important circuits used electrical and electronic circuits. They can be found in various forms such as in AC mains filters, noise filters and also in radio and television tuning circuits producing a very selective tuning circuit for the receiving of the different frequency channels. Consider the simple series RLC circuit below.

Series RLC Circuit

Firstly, let us define what we already know about series RLC circuits.

From the above equation for inductive reactance, if either the Frequency or the Inductance is increased the overall inductive reactance value of the inductor would also increase. As the frequency approaches infinity the inductors reactance would also increase towards infinity with the circuit element acting like an open circuit.

However, as the frequency approaches zero or DC, the inductors reactance would decrease to zero, causing the opposite effect acting like a short circuit. This means then that inductive reactance is “Proportional” to frequency and is small at low frequencies and high at higher frequencies and this demonstrated in the following curve:

Inductive Reactance against Frequency

The graph of inductive reactance against frequency is a straight line linear curve. The inductive reactance value of an inductor increases linearly as the frequency across it increases. Therefore, inductive reactance is positive and is directly proportional to frequency ( XL ∝ ƒ )

The same is also true for the capacitive reactance formula above but in reverse. If either the Frequency or the Capacitance is increased the overall capacitive reactance would decrease. As the frequency approaches infinity the capacitors reactance would reduce to practically zero causing the circuit element to act like a perfect conductor of 0Ω.

But as the frequency approaches zero or DC level, the capacitors reactance would rapidly increase up to infinity causing it to act like a very large resistance, becoming more like an open circuit condition. This means then that capacitive reactance is “Inversely proportional” to frequency for any given value of capacitance and this shown below:

Capacitive Reactance against Frequency

The graph of capacitive reactance against frequency is a hyperbolic curve. The Reactance value of a capacitor has a very high value at low frequencies but quickly decreases as the frequency across it increases. Therefore, capacitive reactance is negative and is inversely proportional to frequency ( XC ∝ ƒ -1 )

We can see that the values of these resistances depends upon the frequency of the supply. At a higher frequency XL is high and at a low frequency XC is high. Then there must be a frequency point were the value of XL is the same as the value of XC and there is. If we now place the curve for inductive reactance on top of the curve for capacitive reactance so that both curves are on the same axes, the point of intersection will give us the series resonance frequency point, ( ƒr or ωr ) as

Answer 2

$\underline{Text}$

$\boxed{Text}$

$\coloured{red}{Text}$

$\hug{Text}$

$\small{Text}$

$\large{Text}$

_______♦♦☺♦♦________

$\huge{\red{Hello...frd}}$

$\bf{here..is..ur.. answer}$

_______♦♦☺♦♦________

$<b>$

_______♦♦☺♦♦________

$\bf{By-RakeshPatel}$

_______♦♦☺♦♦________

Title:

Content:

_______♦♦☺♦♦________

$\mathbb{\red{\huge{\:\:\:HAPPY\:\:\:HOLI}$

_______▶▶♦⚪♦▶▶________

$\huge\mathfrak\purple{hello\:frd}$

$\bf\underline{here..is..ur.. answer}$

_______♦♦☺♦♦________

_______♦♦☺♦♦________

$\bf\underline{By-RakeshPatel}$

$\sf{I\: Hope\: It\: Is\: Helpful\: For\: you!}$

$\huge\mathfrak\red{thank\:u}$

_______♦♦♥♦♦________

_______♦♦♥♦♦________

$<b>____________♦♦♥♦♦____________$

$\huge\mathfrak\purple{\:\:\:\:\:\:\:\:hello\:\:\:mate}$

$<b>____________♦♦♥♦♦____________$

$\tt{\pink{\huge{here\:is\:your\:answer}}}$

$<b>____________♦♦♥♦♦____________$

Cisplatin is a chemotherapy medication used to treat a number of cancers.

Synonyms: Cisplatinum, platamin, neoplatin, cismaplat.

$<b>____________♦♦☺♦♦____________$

$\sf\purple{By-\:\:\:RAKESH\:\:PATEL}$

$<b>________▶▶▶⭐⭐⭐◀◀◀________$

_______♦♦☺♦♦________

$\mathbb{\red{\huge{U\:MEAN\:SO\:MUCH\:2\: ME}}}$

_______▶▶♦⚪♦▶▶________

$\huge\mathfrak\purple{hello\:frd}$

$\bf\underline{here..is..ur.. answer}$

_______♦♦☺♦♦________

$<b>$

_______♦♦☺♦♦________

$\bf\underline{By-RakeshPatel}$

$\bf\underline{Life\:Is\:Too\:Short.\:Don't\:Waste\:It\:Reading\:My\:Status...!!}$

$\sf{I\: Hope\: It\: Is\: Helpful\: For\: you!}$

$\huge\mathfrak\red{thank\:u}$

_______♦♦♥♦♦________

_______♦♦♥♦♦________

Title:

Content:

$\huge\{\boxed{\boxed{\red{hello\:frd}$

$\tt\pink{huge{hello\:frd}}$

$\sf{hello\:frd}$

$\mathfrak{huge{underline {hello\:frd}}}$

Title:

Brainly Fonts

Content:

$\huge\mahfrak{Hello\:Mate}$

$\huge\red{hello}$