Consider a RLC circuit in which resistor, inductor and capacitor are connected in series across a voltage supply. This series RLC circuit has an distinguishing property of resonating at a specific frequency called resonant frequency. In this circuit containing inductor and capacitor the energy is stored in two different ways.
We know that inductive reactance XL = 2πfL it means that inductive reactance is directly proportional to frequency ( XL ∝ ƒ ). When the frequency is zero or in case of DC, inductive reactance is also zero, the circuit act as a short circuit but when frequency increases inductive reactance also increases. At infinite frequency inductive reactance become infinity and circuit behave as open circuit. It means that when frequency increases inductive reactance also increase and when frequency decrease inductive reactance also decrease. So if we plot a graph between inductive reactance and frequency, it is a straight line linear curve passing through origin as shown in figure above
It is clear from the formula of capacitive reactance, XC = 1 / 2πfC that frequency and capacitive reactance are inversely proportional to each other. In case of DC or when frequency is zero, capacitive reactance becomes infinity and circuit behave as open circuit and when frequency increases and become infinite, capacitive reactance decrease and become zero at infinite frequency, at that point the circuit act as short circuit so, the capacitive reactance increases with decease in frequency and if we plot a graph between capacitive reactance and frequency it is an hyperbolic curve as shown in figure above.
From the above discussion it is concluded the inductive reactance is directly proportional to frequency and capacitive reactance is inversely proportional to frequency i.e at low frequency XL is low and XC is high but there must be a frequency, where the value of inductive reactance becomes equal to capacitive reactance. Now if we plot a single graph of inductive reactance vs frequency and capacitive reactance vs frequency then there must occur a point where these two graph cut each other. At that point of intersection the inductive and capacitive reactance becomes the equal and the frequency at which these two reactances become equal is called resonant frequency, fr
At resonant frequency, XL = XL
At resonance in series RLC circuit, two reactances become equal and cancel each other. So in resonant series RLC circuit, the opposition to the flow of current is due to resistance only. At resonance the total impedance of series RLC circuit is equal to resistance i.e Z = R, impedance has only real part but no imaginary part and this impedance at resonant frequency is called dynamic impedance and this dynamic impedance is always less than impedance of series RLC circuit. Before series resonance i.e before frequency, fr capacitive reactance dominate and after resonance inductive reactance dominate and at resonance the circuit act as purely resistive circuit causing a large amount of current to circulate through the circuit.
In series RLC circuit the total voltage is the phasor sum of voltage across resistor, inductor and capacitor. At resonance in series RLC circuit both inductive and capacitive reactance cancel each other and we know that in series circuit the current flowing through all the elements is same So, the voltage across inductor and capacitor is equal in magnitude and opposite in direction and thereby they cancel each other. So, in a series resonant circuit voltage across resistor is equal to supply voltage i.e V = Vr
In series RLC circuit current, I = V / Z but at resonance current I = V / R , therefore the current at resonant frequency is maximum as at resonance in , impedance of circuit is resistance only and is minimum.
At resonance, the inductive reactance is equal to capacitive reactance and hence the voltage across inductor and capacitor cancel each other. The total impedance of circuit is resistance only. So, the circuit behave like a pure resistive circuit and we know that in pure resistive circuit voltage and the circuit current are in same phase i.e Vr , V and I are in same phase direction . Therefore the phase angle between voltage and current is zero and the power factor is unity.
• When a current flows in a inductor ,energy is stored in magnetic field.
• When a capacitor is charged, energy is stored in electric field.
• When a capacitor is charged, energy is stored in electric field.
The magnetic field in the inductor is build by the electric current, which is provided by the discharging capacitor. Similarly the capacitor is charged by the electric current produced by collapsing magnetic field of inductor and this process continue on and on, causing electrical energy to oscillate between the magnetic field and the electric field .In some cases at certain frequency called resonant frequency, the inductive reactance of the circuit becomes equal to capacitive reactance which cause the electrical energy to oscillate between the electric field of the capacitor and magnetic field of the inductor. This forms a harmonic oscillator for current. In RLC circuit, the presence of resistor causes these oscillation to die out over period of time and is called damping effect of resistor.
Variation in Inductive Reactance and Capacitive Reactance with Frequency
Variation of Inductive Reactance Vs Frequency
Variation of Inductive Reactance Vs Frequency
We know that inductive reactance XL = 2πfL it means that inductive reactance is directly proportional to frequency ( XL ∝ ƒ ). When the frequency is zero or in case of DC, inductive reactance is also zero, the circuit act as a short circuit but when frequency increases inductive reactance also increases. At infinite frequency inductive reactance become infinity and circuit behave as open circuit. It means that when frequency increases inductive reactance also increase and when frequency decrease inductive reactance also decrease. So if we plot a graph between inductive reactance and frequency, it is a straight line linear curve passing through origin as shown in figure above
Variation of Capacitive Reactance Vs Frequency
Variation of Capacitive Reactance Vs Frequency
It is clear from the formula of capacitive reactance, XC = 1 / 2πfC that frequency and capacitive reactance are inversely proportional to each other. In case of DC or when frequency is zero, capacitive reactance becomes infinity and circuit behave as open circuit and when frequency increases and become infinite, capacitive reactance decrease and become zero at infinite frequency, at that point the circuit act as short circuit so, the capacitive reactance increases with decease in frequency and if we plot a graph between capacitive reactance and frequency it is an hyperbolic curve as shown in figure above.
Inductive Reactance and Capacitive Reactance Vs Frequency
Inductive Reactance and Capacitive Reactance Vs Frequency
From the above discussion it is concluded the inductive reactance is directly proportional to frequency and capacitive reactance is inversely proportional to frequency i.e at low frequency XL is low and XC is high but there must be a frequency, where the value of inductive reactance becomes equal to capacitive reactance. Now if we plot a single graph of inductive reactance vs frequency and capacitive reactance vs frequency then there must occur a point where these two graph cut each other. At that point of intersection the inductive and capacitive reactance becomes the equal and the frequency at which these two reactances become equal is called resonant frequency, fr
At resonant frequency, XL = XL
At resonance f = fr and on solving above equation we get,
Variation of Impedance Vs Frequency
Variation of Impedance Vs Frequency
At resonance in series RLC circuit, two reactances become equal and cancel each other. So in resonant series RLC circuit, the opposition to the flow of current is due to resistance only. At resonance the total impedance of series RLC circuit is equal to resistance i.e Z = R, impedance has only real part but no imaginary part and this impedance at resonant frequency is called dynamic impedance and this dynamic impedance is always less than impedance of series RLC circuit. Before series resonance i.e before frequency, fr capacitive reactance dominate and after resonance inductive reactance dominate and at resonance the circuit act as purely resistive circuit causing a large amount of current to circulate through the circuit.
Resonant Current
Resonant Current
In series RLC circuit the total voltage is the phasor sum of voltage across resistor, inductor and capacitor. At resonance in series RLC circuit both inductive and capacitive reactance cancel each other and we know that in series circuit the current flowing through all the elements is same So, the voltage across inductor and capacitor is equal in magnitude and opposite in direction and thereby they cancel each other. So, in a series resonant circuit voltage across resistor is equal to supply voltage i.e V = Vr
In series RLC circuit current, I = V / Z but at resonance current I = V / R , therefore the current at resonant frequency is maximum as at resonance in , impedance of circuit is resistance only and is minimum.
The above graph shows the plot between circuit current and frequency. At starting when the frequency increases, the impedance Zc decrease and hence the circuit current increases. After some time frequency becomes equal to resonant frequency at that point inductive reactance become equal to capacitive reactance and the impedance of circuit reduces and is equal to circuit resistance only. So at this point the circuit current becomes maximum I = V / R. Now when the frequency is further increased, ZL increases and with increase in ZL , the circuit current reduces and then the current drop finally to zero as frequency becomes infinite.
Power Factor at Resonance
Power Factor at Resonance
At resonance, the inductive reactance is equal to capacitive reactance and hence the voltage across inductor and capacitor cancel each other. The total impedance of circuit is resistance only. So, the circuit behave like a pure resistive circuit and we know that in pure resistive circuit voltage and the circuit current are in same phase i.e Vr , V and I are in same phase direction . Therefore the phase angle between voltage and current is zero and the power factor is unity.
Application of Series RLC Resonant circuit
Since resonance in Series RLC Circuit occurs at particular frequency so, it is used for filtering and tuning purpose as it does not allow unwanted oscillations that would otherwise cause signal distortion , noise and damage to circuit to pass through it.
Summary
For a series RLC circuit at certain frequency called resonant frequency, the following points must be remembered. So at resonance :
For a series RLC circuit at certain frequency called resonant frequency, the following points must be remembered. So at resonance :
Inductive reactance XL is equal to capacitive reactance XC
Total impedance of circuit becomes minimum which is equal to R i.e Z = R
Circuit current becomes maximum as impedance reduces, I = V / R
voltage across inductor and capacitor cancels each other, so voltage across resistor Vr = V, supply voltage
Since net reactance is zero, circuit becomes purely resistive circuit and hence the voltage and the current are in same phase , so the phase angle between them is zero
Power factor is unity
Frequency at which resonance in series RLC circuit occur is given by
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