RLC Series Resonance

A brief session including the introduction about the RLC resonance circuit placed in series. Also discussed about the mathematical calculations and verification of formula. Circuit diagrams are included as well as the graphs to enhance the better view for the viewer. This is a wholesome package for the engineers at the first stage of their aim.

1. Resonance in R-L-C
Series Circuit
Represented By: Arijit Dhali –Ipsita Raha

2. 01
Epilogue &
Introduction
Information
& Proof
02
Formulas
& Graph
03
Conclusion
& References
04
Table of contents

3. ResonanceinACcircuitsimpliesaspecialfrequency
determinedbythevaluesoftheresistance,capacitance,and
inductance.Forseriesresonancetheconditionofresonanceis
straightforwardanditischaracterizedbyminimumimpedance,
maximumcurrentandzerophase.
Introduction

4. R-L-C Series Circuit
Let us consider a R-L-C series circuit mentioned below.
The circuit shown in figure is connected to an a.c. source of constant supply voltage Vbut having
variable frequency. The frequency can be varied from zero, increasing and approaching infinity.

5. Observation on Impedance
The Impedance of Circuit is given by,
|Z| = ??????
2
+(????????????−????????????)
2
Since X
Land X
Care functions of frequency, at a particular frequency of the
applied voltage, X
Land X
Cwill became equal in magnitude.
Since, X
L= X
C
X
L–X
C= 0
∴Z = ??????
2
+0= R
The circuit, when X
L= X
Cand hence ??????= ??????, is said to be in resonance. In a series
circuit current Iremains the same throughout we can write,
IX
L= IX
C
i.eV
L = V
C
Where, X
L= 2??????fL
X
C=ൗ
1
2??????fL

6. Observation on Current & Voltage
In a series circuit current Iremains the same throughout we can write,
IX
L= IX
C
i.e. V
L = V
C
So, at resonance VL and VC will cancel out each other.
∴The supply voltage ,
V = ??????
??????
2
+(??????
??????−??????
??????)
2
V = ??????
??????
2
V =??????
??????
i.e. The entire supply voltage will drop across the resistor R

7. Q-Factor in R-L-C Series Circuit
Across Inductor
Q-Factor:
Across Capacitor
Voltage across Inductor = I
mX
L
Voltage Magnification = Τ
??????
??????
??????
= Τ
??????
??????
??????
??????
??????
??????
�= Τ
??????
??????
??????
In case of R-L-C series circuit Q-Factor is defined as the voltage
magnification of the circuit at resonance.
Q-factor = Τ
1
�xΤ
??????
??????
Current at resonance is given by,
⇒I
m= Τ
??????
�
⇒V = I
mR
Voltage across Capacitor = I
mX
C
Voltage Magnification = Τ
??????
??????
??????
= Τ
??????
??????
??????
??????
??????
??????
�= Τ
??????
??????
??????

8. = Τ
??????
??????
�
= ൗ
??????
rL
�
= ൗ
2??????f
rL
�
= ൗ
2??????L
2??????�????????????
= Τ
1
�xΤ
??????
2
????????????
= Τ
1
�xΤ
??????
??????
Q-Factor of Inductor and Capacitor
Solving with
Inductance
Solving with
Capacitance
= Τ
??????
??????
�
= Τ
1
??????
r
??????�
= Τ
1
2??????f
r
??????�
= Τ
1
2??????xൗ
1
2????????????�????????????
= Τ
1
�xΤ
????????????
??????
2
= Τ
1
�xΤ
??????
??????

9. Resonant Frequency
Finding F
r
At resonance,
X
L= X
C
∴2??????f
rL= ൗ
1
2??????f
rC
(f
ris the resonant frequency)
∴f
r
2
=ൗ
1
(2??????)
2
LC
∴f
r
2
=ൗ
1
2??????????????????
Where Lis the inductance in henry, Cis the
capacitance in farad and f
rthe resonant
frequency in Hz.
At resonance, as net reactance is zero
X = 0 [ X
L–X
C= 0 ]
∴Z = ??????
2
+0= R
This gives minimum value for impedance
and maximum current, where
I
m= Τ
??????
??????= Τ
??????
�(∵Z = R )
Thecircuitbehaveslikeapureresistivecircuit
becausenetreactanceiszero.
So,thecurrentisinphasewithappliedvoltage.
AlsoPowerFactor=1.
Ascurrentismaximumitproduceslargevoltage
dropacrossLandC.

10. Voltage across L and C
Voltage across Inductor
V
L= I
m
Τ
??????
??????
Also,
V
L= V Τ
??????
??????�
2
Similarly voltage across Capacitor
V
C= I
m
Τ
??????
??????
Also,
V
C= V Τ
??????
??????�
2

11. Series Resonance Frequency
Graphs of RLC Series
Resonance
Impedance of RLC Resonance Current of RLC Resonance
Phase Angle of RLC Resonance

12. At series resonance,
•X
L = X
C
•Z = R
•I
rms= Τ
??????
�
�{Maximum current}
•V
L = V
C
•Power factor : cos??????= 1
•F
r= ൗ
1
2πLC
•Q = Τ
1
�xΤ
??????
??????
Characteristics of RLC Resonance Circuit

13. Effects of Resonance
Resistor –Inductor –Capacitor
Series Circuit
When a series in R-L-C circuit attains resonance ??????
??????= ??????
??????i.e.,
the next reactance of the circuit is zero.
??????= ????????????. ??????. , the impedance of the circuit is minimum.
Since Zis minimum, ??????= Τ
??????
??????will be maximum.
Since Iis maximum, the power dissipated would be
maximum ??????= ??????
2
R.
Since ??????
??????= ??????
??????, ??????= ??????
??????. ??????. ??????., the supply voltage is in phase
with the supply current.

14. TheseriesRLCcircuitissimplyanassociationinseriesofthethreeelementarycomponents
ofelectronics:resistor,inductor,andcapacitor.Theimpedanceofaresistorisarealnumber
andtheimpedancesoftheinductorandcapacitorarepureimaginarynumbers,thetotal
impedanceofthecircuitisasumofthesethreeimpedancesandis,therefore,arealnumber.
Conclusion

15. •https://www.electronics-tutorials.ws/accircuits/series-resonance.html
•https://www.slideshare.net/SIDDHI31/resonance-in-r-lc-circuit
•https://www.electrical4u.com/resonance-in-series-rlc-circuit/
•https://www.electronics-lab.com/article/series-rlc-circuit-analysis/
References

16. Thank
You
For your kind attention
to our presentation