1. BASIC ELECTRICAL ENGINEERING
D. C. KULSHRESHTHA,
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2. Chapter 11 Resonance in AC Circuits
D.C. Kulshreshtha
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3. No one will manufacture God won’t give problems
Thought of The DAY
No one will manufacture
a lock without a key.
Similarly,
God won’t give problems
without solutions.
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Ch. 11 Resonance in AC Circuits
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4. Ch. 11 Resonance in AC Circuits
Topics to be Discussed
Introduction.
Series Resonant Circuit.
Effect of Variation of Frequency.
Variation of Voltage across C and L with Frequency.
Quality Factor (Q).
Voltage Magnification.
Resonance Curve.
Relation between f0, f1 and f2.
Bandwidth (BW) in Terms of Circuit Parameters.
Bandwidth (BW) in Terms of Q and f0.
Parallel Resonant Circuit.
Practical Circuit.
Resonance Curve.
Effect of Variation of Frequency.
Some Important Points.
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Ch. 11 Resonance in AC Circuits
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5. Ch. 11 Resonance in AC Circuits
Introduction
Resonance is the condition that exists in ac circuits when the input current is in phase with the input voltage.
When in resonance, the ac circuit is purely resistive and draws power at unity power factor.
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Ch. 11 Resonance in AC Circuits
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6. Series Resonant Circuit
Click
The circuit.
(b) Phasor diagram
at resonance.
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Ch. 11 Resonance in AC Circuits
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7. Ch. 11 Resonance in AC Circuits
The resonance in a series RLC circuit requires that
The frequency of resonance,
Click
The impedance of the circuit assumes a minimum value given as
Click
The current has a maximum value given by
Click
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Ch. 11 Resonance in AC Circuits
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8. Effect of Variation of Frequency
(a) Impedance.
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9. Ch. 11 Resonance in AC Circuits
(b) Current. (c) Power factor angle.
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10. Ch. 11 Resonance in AC Circuits
At f0, the circuit behaves as purely resistive.
Below f0,
X has negative values
(i.e., the circuit is capacitive).
Above f0,
X has positive values
(i.e., the circuit is inductive).
Since the current becomes maximum at resonant frequency, the series RLC resonant circuit is also called an acceptor circuit.
Click
Click
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Ch. 11 Resonance in AC Circuits
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11. Variation of Voltage across C and L with Frequency
We can show that
(a) When R has appreciable value.
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Ch. 11 Resonance in AC Circuits
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12. Ch. 11 Resonance in AC Circuits
(b) When R is very small.
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13. Ch. 11 Resonance in AC Circuits
Quality Factor (Q)
The quality of a resonant circuit to accept current (and power) at the resonant frequency to the exclusion of other frequencies is measured by its quality factor (Q factor), defined below.
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Ch. 11 Resonance in AC Circuits
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14. Ch. 11 Resonance in AC Circuits
Click
Since
By putting,
we get another form
Click
A capacitor usually has no losses.
The Q of a series inductor-capacitor circuit is the same as the Q of the coil used.
In fact, Q of the coil is used as a figure of merit for the coil.
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Ch. 11 Resonance in AC Circuits
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15. Ch. 11 Resonance in AC Circuits
Coils with Q < 10 are described as low-Q coils.
Coils with Q > 10 are described as high-Q coils.
Coils having Q as high as are used in electronic circuits.
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16. Voltage Magnification
Let us determine the voltage drops across each element,
Click
Click
Click
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Ch. 11 Resonance in AC Circuits
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17. Ch. 11 Resonance in AC Circuits
The entire supply voltage V appears across R.
The voltage VL (and voltage VC) is Q times V.
This is often called Q gain in electronics.
The circuit is called voltage resonant circuit.
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Ch. 11 Resonance in AC Circuits

18. Ch. 11 Resonance in AC Circuits
Example 1
A series combination of a resistance of 4 Ω, an inductance of 0.5 H and a variable capacitance is connected across a 100-V, 50-Hz supply. Calculate
(a) the capacitance to give resonance,
(b) the voltage across the inductance and the capacitance, and
(c) the Q factor of the circuit.
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19. Ch. 11 Resonance in AC Circuits
Solution :
Click
(a) For resonance,
Click
(b) At resonance,
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20. Ch. 11 Resonance in AC Circuits
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21. Ch. 11 Resonance in AC Circuits
Resonance Curve
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Ch. 11 Resonance in AC Circuits
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22. Ch. 11 Resonance in AC Circuits
The frequencies f1 and f2 are often called lower and upper cutoff frequencies.
At these frequencies, the current reduces to 0.707I0.
These frequencies are also called half-power frequencies.
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Ch. 11 Resonance in AC Circuits
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23. Ch. 11 Resonance in AC Circuits
The lower the value of R, the sharper is the resonance curve.
We say that such a resonant circuit has high selectivity.
For larger values of R, not only the peak value of current falls, but even the response curve becomes less sharp (i.e., low selectivity).
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Ch. 11 Resonance in AC Circuits
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24. Relation between f0, f1 and f2
Click
At f1 and f2,
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25. Ch. 11 Resonance in AC Circuits
For XL < XC, we get
Click
Click
For XL > XC, we get
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Ch. 11 Resonance in AC Circuits

26. Ch. 11 Resonance in AC Circuits
Multiplying these two equations,
Click
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27. Bandwidth (BW) in Terms of Circuit Parameters
Click
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28. Bandwidth (BW) in Terms of Q and f0
We know that
Click
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29. Ch. 11 Resonance in AC Circuits
Comments on Resonance
The f0 is not centrally located with respect to f1 and f2, especially when the Q is small.
Actually, f0 is the geometric mean of f1 and f2.
The geometric mean is always less than the arithmetic mean.
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Ch. 11 Resonance in AC Circuits
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30. Ch. 11 Resonance in AC Circuits
However, if Q > 10, f0 is sufficiently centered with respect to f1 and f2.
Hence, we can write
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Ch. 11 Resonance in AC Circuits

31. Ch. 11 Resonance in AC Circuits
Example 2
A series ac circuit has a resonance frequency of 150 kHz and a bandwidth of 75 kHz. Determine its half-power frequencies.
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Solution : Let us calculate the Q of the circuit,
Hence, we cannot use the approximate relations. Using the exact relations, and working in kHz,
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Ch. 11 Resonance in AC Circuits
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32. Ch. 11 Resonance in AC Circuits
Eliminating f2 between the two equations, we get
Click
Ignoring the negative value, we have f1 = kHz.
Hence, f2 = 75 + f1 = kHz.
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33. Parallel Resonant Circuit
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34. Ch. 11 Resonance in AC Circuits
The losses in inductor and capacitor are accounted for by equivalent resistances R1 and R2.
Resonant condition reaches when the reactive (or wattless) component of line current I reduces to zero.
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Ch. 11 Resonance in AC Circuits
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35. Ch. 11 Resonance in AC Circuits
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36. Ch. 11 Resonance in AC Circuits
Under resonance condition, the reactive components of these two currents are equal in magnitude (but opposite in phase). That is,
Click
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Ch. 11 Resonance in AC Circuits

37. Ch. 11 Resonance in AC Circuits
Click
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38. Practical Parallel Resonance Circuit
It is possible to get a capacitor having negligible losses.
It means that the resistance R2 in series with capacitor C can be ignored.
However, in practice an inductor does have some losses.
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Ch. 11 Resonance in AC Circuits
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39. Ch. 11 Resonance in AC Circuits
(a) A practical circuit.
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40. Ch. 11 Resonance in AC Circuits
(b) Its phasor diagram.
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41. Ch. 11 Resonance in AC Circuits
By putting R2 = 0 and R1 = R,
Click
Note that if
Click
f0 is imaginary, and therefore the resonance cannot occur.
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Ch. 11 Resonance in AC Circuits
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42. Ideal Parallel Resonant Circuit
If it were possible, it would have been ideal if the inductor too were lossless. In such a case, we could ignore resistance R,
Click
(c) An ideal circuit.
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Ch. 11 Resonance in AC Circuits
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43. Ch. 11 Resonance in AC Circuits
Current at Resonance
For the practical circuit, at resonance we have
Click
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44. Ch. 11 Resonance in AC Circuits
Equating the reactive components of the two currents, we get
Click
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45. Ch. 11 Resonance in AC Circuits
Since the reactive components of the two currents cancel each other, the line current is
Thus, the effective or equivalent or dynamic impedance of the parallel resonance circuit is given as
Click
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46. Effect of Variation of Frequency
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The conductance and susceptance of the inductive branch,
Click
Click
For the capacitive branch,
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47. Ch. 11 Resonance in AC Circuits
Total admittance of the circuit,
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48. Ch. 11 Resonance in AC Circuits
It can be seen that
As f increases, G decreases.
The BL is considered negative, since –j is associated with it.
For low f, ωL < R and (ωL)2 << R2, hence BL is directly proportional to f.
For high f, ωL > R and (ωL)2 >> R2, hence BL is inversely proportional to f.
Consequently, the plot is a rectangular hyperbola.
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49. Ch. 11 Resonance in AC Circuits
(a) Admittance versus frequency.
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50. Ch. 11 Resonance in AC Circuits
(b) Current versus frequency.
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51. Ch. 11 Resonance in AC Circuits
At resonance frequency, the line current is seen to have minimum,
Click
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Ch. 11 Resonance in AC Circuits
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52. Ch. 11 Resonance in AC Circuits
Some Important Points
Since the circuit rejects the current at resonance (i.e., it has minimum value), the parallel resonant circuit is also called rejector circuit or anti-resonant circuit.
Since the circulating current between the two branches is many times the line current, the parallel resonant circuit is also called current resonant circuit.
The circuit is also called a tank circuit.
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Ch. 11 Resonance in AC Circuits
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53. Ch. 11 Resonance in AC Circuits
Review
Introduction.
Series Resonant Circuit.
Effect of Variation of Frequency.
Variation of Voltage across C and L with Frequency.
Quality Factor (Q).
Voltage Magnification.
Resonance Curve.
Relation between f0, f1 and f2.
Bandwidth (BW) in Terms of Circuit Parameters.
Bandwidth (BW) in Terms of Q and f0.
Parallel Resonant Circuit.
Practical Circuit.
Resonance Curve.
Effect of Variation of Frequency.
Some Important Points.
Friday, December 07, 2018
Ch. 11 Resonance in AC Circuits
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