UNIT9 Resonance in AC Circuits Ch 11 Resonance

UNIT-9 Resonance in AC Circuits Ch. 11 Resonance in AC Circuits Next

9. 1 Introduction : n Resonance is the condition that exists in ac circuits when the input current is in phase with the input voltage. n When in resonance, the ac circuit is purely resistive and draws power at unity power factor. 10/31/2020 Ch. 11 Resonance in AC Circuits Next 2

9. 2 Series Resonant Circuit: - Click (a) The circuit. 10/31/2020 (b) Phasor diagram at resonance. Ch. 11 Resonance in AC Circuits Next 3

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 10/31/2020 Ch. 11 Resonance in AC Circuits Next 4

9. 3 Effect of Variation of Frequency: - (a) Impedance. 10/31/2020 Ch. 11 Resonance in AC Circuits Next 5

(b) Current. 10/31/2020 (c) Power factor angle. Ch. 11 Resonance in AC Circuits Next 6

• At f 0, the circuit behaves as purely resistive. • Below f 0, Click X has negative values (i. e. , the circuit is capacitive). • Above f 0, X has positive values Click (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. 10/31/2020 Ch. 11 Resonance in AC Circuits Next 7

9. 4 Variation of Voltage across C and L with Frequency : We can show that (a) When R has appreciable value. 10/31/2020 Ch. 11 Resonance in AC Circuits Next 8

(b) When R is very small. 10/31/2020 Ch. 11 Resonance in AC Circuits Next 9

9. 5 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. 10/31/2020 Ch. 11 Resonance in AC Circuits Next 10

Click Since By putting, we get another form for Q : 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. 10/31/2020 Ch. 11 Resonance in AC Circuits Next 11

• 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 200 -300 are used in electronic circuits. 10/31/2020 Ch. 11 Resonance in AC Circuits Next 12

9. 6 Voltage Magnification : Let us determine the voltage drops across each element, Click • • 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. 10/31/2020 Ch. 11 Resonance in AC Circuits Next 13

Example 1 n 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. 10/31/2020 Ch. 11 Resonance in AC Circuits Next 14

Solution : Click (a) For resonance, Click (b) At resonance, Click (c) 10/31/2020 Ch. 11 Resonance in AC Circuits Next 15

9. 7 Resonance Curve: - 10/31/2020 Ch. 11 Resonance in AC Circuits Next 16

n n n The frequencies f 1 and f 2 are often called lower and upper cutoff frequencies. At these frequencies, the current reduces to 0. 707 I 0. These frequencies are also called half-power frequencies. 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). 10/31/2020 Ch. 11 Resonance in AC Circuits Next 17

9. 8 Relation between f 0, f 1 and f 2: Click At f 1 and f 2, Click For XL < XC, we get For XL > XC, we get 10/31/2020 Click Ch. 11 Resonance in AC Circuits Next 18

Multiplying these two equations, 10/31/2020 Ch. 11 Resonance in AC Circuits Click Next 19

9. 9 Bandwidth (BW) in Terms of Circuit Parameters: - 10/31/2020 Ch. 11 Resonance in AC Circuits Next Click 20

9. 10 Bandwidth (BW) in Terms of Q and f 0: We know that Click 10/31/2020 Ch. 11 Resonance in AC Circuits Next 21

Comments on Resonance n n n The f 0 is not centrally located with respect to f 1 and f 2, especially when the Q is small. Actually, f 0 is the geometric mean of f 1 and f 2. The geometric mean is always less than the arithmetic mean. However, if Q > 10, f 0 is sufficiently centered with respect to f 1 and f 2. Hence, we can write 10/31/2020 Ch. 11 Resonance in AC Circuits Next 22

Example 2 n A series ac circuit has a resonance frequency of 150 k. Hz and a bandwidth of 75 k. Hz. Determine its half-power frequencies. Solution : Let us calculate the Q of the circuit, Click Hence, we cannot use the approximate relations. Using the exact relations, and working in k. Hz, 10/31/2020 Ch. 11 Resonance in AC Circuits Next 23

Eliminating f 2 between the two equations, we get Click • Ignoring the negative value, we have f 1 = 117. 1 k. Hz. • Hence, f 2 = 75 + f 1 = 192. 1 k. Hz. 10/31/2020 Ch. 11 Resonance in AC Circuits Next 24

9. 11 Parallel Resonant Circuit: - • The losses in inductor and capacitor are accounted for by equivalent resistances R 1 and R 2. • Resonant condition reaches when the reactive (or wattless) component of line current I reduces to zero. 10/31/2020 Ch. 11 Resonance in AC Circuits Next 25

Under resonance condition, the reactive components of these two currents are equal in magnitude (but opposite in phase). That is, 10/31/2020 Ch. 11 Resonance in AC Circuits Next 26

Click 10/31/2020 Ch. 11 Resonance in AC Circuits Next 27

Practical Parallel Resonance Circuit: n n n It is possible to get a capacitor having negligible losses. It means that the resistance R 2 in series with capacitor C can be ignored. However, in practice an inductor does have some losses. 10/31/2020 Ch. 11 Resonance in AC Circuits Next 28

Click (a) A practical circuit. (b) Its phasor diagram. By putting R 2 = 0 and R 1 = R, Note that if f 0 is imaginary, and therefore the resonance cannot occur. 10/31/2020 Ch. 11 Resonance in AC Circuits Next 29

9. 13 Ideal Parallel Resonant Circuit: n If it were possible, it would have been ideal if the inductor too were lossless. In such a case, we could ignore resistance R, The resonant frequency is Click (c) An ideal circuit. 10/31/2020 Ch. 11 Resonance in AC Circuits Next 30

9. 14 Current at Resonance: For the practical circuit, at resonance we have Equating the reactive components of the two currents, we get 10/31/2020 Ch. 11 Resonance in AC Circuits Next Click 31

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 10/31/2020 Ch. 11 Resonance in AC Circuits Next 32

9. 15 Effect of Variation of Frequency: Click The conductance and susceptance of the inductive branch, Click For the capacitive branch, 10/31/2020 Ch. 11 Resonance in AC Circuits Click Next 33

Total admittance of the circuit, 10/31/2020 Ch. 11 Resonance in AC Circuits Next 34

It can be seen that 1. As f increases, G decreases. 2. The BL is considered negative, since –j is associated with it. 3. For low f, ωL < R and (ωL)2 << R 2, hence BL is directly proportional to f. 4. For high f, ωL > R and (ωL)2 >> R 2, hence BL is inversely proportional to f. 5. Consequently, the plot is a rectangular hyperbola. 10/31/2020 Ch. 11 Resonance in AC Circuits Next 35

(a) Admittance versus frequency. 10/31/2020 Ch. 11 Resonance in AC Circuits Next 36

(b) Current versus frequency. At resonance frequency, the line current is seen to have minimum, Click 10/31/2020 Ch. 11 Resonance in AC Circuits Next 37

Some Important Points n n n Since the circuit rejects the current at resonance (i. e. , it has minimum value), the parallel resonant circuit is also called rejector circuit or antiresonant 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. 10/31/2020 Ch. 11 Resonance in AC Circuits Next 38