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Chapter 11 Resonance in AC Circuits D.C. Kulshreshtha Next

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. Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

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. Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

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. Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

Series Resonant Circuit Click The circuit. (b) Phasor diagram at resonance. Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

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 Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

Effect of Variation of Frequency (a) Impedance. Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

Ch. 11 Resonance in AC Circuits (b) Current. (c) Power factor angle. Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

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 Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

Variation of Voltage across C and L with Frequency We can show that (a) When R has appreciable value. Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

Ch. 11 Resonance in AC Circuits (b) When R is very small. Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

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. Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

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. Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

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 200-300 are used in electronic circuits. Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

Voltage Magnification Let us determine the voltage drops across each element, Click Click Click Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

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. Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits

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. Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

Ch. 11 Resonance in AC Circuits Solution : Click (a) For resonance, Click (b) At resonance, Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

Ch. 11 Resonance in AC Circuits Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

Ch. 11 Resonance in AC Circuits Resonance Curve Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

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. Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

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). Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

Relation between f0, f1 and f2 Click At f1 and f2, Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

Ch. 11 Resonance in AC Circuits For XL < XC, we get Click Click For XL > XC, we get Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits

Ch. 11 Resonance in AC Circuits Multiplying these two equations, Click Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

Bandwidth (BW) in Terms of Circuit Parameters Click Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

Bandwidth (BW) in Terms of Q and f0 We know that Click Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

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. Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

Ch. 11 Resonance in AC Circuits However, if Q > 10, f0 is sufficiently centered with respect to f1 and f2. Hence, we can write Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits

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. Click Solution : Let us calculate the Q of the circuit, Hence, we cannot use the approximate relations. Using the exact relations, and working in kHz, Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

Ch. 11 Resonance in AC Circuits Eliminating f2 between the two equations, we get Click Ignoring the negative value, we have f1 = 117.1 kHz. Hence, f2 = 75 + f1 = 192.1 kHz. Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

Parallel Resonant Circuit Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

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. Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

Ch. 11 Resonance in AC Circuits Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

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 Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits

Ch. 11 Resonance in AC Circuits Click Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

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. Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

Ch. 11 Resonance in AC Circuits (a) A practical circuit. Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

Ch. 11 Resonance in AC Circuits (b) Its phasor diagram. Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

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. Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

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. Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

Ch. 11 Resonance in AC Circuits Current at Resonance For the practical circuit, at resonance we have Click Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

Ch. 11 Resonance in AC Circuits Equating the reactive components of the two currents, we get Click Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

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 Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

Effect of Variation of Frequency Click The conductance and susceptance of the inductive branch, Click Click For the capacitive branch, Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

Ch. 11 Resonance in AC Circuits Total admittance of the circuit, Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

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. Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

Ch. 11 Resonance in AC Circuits (a) Admittance versus frequency. Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

Ch. 11 Resonance in AC Circuits (b) Current versus frequency. Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

Ch. 11 Resonance in AC Circuits At resonance frequency, the line current is seen to have minimum, Click Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

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. Friday, December 07, 2018 Ch. 11 Resonance in AC Circuits Next

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 Next