A series circuit consists of an ac source of variable frequency, a 115 resistor,

Question

A series circuit consists of an ac source of variable frequency, a 115 resistor, a 1.25 F capacitor, and a 4.50 mH inductor.

a)Find the impedance of this circuit when the angular frequency of the ac source is adjusted to the resonance angular frequency.
Z=_____________________

b)Find the impedance of this circuit when the angular frequency of the ac source is adjusted to twice the resonance angular frequency.

Z=_____________________

c)Find the impedance of this circuit when the angular frequency of the ac source is adjusted to half the resonance angular frequency.

Z=_____________________

Explanation / Answer

resonance frequency f = 1 / 2LC                                   = 1 / {(2)(3.14)(4.5*10-3 H)(1.25*10-6 F)}                                   = 2123.14 Hz a) at resonance XL = XC impedance of this circuit is      Z = {R2 + (XL - XC)2}       Z = {R2 + (XC - XC)2}        Z = R           = 115 b) if frequency is twice , f' = 2f                                     = 4246.28 Hz inductive reactance XL = 2f'L                                    = (2)(3.14)(4246.28 Hz)(4.5*10-3 H)                                    = 120 capacitive reactance XC = 1/2f'C                                      = 1 / (2)(3.14)(4246.28 Hz)(1.25*10-6 F)                                      = 30 impedance of this circuit is      Z = {R2 + (XL - XC)2}         = {(115 )2 + (120 - 30 )2}         = 146.03 c) if frequency is half, f'' = f/2                                  = 1061.57 Hz inductive reactance XL = 2f''L                                    = (2)(3.14)(1061.57 Hz)(4.5*10-3 H)                                    = 30 capacitive reactance XC = 1/2f''C                                      = 1 / (2)(3.14)(1061.57 Hz)(1.25*10-6 F)                                      = 120 impedance of this circuit is      Z = {R2 + (XL - XC)2}         = {(115 )2 + (30 - 120 )2}         = 146.03       inductive reactance XL = 2f'L                                    = (2)(3.14)(4246.28 Hz)(4.5*10-3 H)                                    = 120 capacitive reactance XC = 1/2f'C                                      = 1 / (2)(3.14)(4246.28 Hz)(1.25*10-6 F)                                      = 30 impedance of this circuit is      Z = {R2 + (XL - XC)2}         = {(115 )2 + (120 - 30 )2}         = 146.03 c) if frequency is half, f'' = f/2                                  = 1061.57 Hz inductive reactance XL = 2f''L                                    = (2)(3.14)(1061.57 Hz)(4.5*10-3 H)                                    = 30 capacitive reactance XC = 1/2f''C                                      = 1 / (2)(3.14)(1061.57 Hz)(1.25*10-6 F)                                      = 120 impedance of this circuit is      Z = {R2 + (XL - XC)2}         = {(115 )2 + (30 - 120 )2}         = 146.03            Z = {R2 + (XL - XC)2}         = {(115 )2 + (120 - 30 )2}         = 146.03 c) if frequency is half, f'' = f/2                                  = 1061.57 Hz inductive reactance XL = 2f''L                                    = (2)(3.14)(1061.57 Hz)(4.5*10-3 H)                                    = 30 capacitive reactance XC = 1/2f''C                                      = 1 / (2)(3.14)(1061.57 Hz)(1.25*10-6 F)                                      = 120 impedance of this circuit is      Z = {R2 + (XL - XC)2}         = {(115 )2 + (30 - 120 )2}         = 146.03 inductive reactance XL = 2f''L                                    = (2)(3.14)(1061.57 Hz)(4.5*10-3 H)                                    = 30 capacitive reactance XC = 1/2f''C                                      = 1 / (2)(3.14)(1061.57 Hz)(1.25*10-6 F)                                      = 120 impedance of this circuit is      Z = {R2 + (XL - XC)2}         = {(115 )2 + (30 - 120 )2}         = 146.03 capacitive reactance XC = 1/2f''C                                      = 1 / (2)(3.14)(1061.57 Hz)(1.25*10-6 F)                                      = 120 impedance of this circuit is      Z = {R2 + (XL - XC)2}         = {(115 )2 + (30 - 120 )2}         = 146.03      Z = {R2 + (XL - XC)2}         = {(115 )2 + (30 - 120 )2}         = 146.03      

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