Question

# viii. An ac circuit with a 80 μF capacitor in series with a coil of resistance 16Ω and inductance 160mH is connected to a 100V, 100 Hz supply is shown below. Calculate 7. the inductive reactance 8. the capacitive reactance 9. the circuit impedance and V-I phase angle θ 10. the circuit current I 11. the phasor voltages VR, VL, VC and VS 12. the resonance circuit frequency Also construct a fully labeled and appropriately ‘scaled’ voltage phasor diagram.

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## Answer to a math question viii. An ac circuit with a 80 μF capacitor in series with a coil of resistance 16Ω and inductance 160mH is connected to a 100V, 100 Hz supply is shown below. Calculate 7. the inductive reactance 8. the capacitive reactance 9. the circuit impedance and V-I phase angle θ 10. the circuit current I 11. the phasor voltages VR, VL, VC and VS 12. the resonance circuit frequency Also construct a fully labeled and appropriately ‘scaled’ voltage phasor diagram.

Frederik
4.6
To calculate the values requested, we will use the following formulas:

1. Inductive reactance $XL$ is given by the formula XL = 2πfL, where f is the frequency and L is the inductance.
2. Capacitive reactance $XC$ is given by the formula XC = 1 / $2πfC$, where f is the frequency and C is the capacitance.
3. Circuit impedance $Z$ is given by the formula Z = √$R^2 + (XL - XC$^2), where R is the resistance, XL is the inductive reactance, and XC is the capacitive reactance.
4. V-I phase angle $θ$ is given by the formula θ = atan$(XL - XC$/R), where XL is the inductive reactance, XC is the capacitive reactance, and R is the resistance.
5. Circuit current $I$ is given by the formula I = V / Z, where V is the supply voltage and Z is the circuit impedance.
6. Phasor voltages are calculated by multiplying the circuit current by the respective reactance values $VR = I * R, VL = I * XL, VC = I * XC, VS = I * Z$.
7. Resonance frequency $fr$ is given by the formula fr = 1 / $2π√(LC$), where L is the inductance and C is the capacitance.

Now let's calculate the values step by step.

7. Inductive Reactance $XL$:
XL = 2πfL
= 2π * 100 Hz * 160 mH $converting 160 mH to Henries$
= 0.1 * π * 16 ohms
= 5π ohms

8. Capacitive Reactance $XC$:
XC = 1 / $2πfC$
= 1 / $2π * 100 Hz * 80 μF (converting 80 μF to Farads$
= 1 / $0.1 * π * 80 ohms$
= 1 / $8π ohms$

Answer: XC = 1 / $8π ohms$

9. Circuit Impedance $Z$ and V-I Phase Angle $θ$:
Z = √$R^2 + (XL - XC$^2)
= √$(16 ohms$^2 + $5π ohms - 1/(8π ohms$)^2)
≈ 21.01 ohms

θ = atan$(XL - XC$/R)
= atan$(5π ohms - 1/(8π ohms$)/16 ohms)

10. Circuit Current $I$:
I = V / Z
= 100V / 21.01 ohms
≈ 4.76 A

11. Phasor Voltages $VR, VL, VC, VS$:
VR = I * R
= 4.76 A * 16 ohms
= 76.16 V

VL = I * XL
= 4.76 A * 5π ohms
≈ 15π V

VC = I * XC
= 4.76 A * 1/$8π ohms$
≈ 0.15π V

VS = I * Z
= 4.76 A * 21.01 ohms
≈ 100 V

Answer: VR ≈ 76.16 V, VL ≈ 15π V, VC ≈ 0.15π V, VS ≈ 100 V

12. Resonance Circuit Frequency $fr$:
fr = 1 / $2π√(LC$)
= 1 / $2π√((160 mH$ * $80 μF$))
= 1 / $2π√(0.16 H * 0.08 F$)
= 1 / $2π√(0.0128$)
≈ 9.86 Hz

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