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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Frederik

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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

Answer: XL = 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)

≈ 1.04 radians

Answer: Z ≈ 21.01 ohms, θ ≈ 1.04 radians

10. Circuit Current (I):

I = V / Z

= 100V / 21.01 ohms

≈ 4.76 A

Answer: I ≈ 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

Answer: fr ≈ 9.86 Hz

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

Answer: XL = 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)

≈ 1.04 radians

Answer: Z ≈ 21.01 ohms, θ ≈ 1.04 radians

10. Circuit Current (I):

I = V / Z

= 100V / 21.01 ohms

≈ 4.76 A

Answer: I ≈ 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

Answer: fr ≈ 9.86 Hz

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