Question

An AC voltage of the form Δv = 95 sin 275t where Δv is in volts and t is in seconds, is applied to a series RLC circuit. If R = 38.0 Ω, C = 26.0 µF, and L = 0.240 H, find the following. (a) impedance of the circuit
1 Ω

(b) rms current in the circuit
2 A

(c) average power delivered to the circuit
3 W

Answer 1

Answer:

(a) 83.06Ω

(b) 0.81A

(c) 25.0W

Explanation:

Comparing Δv = 95 sin 275t with Δv = Vmaxsinωt  

                        ω = 275

Inductive reactance Χ = ωL

                        = 275 × 0.240

                        = 66 Ω

Capacitive reactance Χ = 1/ωc

                        = 1/ (275 × 26 x 10^-6)

                        = 139.86Ω

Impedance Z = $\sqrt{R^{2} + (wL - \frac{1}{wc} )^{2}$

                      = $\sqrt{R^{2} + (X_{l} - X_{c} )^{2}$

                      = $\sqrt{38^{2} + (66 - 139.86)^{2}$

                      = 83.06Ω  

(b) $I_{rms} = \frac{V_{rms}}{Z}$

solving for $V_{rms}$,

                     $V_{rms} = \frac{V_{max}}{\sqrt{2}}$

                     $V_{rms} = \frac{95}{\sqrt{2}}$  

                     $V_{rms} = 67.2V$  

substituting the value of $V_{rms}$ and Z into $I_{rms}$ equation, we have;

                    $I_{rms} = \frac{67.2}{83.06}$  

                    $I_{rms} = 0.81A$  

(c) Average power P = $I_{rms}[\tex][tex]V_{rms}$cos∅

To get the average power, we first solve for ∅ since it was not given.

                   ∅ $= tan^{-1}\frac{X_{l} - X_{c}}{R}$  

                   ∅ $= tan^{-1}\frac{66 - 139.86}{38}$  

                   ∅ $= tan^{-1}\frac{-73.86}{38}$  

                   ∅ $= tan^{-1} -1.9437$  

                   ∅ = -62.77°  

Average power P = 0.81 × 67.2 × cos-62.77

                          P = 0.81 × 67.2 × 0.46  

                          P = 25.03872W              

                          P = 25.0W