Problem 15.124 Challenge

กำหนดให้ \(ω = 1,000 rad/s, ΔV_{rms}= 100 V, R = 40.0 Ω,\)

\(L = 50.0 mH and C = 50.0 \mu F\)

a) จาก \(X_{L}= \omega L = 50.0 Ω\)

จาก \(X_{C}=\frac{1}{\omega C}= 20.0 Ω\)

จาก \(Z = \sqrt{R^{2}+(X_{L}-X_{C})^{2}}= 50.0 Ω\)

จาก \(ΔV_{rms}= I_{rms}Z\)

\(_{rms}\)

\(I_{rms}=\frac{\Delta V}{Z}\)

\(I_{rms}= 2.00 A \;\blacksquare\)

b) จาก \(tanφ = \frac{X_{L}-X_{C}}{R}\)

\(φ = \tan ^{-1}(\frac{X_{L}-X_{C}}{R})\)

\(φ = 36.9° \;\blacksquare\)

c) จาก \(Power Factor = cos(φ)\)

\(Power Factor = 0.800 \;\blacksquare\)

d) จาก \(\wp _{av}= I_{rms}\Delta V_{rms}\cos (φ)\)

\(\wp _{av}= 160 W \;\blacksquare\)

e) จาก \(\wp _{av,R}= I_{rms,R}\Delta V_{rms,R}\cos (0°)\)

\(\wp _{av,R}= I_{rms,R}(IR_{rms,R})\)

\(\wp _{av,R}= I_{rms,R}^{2}R_{rms,R}\)

\(\wp _{av,R}= 160 W \;\blacksquare\)

f) จาก \(I = 2I_{rms}= 2.83 A\)

จาก \(ΔV_{R}= IR = 113 V \;\blacksquare\)

จาก \(ΔV_{L}= IX_{L}= 141 V \;\blacksquare\)

จาก \(ΔV_{C}= IX_{C}= 56.6 V \;\blacksquare\)

g) จาก \(Δv_{R}= \Delta V_{R}\sin (\omega t)\)

\(\Delta v_{R}= (113 V)\sin (1,000t) \;\blacksquare\)

จาก \(Δv_{L}= \Delta V_{L}\sin (\omega t+\frac{\pi}{2})\)

\(\Delta v_{L}= \Delta V_{L}\cos (\omega t)\)

\(\Delta v_{L}= (141 V)\cos (1,000t) \;\blacksquare\)

จาก \(Δv_{C}= \Delta V_{C}\sin (\omega t-\frac{\pi}{2})\)

\(\Delta v_{C}= -\Delta V_{C}\cos (\omega t)\)

\(\Delta v_{C}= -(56.6 V)\cos (1,000t) \;\blacksquare\)