 # EE 215 F 2021 Lecture 26.pdf - 12/4/2021 EE215 –...

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12/4/20211©UWECE TC ChenEE215FUNDAMENTALS OFELECTRICAL ENGINEERINGLECTURE 26Tai ChenUniversity of Washington©UWECE TC ChenPARALLEL RLC CIRCUITS:UNDERDAMPED VOLTAGE RESPONSEUnderdamped solutions:𝑣?= ?1?−𝛼??𝑗𝜔𝑑?+ ?2?−𝛼??−𝑗𝜔𝑑?= ?−𝛼??1cos 𝜔?? + ??1sin 𝜔?? + ?2cos 𝜔?? − ??2sin 𝜔??=?−𝛼?dampingfactor?1+ ?2?1cos 𝜔?? +??1− ?2?2sin 𝜔??Note thatB1andB2are real (i.e., not complex!) becauseA1andA2arecomplex conjugates.𝑣?= ?−𝛼??1cos 𝜔?? + ?2sin 𝜔??𝑣0+= ?1= 𝑉0;?𝑣0+??=??0+?= −𝛼?1+ 𝜔??2; 𝑎𝑛? ??0+= −𝐼0𝑉0𝑅
12/4/20212©UWECE TC ChenUNDERDAMPEDPARALLEL RLC CIRCUIT EXAMPLE (1)Example: Given𝑅 = 2500 Ω, 𝐿 =189.3 mH, ? = 125 nF, 𝑉0=0 V, 𝐼0= −4 A. Find𝑣(?).RLCv(t)+iCiLiRV0+I01.Find roots of characteristic equation?1,2= −𝛼 ±𝛼2− 𝜔02𝛼 =12𝑅?=12 × 2500 × 125 × 10−9= 1600 rad/sec𝜔0=1𝐿?=1189.3 × 10−3× 125 × 10−9= 6500radsec𝜔?=𝜔02− 𝛼2= 100652− 162= 6300radsec?1= −1600 + ?6300rad/sec;?2= −1600 − ?6300rad/sec©UWECE TC ChenUNDERDAMPEDPARALLEL RLC CIRCUIT EXAMPLE (2)2. Find𝑣0+and?𝑣0+??𝑣0+= 𝑉0= 0 ;?𝑣0+??=1???0+??0+= −𝑉0𝑅− 𝐼0= −05200−4= 4 AThis yields,?𝑣0+??

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