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

12/4/2021
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©UWECE TC Chen
EE215 – FUNDAMENTALS OF
ELECTRICAL ENGINEERING
LECTURE 26
Tai Chen
University of Washington
©UWECE TC Chen
PARALLEL RLC CIRCUITS:
UNDERDAMPED VOLTAGE RESPONSE
• Underdamped solutions:
𝑣 ? = ?
1
?
−𝛼?
?
𝑗𝜔
𝑑
?
+ ?
2
?
−𝛼?
?
−𝑗𝜔
𝑑
?
= ?
−𝛼?
?
1
cos 𝜔
?
? + ??
1
sin 𝜔
?
? + ?
2
cos 𝜔
?
? − ??
2
sin 𝜔
?
?
= ถ ?
−𝛼?
damping
factor
?
1
+ ?
2
?
1
cos 𝜔
?
? + ? ?
1
− ?
2
?
2
sin 𝜔
?
?
Note that B
1
and B
2
are real (i.e., not complex!) because A
1
and A
2
are
complex conjugates.
𝑣 ? = ?
−𝛼?
?
1
cos 𝜔
?
? + ?
2
sin 𝜔
?
?
𝑣 0
+
= ?
1
= 𝑉
0
;
?𝑣 0
+
??
=
?
?
0
+
?
= −𝛼?
1
+ 𝜔
?
?
2
; 𝑎𝑛? ?
?
0
+
= −𝐼
0
−
𝑉
0
𝑅

12/4/2021
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©UWECE TC Chen
UNDERDAMPED
PARALLEL RLC CIRCUIT EXAMPLE (1)
• Example: Given
𝑅 = 2500 Ω, 𝐿 =
189.3 mH, ? = 125 nF, 𝑉
0
=
0 V, 𝐼
0
= −4 A . Find 𝑣(?) .
R L C
v(t)
–
+
i
C
i
L
i
R
V
0
–
+
I
0
1. Find roots of characteristic equation
?
1,2
= −𝛼 ± 𝛼
2
− 𝜔
0
2
𝛼 =
1
2𝑅?
=
1
2 × 2500 × 125 × 10
−9
= 1600 rad/sec
𝜔
0
=
1
𝐿?
=
1
189.3 × 10
−3
× 125 × 10
−9
= 6500
rad
sec
𝜔
?
= 𝜔
0
2
− 𝛼 2 = 100 65 2 − 16 2 = 6300
rad
sec
?
1
= −1600 + ?6300 rad/sec ; ?
2
= −1600 − ?6300 rad/sec
©UWECE TC Chen
UNDERDAMPED
PARALLEL RLC CIRCUIT EXAMPLE (2)
2. Find 𝑣 0
+
and
?𝑣 0
+
??
𝑣 0
+
= 𝑉
0
= 0 ;
?𝑣 0
+
??
=
1
?
?
?
0
+
?
?
0
+
= −
𝑉
0
𝑅
− 𝐼
0
= −
0
5200
− −4 = 4 A
This yields,
?𝑣 0
+
??