Series RC, RL, and RLC Circuits
Parallel RC, RL, and RLC Circuits
by Prof. Townsend
MTH 352 Fall 2005
If you want a good description of the analysis of these circuits, go to the Wikipedia web site, for
example http://en.wikipedia.org/wiki/RL_circuit.
Analyses for series RC, parallel RL, and series RLC circuits were taken from class notes for
Berkeley’s EE 40, Introduction to Microelectronic Circuits. The lecture notes for this course are
very well written. http://laser.eecs.berkeley.edu/ee40/lectures/cch-Lec09-021505-2-6p.pdf.
General Information
i=
dQ
dt
vR = RiR
Q 1 t
=
iC (τ ) dτ
C C ∫−∞
di
vL = L L
dt
vC =
Time Constants
L
τ RL =
R
(1a)
(2a)
(3a)
(4a)
(5a)
Q = ∫ i (τ ) dτ
t
−∞
1
vR
R
dv
iC = C C
dt
1 t
iL = ∫ vL (τ ) dτ
L −∞
(3b)
τ RC = RC
(5b)
iR =
ω02 =
Natural frequency
(1b)
1
LC
(2b)
(4b)
(6)
Kirchhoff’s voltage law (KVL): the net voltage (potential) change around a circuit is 0. You
end up where you started.
Kirchhoff’s current law (KCL): the total current into a node = the total current leaving the node.
The electrons have to go somewhere.
R, L, C Circuits
Prof. Townsend
Page 1 of 6
First Order Series RC circuit
ι
νs
R
C
+
-
KVL:
Write in terms of vC :
From equation (2a)
There is only one current in the loop so
vR + vC = vs
(7)
vR = RiR
iR = iC = i
(8)
(9)
⎛ dv ⎞
vR = RiC = R ⎜ C C ⎟
⎝ dt ⎠
dv
RC C + vC = vs
dt
dv
τ RC C + vC = vs
dt
From equation (3b)
Plug into equation (7)
Rewrite using equation (5a)
(10)
(11a)
(11b)
First Order Series RL circuit
ι
R
+
-
L
νs
KVL:
Write in terms of vC :
From equation (2a)
From equation (4a)
There is only one current in the loop so
Plug into equation (12)
Divide by L
Rewrite using equation (5a)
R, L, C Circuits
Prof. Townsend
v R + v L = vs
(12)
vR = RiR
di
vL = L L
dt
iR = iL = i
di
L + Ri = vs
dt
di R
1
+ i = vs
dt L
L
di 1
1
i = vs
+
dt τ RL
L
(13)
(14)
(15)
(16)
(17a)
(17b)
Page 2 of 6
First Order Parallel RC circuit
ιs
ιR
ιs
ιC
C
R
KCL:
Write in terms of v :
There is only one voltage drop so
From equation (2b)
From equation (3b)
Plug into equation (18)
Divide by C
Rewrite using equation (5b)
iC + iR = is
(18)
vR = vC = v
1
iR = v
R
dv
iC = C
dt
dv 1
C + v = is
dt R
dv 1
1
+
v = is
dt RC
C
dv 1
1
v = is
+
dt τ RC
C
(19)
(20)
(21)
(22)
(23a)
(23b)
First Order Parallel RL circuit
ιs
ιs
ιR
ιL
R
L
iL + iR = is
KCL:
(24)
Write in terms of iL :
1
vR .
R
vR = vL = v
iR =
From equation (2b)
There is only one voltage drop so
iR =
From equation (4a)
Plug into equation (24)
Rewrite using equation (5a)
R, L, C Circuits
Prof. Townsend
1
1 ⎛ di ⎞
vL = ⎜ L L ⎟
R
R ⎝ dt ⎠
L diL
iL +
= is
R dt
di
iL + τ RL L = is
dt
(25)
(26)
(27)
(28a)
(28b)
Page 3 of 6
Second Order Series RLC circuit
ι
νs
R
C
+
L
vL + vR + vC = vs
KVL:
Write in terms of the loop current i , (equations (2a). (3a), (4a)).
di
1 t
vL = L
vR = Ri
vC = ∫ i (τ ) dτ
dt
C −∞
Plug into (29)
(29)
(30)
vL + vR + vC = vs :
di
1 t
L + Ri + ∫ i (τ ) dτ = vs
dt
C −∞
(31)
The integral is a problem so take the time derivative of every term in (31)
dv
d 2i
di 1
L 2 +R + i= s
dt
dt C
dt
(32)
Divide by L
1
1 dvs
d 2i R di
i=
+
+
2
dt
L dt LC
L dt
Rewrite using equations (5a) and (6)
1 dvs
d 2i 1 di
+
+ ω02i =
2
dt τ RL dt
L dt
R, L, C Circuits
Prof. Townsend
(33a)
(33b)
Page 4 of 6
Second Order Parallel RLC circuit
ιs
ιs
KCL:
ιR
ιLC
R
ιL
L
iLC + iR = is and iL + iC = iLC
ιC
C
Combine to give
iL + iR + iC = is
(34)
Write in terms of v , the voltage across all components (equations (2b). (3b), (4b)).
dv
1 t
1
iC = C C
iL = ∫ vL (τ ) dτ
iR = vR
−∞
dt
L
R
(35)
Plug into (34)
iL + iR + iC = is :
dv
1 t
1
vL (τ ) dτ + vR + C C = is
∫
−∞
L
R
dt
(36)
The integral is a problem so take the time derivative of every term in (36). Reorder.
di
d 2 v 1 dv 1
C 2 +
+ v= s
dt
R dt L
dt
(37)
Divide by C
1 dis
d 2 v 1 dv 1
v=
+
+
2
dt
RC dt LC
C dt
Rewrite using equations (5b) and (6)
1 dis
d 2 v 1 dv
+
+ ω02 v =
2
τ RC dt
dt
C dt
R, L, C Circuits
Prof. Townsend
(38a)
(38b)
Page 5 of 6
Summary
Circuit
Differential Equation Form
First Order Series RC circuit
dvC
1
1
vC =
v
+
dt τ RC
τ RC s
di 1
1
i = vs
+
dt τ RL
L
dv 1
1
v = is
+
dt τ RC
C
diL
1
1
iL =
i
+
dt τ RL
τ RL s
First Order Series RL circuit
First Order Parallel RC circuit
First Order Parallel RL circuit
(11b)
(17b)
(23b)
(28b)
dx 1
+ x = f (t )
dt τ
First Order General Form
Second Order Series RLC circuit
1 dvs
d 2i 1 di
+
+ ω02i =
2
dt τ RL dt
L dt
(33b)
Second Order Parallel RLC circuit
1 dis
d 2 v 1 dv
+
+ ω02 v =
2
τ RC dt
dt
C dt
(38b)
d 2 x 1 dx
+
+ ω02 x = f ( t )
2
τ dt
dt
Second Order General Form
Time Constants
τ RL =
L
R
τ RC = RC
Natural frequency
R, L, C Circuits
ω02 =
Prof. Townsend
1
LC
(5a)
(5b)
(6)
Page 6 of 6