Electrical element modeling dynamic system examples

1. R, L, C Circuits Prof. Townsend Page 1 of 6
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
dQ
i
dt
= (1a) ( )
t
Q i dτ τ
−∞
= ∫ (1b)
R Rv Ri= (2a)
1
R Ri v
R
= (2b)
( )
1 t
C C
Q
v i d
C C
τ τ
−∞
= = ∫ (3a) C
C
dv
i C
dt
= (3b)
L
L
di
v L
dt
= (4a) ( )
1 t
L Li v d
L
τ τ
−∞
= ∫ (4b)
Time Constants
RL
L
R
τ = (5a) RC RCτ = (5b)
Natural frequency 2
0
1
LC
ω = (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.

2. R, L, C Circuits Prof. Townsend Page 2 of 6
First Order Series RC circuit
+
-
νs
C
R
ι
KVL: R C sv v v+ = (7)
Write in terms of Cv :
From equation (2a) R Rv Ri= (8)
There is only one current in the loop so R Ci i i= = (9)
From equation (3b) C
R C
dv
v Ri R C
dt
⎛ ⎞
= = ⎜ ⎟
⎝ ⎠
(10)
Plug into equation (7) C
C s
dv
RC v v
dt
+ = (11a)
Rewrite using equation (5a) C
RC C s
dv
v v
dt
τ + = (11b)
First Order Series RL circuit
+
-
R
L
ι
νs
KVL: R L sv v v+ = (12)
Write in terms of Cv :
From equation (2a) R Rv Ri= (13)
From equation (4a) L
L
di
v L
dt
= (14)
There is only one current in the loop so R Li i i= = (15)
Plug into equation (12) s
di
L Ri v
dt
+ = (16)
Divide by L
1
s
di R
i v
dt L L
+ = (17a)
Rewrite using equation (5a)
1 1
s
RL
di
i v
dt Lτ
+ = (17b)

3. R, L, C Circuits Prof. Townsend Page 3 of 6
First Order Parallel RC circuit
ιs
CR
ιs ιR ιC
KCL: C R si i i+ = (18)
Write in terms of v :
There is only one voltage drop so R Cv v v= = (19)
From equation (2b)
1
Ri v
R
= (20)
From equation (3b) C
dv
i C
dt
= (21)
Plug into equation (18)
1
s
dv
C v i
dt R
+ = (22)
Divide by C
1 1
s
dv
v i
dt RC C
+ = (23a)
Rewrite using equation (5b)
1 1
s
RC
dv
v i
dt Cτ
+ = (23b)
First Order Parallel RL circuit
ιs
LR
ιs ιLιR
KCL: L R si i i+ = (24)
Write in terms of Li :
From equation (2b)
1
R Ri v
R
= . (25)
There is only one voltage drop so R Lv v v= = (26)
From equation (4a)
1 1 L
R L
di
i v L
R R dt
⎛ ⎞
= = ⎜ ⎟
⎝ ⎠
(27)
Plug into equation (24) L
L s
diL
i i
R dt
+ = (28a)
Rewrite using equation (5a) L
L RL s
di
i i
dt
τ+ = (28b)

4. R, L, C Circuits Prof. Townsend Page 4 of 6
Second Order Series RLC circuit
+
-
C
R
L
ι
νs
KVL: L R C sv v v v+ + = (29)
Write in terms of the loop current i , (equations (2a). (3a), (4a)).
L
di
v L
dt
= Rv Ri= ( )
1 t
Cv i d
C
τ τ
−∞
= ∫ (30)
Plug into (29) L R C sv v v v+ + = :
( )
1 t
s
di
L Ri i d v
dt C
τ τ
−∞
+ + =∫ (31)
The integral is a problem so take the time derivative of every term in (31)
2
2
1 sdvd i di
L R i
dt dt C dt
+ + = (32)
Divide by L
2
2
1 1 sdvd i R di
i
dt L dt LC L dt
+ + = (33a)
Rewrite using equations (5a) and (6)
2
2
02
1 1 s
RL
dvd i di
i
dt dt L dt
ω
τ
+ + = (33b)

5. R, L, C Circuits Prof. Townsend Page 5 of 6
Second Order Parallel RLC circuit
ιs
CLR
ιs ιLιR
ιCιLC
KCL: LC R si i i+ = and L C LCi i i+ = Combine to give L R C si i i i+ + = (34)
Write in terms of v , the voltage across all components (equations (2b). (3b), (4b)).
( )
1 t
L Li v d
L
τ τ
−∞
= ∫
1
R Ri v
R
= C
C
dv
i C
dt
=
(35)
Plug into (34) L R C si i i i+ + = :
( )
1 1t
C
L R s
dv
v d v C i
L R dt
τ τ
−∞
+ + =∫ (36)
The integral is a problem so take the time derivative of every term in (36). Reorder.
2
2
1 1 sdid v dv
C v
dt R dt L dt
+ + =
(37)
Divide by C
2
2
1 1 1 sdid v dv
v
dt RC dt LC C dt
+ + =
(38a)
Rewrite using equations (5b) and (6)
2
2
02
1 1 s
RC
did v dv
v
dt dt C dt
ω
τ
+ + =
(38b)

6. R, L, C Circuits Prof. Townsend Page 6 of 6
Summary
Circuit Differential Equation Form
First Order Series RC circuit 1 1C
C s
RC RC
dv
v v
dt τ τ
+ = (11b)
First Order Series RL circuit 1 1
s
RL
di
i v
dt Lτ
+ = (17b)
First Order Parallel RC circuit 1 1
s
RC
dv
v i
dt Cτ
+ = (23b)
First Order Parallel RL circuit 1 1L
L s
RL RL
di
i i
dt τ τ
+ = (28b)
First Order General Form ( )
1dx
x f t
dt τ
+ =
Second Order Series RLC circuit 2
2
02
1 1 s
RL
dvd i di
i
dt dt L dt
ω
τ
+ + = (33b)
Second Order Parallel RLC circuit 2
2
02
1 1 s
RC
did v dv
v
dt dt C dt
ω
τ
+ + = (38b)
Second Order General Form
( )
2
2
02
1d x dx
x f t
dt dt
ω
τ
+ + =
Time Constants
RL
L
R
τ = (5a)
RC RCτ = (5b)
Natural frequency 2
0
1
LC
ω = (6)