series and parallel equations

8/9/2019 Series and Parallel Equations

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R, L, C Circuits Prof. Townsend Page 1 of 6

Series RC, RL, and RLC Circuits

Parallel RC, RL, and RLC Circuitsby 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

Berkeleys 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

dQi

dt= (1a) ( )

t

Q i d

= (1b)

R Rv Ri= (2a)1

R Ri vR

= (2b)

( )1 t

C C

Qv i d

C C

= = (3a) CC

dvi C

dt= (3b)

LL

div L

dt= (4a) ( )

1 tL Li v d

L

= (4b)

Time Constants

RL LR

= (5a) RC RC = (5b)

Natural frequency2

0

1

LC = (6)

Kirchhoffs voltage law (KVL): the net voltage (potential) change around a circuit is 0. You

end up where you started.

Kirchhoffs current law (KCL): the total current into a node = the total current leaving the node.

The electrons have to go somewhere.


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First Order Series RC circuit

+

-

s

C

R

KVL: R C sv v v+ = (7)

Write in terms of Cv :

From equation (2a)R R

v Ri= (8)

There is only one current in the loop so R Ci i i= = (9)

From equation (3b) CR Cdv

v Ri R C dt

= =

(10)

Plug into equation (7)C

C s

dv

RC v vdt + = (11a)

Rewrite using equation (5a) CRC C sdv

v vdt

+ = (11b)

First Order Series RL circuit

+

-

R

L

s

KVL: R L sv v v+ = (12)

Write in terms ofC

v :

From equation (2a) R Rv Ri= (13)

From equation (4a) LL

div L

dt= (14)

There is only one current in the loop so R Li i i= = (15)

Plug into equation (12) sdi

L Ri vdt

+ = (16)

Divide by L1

s

di Ri v

dt L L+ = (17a)

Rewrite using equation (5a)1 1

s

RL

dii v

dt L+ = (17b)


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First Order Parallel RC circuit

s

CR

s

R

C

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 vR

= (20)

From equation (3b) Cdv

i Cdt

= (21)

Plug into equation (18)1

s

dvC v i

dt R+ = (22)

Divide by C1 1

s

dvv i

dt RC C + = (23a)

Rewrite using equation (5b)1 1

s

RC

dvv i

dt C+ = (23b)

First Order Parallel RL circuit

s

LR

s

LR

KCL: L R si i i+ = (24)

Write in terms of Li :

From equation (2b)1

R Ri vR

= . (25)

There is only one voltage drop so R Lv v v= = (26)

From equation (4a)1 1 L

R L

dii v L

R R dt

= =

(27)

Plug into equation (24) LL sdiL

i iR dt

+ = (28a)

Rewrite using equation (5a) LL RL sdi

i idt

+ = (28b)


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Second Order Series RLC circuit

+

-

C

R

L

s

KVL: L R C sv v v v+ + = (29)

Write in terms of the loop current i , (equations (2a). (3a), (4a)).

L

div L

dt= Rv Ri= ( )

1 tCv i d

C

= (30)

Plug into (29)L R C s

v v v v+ + = :

( )1 t

s

diL Ri i d v

dt C

+ + = (31)

The integral is a problem so take the time derivative of every term in (31)2

2

1 sdvd i diL R idt dt C dt

+ + = (32)

Divide by L2

2

1 1 sdvd i R di idt L dt LC L dt

+ + = (33a)

Rewrite using equations (5a) and (6)22

02

1 1 s

RL

dvd i dii

dt dt L dt

+ + = (33b)


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Second Order Parallel RLC circuit

s

CLR

s

LR

C

LC

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 dL

=

1R Ri v

R=

CC

dvi C

dt=

(35)

Plug into (34) L R C si i i i+ + =

:

( )

1 1t CL R s

dv

v d v C iL R dt

+ + = (36)

The integral is a problem so take the time derivative of every term in (36). Reorder.2

2

1 1 sdid v dvC vdt R dt L dt

+ + = (37)

Divide by C2

2

1 1 1 sdid v dv vdt RC dt LC C dt

+ + = (38a)

Rewrite using equations (5b) and (6)2

2

02

1 1 s

RC

did v dvv

dt dt C dt

+ + = (38b)


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Summary

Circuit Differential Equation Form

First Order Series RC circuit 1 1CC s

RC RC

dvv v

dt + = (11b)

First Order Series RL circuit 1 1s

RL

dii v

dt L+ = (17b)

First Order Parallel RC circuit 1 1s

RC

dvv i

dt C+ = (23b)

First Order Parallel RL circuit 1 1LL s

RL RL

dii i

dt + = (28b)

First Order General Form

( )

1dxx f t

dt + =

Second Order Series RLC circuit 22

02

1 1 s

RL

dvd i dii

dt dt L dt

+ + = (33b)

Second Order Parallel RLC circuit 22

02

1 1 s

RC

did v dvv

dt dt C dt

+ + = (38b)

Second Order General Form

( )

22

02

1d x dxx f t

dt dt

+ + =

Time ConstantsRL

L

R = (5a)

RC RC = (5b)

Natural frequency 20

1

LC = (6)

series and parallel equations

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