8/11/2019 amt026.pdf

1/5

www.NewCircuits.com

R L C circuit with one switchProblem #1

#1) In the circuit showed in figure 1, answer the questions below:

#1-1) Find so that voltage of the capacitor becomes critically damped.1R

#1-2) Find so that2

R (0) 100 Vv =

#1-1) Find ( )v t in the point 1 mst=

Figure 1(Solution in the next page)

http://www.newcircuits.com/http://www.newcircuits.com/http://www.newcircuits.com/

8/11/2019 amt026.pdf

2/5

Solution:

For the current source delivers 0 Ampere and the switch is closed (see

figure 2). For the current source has been delivering 0.5 Amperes for a

long time to the circuit (see figure 3).

0t>0t

1R 2R

( )

2

2 3 6 6

23 3

2

2 3 3

2

1 1

010 10 4 10

10 250 10 0

10 250 10 0

500 0

500

d v dv

vdt dt

d v dvv

dt dt

p p

p

p

+ +

+ + =

+ + =

+ =

=

=

So the homogeneous response becomes,

( ) 5001 2( ) t

v t A A t e= + (3)

There are two conditions, the first one is (0) 100 Vv = ,

( ) 500 01 2 1(0) 100 0 100v A A e A = = + =

For the second condition,

6 500

2

6 500 500

2 2

6 500 0 500 0

2 2

10 (100 )

( ) 10 500 (100 )

(0 ) 10 500 (100 0)

t

C

t t

C

C

dv dI C A t e

dt dt

I t A e A t e

I A e A e

+

= = +

= +

= +

6 3

2(0 ) 10 50 10

CI A+ = (4)

Here we need to discuss about the value of (0 )CI + . Although its value is

definitely zero for , but is it zero for 00t< + ? First, we calculate the (0 )LI and

:1(0 )RI

1(0 ) (0 ) (0 ) 0.5C L RI I I A

+ + =

A

10 (0 ) (0 ) 0.5L RI I

+ + = (5)

And,

8/11/2019 amt026.pdf

5/5

12 1(0 ) (0 )

L RR I R I =

1250 (0 ) 1000 (0 )

L RI I = (6)

(5) and (6) yield,

1

(0 ) 0.4

(0 ) 0.1

L

R

I A

I A

=

=

We know that the current of an inductor won't change immediately. Let's say

the current of the resistor for 01R + can be changed so that the current of the

capacitor remains zero for 0 , in this case the current of the inductor must pass

through the resistor so we will lose 0.1 Amperes in the circuit immediately

and this is not possible. Therefore the currents

+

1R

(0 )LI

and cannot be

changed and must pass through the capacitor, so

1(0 )

RI

1(0 ) (0 ) (0 ) 0.5C L RI I I

+ + += + = A (7)

(4) and (7) yield,6 3

20.5 10 50 10A =

3

2450 10A =

Then,

( )3 50( ) 100 450 10 t0v t t = e For t=1 ms,

( )33 3 500 10(1ms) 100 450 10 10v e

= 0.5(1ms) 350v e =

(1 ) 212 Vv ms =

Last updated on April 9, 2009

Rasoul Mojtahedzadeh

Back to Articles

http://www.newcircuits.com/articles.phphttp://www.newcircuits.com/articles.phphttp://www.newcircuits.com/articles.phphttp://www.newcircuits.com/articles.php