# Handouts 6002 L18 Oei12 Gaps Annotated

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• 1

6.002x CIRCUITS AND ELECTRONICS

Damped Second-Order

Systems

• Review

C A B

5V

+!!

5V

CGS large loop

2K 50

2K S

LvA 5

0

vB

0

vC

0

50

t

t

t

5

5

• 3

In the last lecture, we started by analyzing the simpler LC circuit to build intuition

• 4

And for initial conditions

We solved

For input

In the last lecture LC circuit

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In the last lecture LC circuit

Total solution

+!! C L +!

!)(tv

)(tvI

)(ti

• 6 See A&L Section 13.6

Today, we will close the loop on our observations in the demo by analyzing the RLC circuit

+!! C

L

+!

!

i(t)

v(t) vI (t) LC

I v

v(t) 2VI

t

VI

0

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Recall element rules

Lets analyze the RLC network

L:

C:

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Set up the differential equation vA

+!! C L +!

!

R i(t)

v(t) vI(t) Need to get rid of vA

v

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Set up the differential equation differently

v +!!

C

L +!

!

R i

vI

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Solve IvLCvLCdtdv

LR

dtvd 112

2

=++

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Lets solve IvLC

vLCdt

dvLR

dtvd 112

2

=++

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1 Particular solution IvLC

vLCdt

dvLR

dtvd 112

2

=++

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1 Particular solution IvLC

vLCdt

dvLR

dtvd 112

2

=++

IPPP V

LCv

LCdtdv

LR

dtvd 112

2

=++

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Homogeneous solution 2 IvLC

vLCdt

dvLR

dtvd 112

2

=++

• 15

Homogeneous solution 2 0vLC1

dtdv

LR

dtvd

HHH =++2

2

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0vLC1

dtdv

LR

dtvd

HHH =++2

2

Solution to Homogeneous solution 2

Assume solution of the form 2A

vH = AeST , A, S = ?

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Homogeneous solution 2 0vLC1

dtdv

LR

dtvd

HHH =++2

2

Solution to

02 =++LC1s

LRs

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Homogeneous solution 2 0vLC1

dtdv

LR

dtvd

HHH =++2

2

Solution to

02 =++LC1s

LRs

Roots o1s22 +=

os 222 =

characteristic equation

LC1

o =

LR2

=

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Total solution 3

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v(t) =VI + A1ete22o( )t + A2ete

22o( )t

Lets stare at this a while longer 3 v(t) =VI + A1e+ 22o( )t + A2e

22o( )t

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v(t) =VI + A1ete22o( )t + A2ete

22o( )t3

Overdamped o

>

Underdamped o <

Critically damped o

=

Overdamped o >

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Underdamped o < 3 v(t) =VI + A1ete22o( )t + A2ete

22o( )t

Overdamped o

>

Underdamped o <

Critically damped o

=

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Underdamped contd o < 3 teKteKVtv d

td

tI

sincos)( 21 ++=

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Underdamped contd o <

Remember, scaled sum of sines (of the same frequency) are also sines! -- Appendix B.7

v(t) =VI VIet cosdt VId

et sindt3

+=+

1

2122

2121 tancossincos A

AAAAA

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Critically damped o =

Section 13.2.3

3

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Remember this? We have now closed the loop

vA 5

0

vB

0

vC

0

50

t

t

t

5

5

• 27 See Sec. 12.7 of A&L textbook

ringing v(t)

VI 0

t

Easy way: Characteristic equation tells the whole story!

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Intuitive Analysis +!! C

L +!

!)(tvIv

)(ti

R

)0(i)0(v

given -ve +ve

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Next, introduce R: RLC Circuits

More in the next sequence! If you are impatient, see A&L Section 13.2

+!! C

L

+!!vI (t)

i(t)

v(t)

v(t)

t

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What about other variables?

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Parallel RLC Characteristic equation says it all