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6.002x CIRCUITS AND ELECTRONICS

Damped Second-Order

Systems

Review

C A B

5V

+!–!

5V

CGS large loop

2KΩ 50Ω

2KΩ S  

LvA 5

0

vB

0

vC

0

50Ω

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

Let’s 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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Let’s 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

=++

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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 ωαα −+−=

os 222 ωαα −−−=

characteristic equation

LC1

o =ω

LR2

=α

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Total solution 3  

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v(t) =VI + A1e−αte

α2−ω2o( )t + A2e−αte− α2−ω2o( )t

Let’s stare at this a while longer… 3 v(t) =VI + A1e−α+ α2−ω2o( )t + A2e

−α− α2−ω2o( )t

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v(t) =VI + A1e−αte

α2−ω2o( )t + A2e−αte− α2−ω2o( )t3

Overdamped o  

ω  α   >  

Underdamped o  ω  α   <  

Critically damped o  

ω  α   =  

Overdamped o  ω  α   >  

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Underdamped o  ω  α  <  3 v(t) =VI + A1e−αte

α2−ω2o( )t + A2e−αte− α2−ω2o( )t

Overdamped o  

ω  α   >  

Underdamped o  ω  α   <  

Critically damped o  

ω  α   =  

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Underdamped contd… o  ω  α  <  3   teKteKVtv d

td

tI ωω

αα sincos)( 21−− ++=

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Underdamped contd… o  ω  α  <  

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

v(t) =VI −VIe−αt cosωdt −VI

αωd

e−αt sinωdt3  

⎟⎟⎠

⎞⎜⎜⎝

⎛−+=+ −

1

2122

2121 tancossincos A

AAAAA θθθ

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Critically damped o  ω  α   =  

Section 13.2.3

3  

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Remember this? We have now closed the loop…

vA 5

0

vB

0

vC

0

50Ω

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