Chapter 5: Energy Storage and Dynamic Circuits

Chapter 5: Outline

Capacitor (Brief)

dtdvCvvip == :power ousinstantane

221 :energy stored ousinstantane Cvw =

Electrical memory

Branch equation

vc is the preferred variable.

∫∫∞−

+==t

tdi

C

ttvdi

Ctv

0

)(1)0()(1)( λλλλ

∫

+

−+−=+ jt

jtdi

Cjtvjtv λλ)(1)()(

Voltage continuity when the current is finite.

Inductor (Brief)

dtdiLivip == :power ousinstantane

221 :energy stored ousinstantane Liw =

Branch equation

iL is the preferred variable.

Electrical memory

Current continuity when the voltage is finite.

∫∫∞−

+==t

tdv

L

ttidv

Lti

0

)(1)0()(1)( λλλλ

Dynamic Circuits

Dynamic Circuits

l A circuit is dynamic when currents or voltages are time-varying.

l Dynamic circuits are described by differential equations.

l Order of the circuit is determined by order of the differential equation.

l The differential equations are derived based on Kirchhoff’s laws and device (branch) equations.

First-Order Dynamic Circuits

RidtdiLv += :formdirect Ri

dtdiLv += :formindirect

)(01 :ial)(different form Generic tfyadtdy

a =+

Forcing function

First-Order Circuits

ivRdt

dvc =+

1∫ ∞−

=+⇒+=t

dtdv

iCdt

diRid

CRiv

1

1λ

KCL KVL

or vvdt

dvRC c

c =+Generally, use vC or iL.

Second-Order Circuits

l Second-order circuits: circuits described by a second-order differential equation.

)(012

2

2 :form Generic tfyadtdya

dt

yda =++

Second-Order Circuits

CRL vvvv ++= :KVL

iCdt

diR

dtid

Ldtdv 1

:form aldifferentiIn 2

2

++=

Example 5.8: Second-Order Amplifier Circuit

inoutout v

dtdv

RCdtvd

LC β−=+2

2

dtdvC

i out

β−=Ri

dtdi

Lvin +=

Response: - Natural, Forced- Transient, Steady state- many more,…

Natural Response

l Natural response yN(t) is the solution of the circuit equation with the forcing function set to zero. It is also known as the complementary solution.

l With the forcing function set to zero, the differential equation becomes homogeneous.

l A homogeneous differential equation can be solved using the characteristic equation along with the initial condition.

l First-order circuits require one initial condition. Second-order circuits require two initial conditions.

First-Order Circuits

001 =+ NyadtNdy

a

0

1

0

01

)0(condition initialby determined is

)(

0 :equation sticCharacteri

1

0

YAyA

AeAety

aa

s

asa

N

taa

stN

==

==

−=⇒

=+

+

−

Second-Order Circuits

0012

2

2 =++ NyadtNdy

adt

Nyda

dtdy

yAA

eAeAty

aaaaa

s

asasa

NN

ttN

)0( and )0( conditions initial by two determined are ,

)(

24

,

0 :equation sticCharacteri

21

21

2

02211

21

012

2

21

++

+=

−±−=⇒

=++

αα

αα

Stable Circuit

l A circuit is stable if the circuit variable yN(t) à0, as tà∞.

l A circuit is exponentially stable if the circuit variable yN(t) à 0, as tà∞ in an exponential form.

Example 5.9: Capacitor Discharge

C=300µFR=2MΩv(0-)=1000V 0,1000)(

y)(continuit 1000)0()0(

)(60011

01

0

600

600/

>=

===

=

−=−=⇒=+

=+

−

−+

−

tVetv

vvA

AetvRC

sRCs

vdt

dvRC

t

N

NN

tN

NN

Forced Response

l Forced response yF(t) is the solution of the inhomogeneous differential equation (i.e., the forcing function is not zero), independent of any initial conditions. The solution is also known as the particular solution.

l Please refer to Table 5.3 for method of undetermined coefficients.

Table 5.3

Example 5.10 Sinusoidal Forced Response

R=4ΩL=0.1HV(t)=25sin30t

Example 5.10 Sinusoidal Forced Response

))30cos()((or 30sin430cos3)(

4,3 have weon,substitutiafter 30sin30cos)(

30sin2541.0

43

43

θ+=+−=

=−=+=

=+

tAtittti

kktktkti

tidtdi

FF

F

FF

If the forcing function contains any term proportional to a component of the natural response (i.e., excitation of a natural frequency), then the term must be multiplied t.

Example 5.11: Exponential Forced Response

btev

ss

vidtdi

−=

−=⇒=+

=+

10

40041.0 :solution shomogeneou

41.0

Aeti

k

ekti

b

tF

tF

20

2

202

5)(

5

)(

20 if

−

−

=

==

=

Ateti

k

tekti

b

tF

tF

40

2

402

100)(

100

)(

40 if

−

−

=

=

=

=

Complete Response

l Complete response is the sum of the natural response and the forced response, i.e., y(t) = yF(t) + yN(t) .

l The constants in yN(t) are evaluated from the initial conditions on with the complete response.

l For a stable circuit, y(t)= yF(t), as tà∞, since yN(t)à0, as tà∞.

l The circuit is in the steady state if yN(t) is negligible compared to yF(t).

l Before arriving the steady state, the circuit is in the transient state.

Example 5.12: Complete Response Calculation (RL)

14140)0(

280sin2280cos14)()()( :response complete

280sin2280cos14)( :solution particular

)( :solution shomogeneou

0for ,280sin400)(

0for 0)( ,41.0

40

40

=+−==

+−=+=

+−==

>=

<==+

+

−

−

AAi

ttAetititi

ttti

Aeti

tttv

ttividtdi

tFN

F

tN

Example 5.12: Complete Response Calculation (RL)

Chapter 5: Problem Set

l 42, 44, 45, 48, 54, 61, 64