Chapter 5: Energy Storage and Dynamic...
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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)(
+
+=+
jt
jtdi
Cjtvjtv )(
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)(

Dynamic Circuits

Dynamic Circuits
l A circuit is dynamic when currents or voltages are timevarying.
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 Kirchhoffs laws and device (branch) equations.

FirstOrder Dynamic Circuits
RidtdiLv += :formdirect Ri
dtdiLv += :formindirect
)(01 :ial)(different form Generic tfyadtdy
a =+
Forcing function

FirstOrder Circuits
ivRdt
dvc =+
1 =++=
t
dtdv
iCdt
diRid
CRiv
1
1
KCL KVL
or vvdt
dvRC c
c =+Generally, use vC or iL.

SecondOrder Circuits
l Secondorder circuits: circuits described by a secondorder differential equation.
)(012
2
2 :form Generic tfyadtdya
dt
yda =++

SecondOrder Circuits
CRL vvvv ++= :KVL
iCdt
diR
dtid
Ldtdv 1
:form aldifferentiIn 22
++=

Example 5.8: SecondOrder Amplifier Circuit
inoutout v
dtdv
RCdtvd
LC =+22
dtdvC
i out
=Ridtdi
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 Firstorder circuits require one initial condition. Secondorder circuits require two initial conditions.

FirstOrder Circuits
001 =+ NyadtNdya
0
1
0
01
)0(condition initialby determined is
)(
0 :equation sticCharacteri
1
0
YAyA
AeAety
aa
s
asa
N
taa
stN
==
==
=
=+
+

SecondOrder Circuits
0012
2
2 =++ NyadtNdya
dtNyda
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
++
+=
=
=++

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

Example 5.9: Capacitor Discharge
C=300FR=2Mv(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=4L=0.1HV(t)=25sin30t

Example 5.10 Sinusoidal Forced Response
))30cos()((or 30sin430cos3)(
4,3 have weon,substitutiafter 30sin30cos)(
30sin2541.0
43
43
+=+=
==+=
=+
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 t, since yN(t)0, as t.
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
=+==
+=+=
+==
>=

Example 5.12: Complete Response Calculation (RL)

Chapter 5: Problem Set
l 42, 44, 45, 48, 54, 61, 64