Chapter 5: Energy Storage and Dynamic...
Transcript of Chapter 5: Energy Storage and Dynamic...
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