06_Transients

Prévia do material em texto

Page 1 
 
 
 
GENERAL NOTES: 
 
1) Vectors are herein denoted by arrows. In addition, the book also 
employs a “^” over the quantity to denote a vector phasor or 
complex vector (i.e., each vector component is a complex 
number or function). Here it is assumed that all vectors are 
phasors (time-harmonic fields or permanent response), unless 
the time dependence is explicitly indicated. 
2) QUICK REVIEW: vector phasor zzyyxx aAaAaAA

++= , 
where Ax = Ax1 + jAx2, Ay = Ay1 + jAy2, and Az = Az1 + jAz2. In 
general 
*2 AAA

•= , where “*” denotes the complex 
conjugate only (not also transposed as in linear algebra). Note 
that it denotes the absolute value with respect to both the real 
and imaginary parts as well as with respect to the coordinate 
directions. For example, If 
( ) ( ) zzxx ajAajAA

2z12x1 AA +++= then 
2
2
2
z1
2
2
2
x1
2
AA zx AAA +++=

. Note that this is different than 
taking the dot-product only: 
2
221
2
z1
2
221
2
x1 2A2A zzzxxx AAjAAAjAAA −++−+=•

 
3) MKS (meter-kilogram-second) system is used here and in the 
book. 
4) All quoted text (“ ”) is the professor’s own, written freely 
and expressing his own views of the subjects, derived from 
experience and discussions but normally not available in 
references. 
 
 Page 2 
TRANSIENTS ON TRANSMISSION LINES 
• Laplace transforms (s-domain) are used to assess the transitory 
response (before reaching the time-harmonic or permanent state 
response analyzed previously, assuming the system is not 
oscillatory and converges to a permanent response: “settling 
time”). 
• All initial conditions are assumed to be zero. 
• Transmission lines: ( )
sCG
sLR
sZ
+
+
=0 and a propagation 
constant of ( ) ( )( )sCGsLRs ++= 
• The voltage and currents at any point x along the transmission 
line are v(x,t) and i(x,t) with transforms V(x,s) and I(x,s). 
• Consider that a forward wave vf(t), with a Laplace transform 
Vf(s), is launched at t=0 toward the load, as shown by Fig. 
10.54. “Note that at t=T, the wave reaches the load (x=d): 
d
f
dx
f eVeV
 −=− =)( . Part of the wave is reflected 
accordingly to the reflection coefficient and starts traveling back 
to the generator: 
)( dxd
fL eeV
−−−  . At t=2T the wave is back 
at the generator and is partially reflected back to the load 
accordingly to ΓG (generator) and so on, until its amplitude is 
reduced to a negligible level. At that instant in time we can say 
that the system has reached its permanent state or response.” 
 Page 3 
Summing all waves we obtain 
 
( ) ( )  xdGLdGLdGLf eeeeVsxV  −−−− ++++= ...1),( 32222
( ) ( )  )(32222 ...1 dxdLdGLdGLdGLf eeeeeV −−−−−− +++++ 
(1) 
 
where the terms within the first pair of brackets represent all the 
forward traveling waves (i.e., from the generator to the load), 
and the remaining terms are related to the backward traveling 
waves. 
 
“A similar expression can be derived for the current. From 
circuit theory we know that V(x,s) = Z0(s) I(x,s), and due to 
the norm used with the direction of current we need to 
introduce a sign inversion for the terms related to the waves 
traveling backwards (i.e., from the load to the generator): 
 
( ) ( )  xdGLdGLdGLf eeee
Z
V
sxI  −−−− ++++= ...1),(
32222
0
( ) ( )  )(32222 ...1 dxdLdGLdGLdGLf eeeeeV −−−−−− ++++− 
(*) 
 
 Page 4 
Expressions (1) and (*) account for all terms in the s-domain. 
Once they are evaluated, they must be transformed back to the 
time-domain in order to assess the time response (transitory) of the 
transmission line. Although there are an infinite number of terms 
in each expression, the higher order terms normally reach 
negligible levels after a certain period of time, provided the system 
is not oscillatory. In addition, we are normally interested to 
evaluate the response within a certain period [0, t0] and at a 
given location x0, and all terms that have not yet reached x0 
before t0 are zeroed after transforming back to the time-
domain.” 
“LISTED BELOW ARE A FEW USEFUL LAPLACE 
TRANSFORMS” 
( ) ( )  ( ) ( ) jwsdtetftfsF st +=== −

 transitory 
0

 
1. Unit Impulse: ( ) ( ) ( ) 10at 1 ==== sFtttf  
2. Unit Step: ( ) ( ) ( )
s
sFttutf 10for 1 ==== 
3. Shift Theorems (1): ( ) ( )asFtfe at +− 
a. Ex.: ( ) ( )ass
as
tue at +→
+
− let just 1 
4. Shift Theorems (2): ( ) ( ) ( )sFeatuatf as−−− 
“u(t-a) to enforce t>a only; unnecessary if f(t-a) is a step.”