ulaby equations
DESCRIPTION
equations from circuits ulaby 2nd edition good pdf annotatedTRANSCRIPT
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Circuitsby Fawwaz T. Ulaby and Michel M. Maharbiz
Equations
Fawwaz T. Ulaby and Michel M. Maharbiz, Circuits c 2013 National Technology Press
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Chapter 1: Circuit Terminology
Chapter 2: Resisitive Circuits
Chapter 3: Analysis Techniques
Chapter 4: Operational Amplifiers
Chapter 5: RC and RL First-Order Circuits
Chapter 6: Circuit Analysis by Laplace Transform
Chapter 7: ac Analysis
Chapter 8: ac Power
Chapter 9: Frequency Response of Circuits and Filters
Chapter 10: Three-Phase Circuits
Chapter 11: Magnetically Coupled Circuits
Chapter 12: Fourier Analysis Techniques
Fawwaz T. Ulaby and Michel M. Maharbiz, Circuits c 2013 National Technology Press
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I =VR
(1.2)
i=dqdt
(A) (1.3)
q(t) = t
i dt (C) (1.6)
p= i (W) (1.9)
n
k=1
pk = 0 (1.10)
Fawwaz T. Ulaby and Michel M. Maharbiz, Circuits c 2013 National Technology Press
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R=`
A=
`
A() (2.2)
= iR (2.3)
p= i = i2R=2
R(W) (2.4)
G=1R
(S) (2.5)
p= i = G2 (W) (2.7)
N
n=1
in = 0 (KCL) (2.8)
N
n=1
n = 0 (KVL) (2.11)
Req =N
i=1
Ri (resistors in series) (2.22a)
i =(
RiReq
)s (2.22b)
1Req
=N
i=1
1Ri
(resistors in parallel) (2.31)
Geq =N
i=1
Gi (conductances in parallel) (2.34)
Fawwaz T. Ulaby and Michel M. Maharbiz, Circuits c 2013 National Technology Press
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R1 = R2 (2.38a)
is =sR1
(2.38b)
R1 =RbRc
Ra+Rb+Rc(2.42a)
R2 =RaRc
Ra+Rb+Rc(2.42b)
R3 =RaRb
Ra+Rb+Rc(2.42c)
Ra =R1R2+R2R3+R1R3
R1(2.43a)
Rb =R1R2+R2R3+R1R3
R2(2.43b)
Rc =R1R2+R2R3+R1R3
R3(2.43c)
R1 = R2 = R3 =Ra3
(if Ra = Rb = Rc) (2.44a)
Ra = Rb = Rc = 3R1 (if R1 = R2 = R3) (2.44b)
Rx =(R2R1
)R3 (balanced condition) (2.47)
Vout ' V04(RR
)(2.48)
Fawwaz T. Ulaby and Michel M. Maharbiz, Circuits c 2013 National Technology Press
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GV = It (3.26)
RI = Vt (3.27)
iN =ThRTh
(3.37a)
RN = RTh (3.37b)
RL Rs (maximum current transfer) (3.39)
RL Rs (maximum voltage transfer) (3.40)
RL = Rs (maximum power transfer) (3.42)
pmax =2s RL
(RL+RL)2=
2s4RL
(3.43)
Fawwaz T. Ulaby and Michel M. Maharbiz, Circuits c 2013 National Technology Press
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ip = in = 0 (ideal op-amp model) (4.16)
p = n (ideal op-amp model) (4.17)
G=os
=(RfRs
)(4.24)
o = G11+G22 (4.31)
o =(RfR
)[1+2] (equal gain) (4.32)
o =(1+2) (inverted adder) (4.33)
o =(RfR1
)1+
(RfR2
)2+ +
(RfRn
)n (4.34)
o =[(
R4R3+R4
)(R1+R2R1
)]2
(R2R1
)1 (4.40)
o =(R2R1
)(21) (equal gain) (4.44)
o =(
1+2RR2
)(21) (4.56)
Fawwaz T. Ulaby and Michel M. Maharbiz, Circuits c 2013 National Technology Press
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u(t) =
{0 for t < 01 for t > 0
(5.2)
r(tT ) ={
0 for t T(tT ) for t T (5.4)
r(t) = t
u(t) dt = t u(t) (5.6)
rect(tT
)
=
0 for t < (T /2)1 for (T /2) t (T + /2)0 for t > (T + /2)
(5.8)
C =q
(F) (any capacitor) (5.20)
C =Ad
(parallel-plate capacitor) (5.21)
C =2pi`
ln(b/a)(coaxial capacitor) (5.22)
(t) = (t0)+1C
tt0i dt (5.24)
(t) =1C
t0i dt
(capacitor uncharged before t = 0)
(5.25)
w(t) =12C 2(t) (J) (5.28)
Fawwaz T. Ulaby and Michel M. Maharbiz, Circuits c 2013 National Technology Press
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1Ceq
=N
i=1
1Ci
=1C1
+1C2
+ + 1CN
(capacitors in series)
(5.35)
Ceq =N
i=1
Ci (capacitors in parallel) (5.40)
C11 =C22 (5.46)
L=N2S`
(solenoid) (5.51)
i(t) = i(t0)+1L
tt0 dt (5.55)
Leq =N
i=1
Li = L1+L2+ +LN
(inductors in series) (5.62)
1Leq
=N
i=1
1Li=
1L1
+1L2
+ + 1LN
(inductors in parallel) (5.65)
ddt
+a = 0 (source-free) (5.69)
(t) = (0) et/ (natural response) (5.77)
= RC (s) (5.78)
Fawwaz T. Ulaby and Michel M. Maharbiz, Circuits c 2013 National Technology Press
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i(t) =VsR
et/ u(t) (for t 0)(natural response) (5.82)
ddt
+a = b (5.87)
(t) = ()+ [(0)()]et/ (for t 0)(switch action at t = 0) (5.95)
(t) = ()+ [(T0)()]e(tT0)/(for t T0) (5.97)
(switch action at t = T0)
i(t) = i(0) et/ (for t 0)(natural response) (5.102)
=1a=
LR
(5.103)
i(t) = i()+ [i(0) i()]et/ (for t 0)(switch action at t = 0) (5.106)
i(t) = i()+ [i(T0) i()]e(tT0)/(for t T0)
(switch action at t = T0)
(5.107)
out(t) = 1RC tt0i(t ) dt +out(t0) (5.128)
out(t) = 1RC t
0i(t ) dt (if out(0) = 0) (5.129)
Fawwaz T. Ulaby and Michel M. Maharbiz, Circuits c 2013 National Technology Press
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out =RC didt (5.130)
tfall =CnD+C
pD
g(5.155)
Fawwaz T. Ulaby and Michel M. Maharbiz, Circuits c 2013 National Technology Press
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damping coefficient =R2L
(Np/s) (6.1a)
resonant frequency 0 =1LC
(rad/s) (6.1b)
(series RLC)
iL+a2iL+b2iL = c2 (6.12)
=1
2RC(parallel RLC) (6.14)
(tT ) = 0 for t 6= T
(tT ) dt = 1(6.15a)
(6.15b)
u(tT ) = t
(T ) dddt
[u(tT )] = (tT )
(6.19a)
(6.19b)
x(t) (tT ) dt = x(T )
(sampling property)
(6.23)
e j = cos + j sin (6.27)
x= |z|cos y= |z|sin|z|=
x2+ y2 = tan1(y/x) (6.30)
z = (x+ jy) = x jy= |z|e j = |z| (6.31)
|z|=
zz (6.32)
Fawwaz T. Ulaby and Michel M. Maharbiz, Circuits c 2013 National Technology Press
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F(s) =LLL[ f (t)] =
0f (t) est dt (6.40)
(t) 1 (6.46)
[cos(t)] u(t)s
s2+2(6.47)
f (at)1a
F( sa
)a> 0
(time-scaling property)
(6.49)
f (tT ) u(tT ) eT s F(s)T 0
(time-shift property)
(6.53)
f =d fdt
s F(s) f (0)(time-differentiation property)
(6.58)
f =d2 fdt2
s2 F(s) s f (0) f (0)(second-derivative property) (6.61)
t0
f (t ) dt 1s
F(s)
(time-integration property)
(6.62)
F(s) =A1
s+ p1+
A2s+ p2
+ + Ans+ pn
=n
i=1
Ais+ pi
(6.83)
Fawwaz T. Ulaby and Michel M. Maharbiz, Circuits c 2013 National Technology Press
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A1 = (s+ pi) F(s)s=pi
i= 1,2, . . . ,n(6.84)
B j ={
1(m j)!
dm j
dsm j[(s+ p)m F(s)]
}s=p
j = 1,2, . . . ,m (6.92)
LLL1[(n1)!(s+a)n
]= tn1eat u(t) (6.94)
= Ri V = RI (6.107)
= Ldidt
V = sLIL i(0) (6.110)
i=Cddt
I = sCVC (0) (6.111)
ZR = R, ZL = sL and ZC =1
sC(6.112)
Fawwaz T. Ulaby and Michel M. Maharbiz, Circuits c 2013 National Technology Press
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= 2pi f (rad/s) (7.3)
T =1f
(s) (7.4)
didt
jI (7.21)
i dt
Ij
(7.23)
Z =VI
() (7.29)
ZR =VRIR
= R (7.30)
ZL =VLIL
= jL (7.35)
ZC =VCIC
=1
jC(7.38)
Zeq =N
i=1
Zi (impedances in series) (7.64)
Yeq =N
i=1
Yi (admittances in parallel) (7.66)
Z1 =ZbZc
Za+Zb+Zc(7.69a)
Fawwaz T. Ulaby and Michel M. Maharbiz, Circuits c 2013 National Technology Press
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Z2 =ZaZc
Za+Zb+Zc(7.69b)
Z3 =ZaZb
Za+Zb+Zc(7.69c)
Za =Z1Z2+Z2Z3+Z1Z3
Z1(7.70a)
Zb =Z1Z2+Z2Z3+Z1Z3
Z2(7.70b)
Zc =Z1Z2+Z2Z3+Z1Z3
Z3(7.70c)
Z1 = Z2 = Z3 =Za3, if Za = Zb = Zc (7.71a)
Za = Zb = Zc = 3Z1 if Z1 = Z2 = Z3 (7.71b)
21
=N2N1
= n (7.120)
i2i1=
N1N2
(7.121)
Fawwaz T. Ulaby and Michel M. Maharbiz, Circuits c 2013 National Technology Press
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Xav =1T
T0
x(t) dt (8.5)
cos2 x=12+
12
cos2x
1T
T0
cos2(
2pintT
+1)
dt =12
and
1T
T0
sin2(
2pintT
+2)
dt =12
(8.10)
Ieff =
1T
T0
i2(t) dt (8.13)
Xrms = Xeff =
1T
T0
x2(t) dt (8.14)
1T
T0
cos(nt+) dt = 0 (n= 1,2, . . .) (8.22)
Pav =VmIm
2cos( i) (W) (8.23)
Pav =VrmsIrms cos( i) (W) (8.24)
Pav =VrmsIrms =V 2rmsR
(purely resistive load)
(8.25)
Pav =VrmsIrms cos90 = 0
(purely reactive load)
(8.26)
Fawwaz T. Ulaby and Michel M. Maharbiz, Circuits c 2013 National Technology Press
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S =12
VI (VA) (8.29)
S = VrmsIrms (VA) (8.32)
Q=VrmsIrms sin( i) (VAR) (8.34)
Pav =Re[S] (average absorbed power) (8.36a)
Q= Im[S] (peak exchanged power) (8.36b)
Pav =Re[S] =12|I|2R= I2rmsR (W) (8.39a)
Q= Im[S] =12|I|2X = I2rmsX (VAR) (8.39b)
n
i=1
Pavi = 0 andn
i=1
Qi = 0 (8.40)
S= |S|=P2av+Q2 =VrmsIrms (8.43)
pf=PavS
= cos( i) (8.44)
pf= cosz (8.49)
pf=
{cosZL for the RL circuit alonecosnew for the compensated circuit
(8.53)
XL =Xs (8.63)
Fawwaz T. Ulaby and Michel M. Maharbiz, Circuits c 2013 National Technology Press
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RL = Rs (8.64)
ZL = Zs (maximum power transfer) (8.65)
Pav(max) =18|Vs|2RL
(8.66)
Fawwaz T. Ulaby and Michel M. Maharbiz, Circuits c 2013 National Technology Press
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M(c) =M0
2= 0.707M0 (9.5)
0 =1LC
(RLC circuit) (9.11)
R = KmR,
L = KmL,
C =CKm
,
and
=
(magnitude scaling only)
(9.23)
R = R,
L =LKf,
C =CKf,
and
= Kf
(frequency scaling only)
(9.25)
R = KmR,
L =KmKf
L,
C =1
KmKfC,
and
= Kf
(magnitude and frequency scaling)
(9.26)
G= XY G [dB] = X [dB]+Y [dB] (9.31)
Fawwaz T. Ulaby and Michel M. Maharbiz, Circuits c 2013 National Technology Press
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G=XY
G [dB] = X [dB]Y [dB] (9.32)
0 =1LC
(9.48)
c1 =R2L
+
(R2L
)2+
1LC
(9.50a)
c2 =R2L
+
(R2L
)2+
1LC
(9.50b)
B= c2c1 =RL
(9.51)
0 =c1c2 (9.52)
Q= 2pi(WstorWdiss
)=0
(9.53)
Q=0LR
(bandpass filter) (9.61)
Q=0B
(bandpass filter) (9.62)
c1 =1RC
(RC filter) (9.72)
Fawwaz T. Ulaby and Michel M. Maharbiz, Circuits c 2013 National Technology Press
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Y-Source Configuration
V1 =VYs0
V2 =VYs120
V3 =VYs240(10.1)
-Source Configuration
V12 = V1V2=VYs0VYs120
=
3VYs30 =Vs30
V23 = V2V3 =Vs90V31 = V3V1 =Vs150
with Vs =
3VYs
(10.3)
VN = 0 (balanced network) (10.8)
Z = 3ZY (10.12)
PT = 3VYLIYL cosYQT = 3VYLIYL sinY
(balanced network)
(10.18a)
(10.18)
ST = PT+ jQT =
3VLILY
(balanced Y-load)
(10.20)
PT(t) = 3VYLIYL cosY (10.27)
PT = P1+P2(any 3-phase load)
(10.41)
QT = 3V 2LZ
sin (10.43)
Fawwaz T. Ulaby and Michel M. Maharbiz, Circuits c 2013 National Technology Press
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QT =
3 (P2P1) (balanced load) (10.45)
Fawwaz T. Ulaby and Michel M. Maharbiz, Circuits c 2013 National Technology Press
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2 =M21 di1dt (11.6)
1 = L1di1dt
+Mdi2dt
and
2 = L2di2dt
+Mdi1dt
(11.8a)
(11.8b)
1 = L1di1dtM di2
dtand
2 = L2di2dtM di1
dt
(11.9a)
(11.9b)
k =ML1L2
(11.21)
M(max) =L1L2 (11.22)
(perfectly coupled transformer with k = 1)
ZR =2M2
R2+ jL2+ZL(11.25)
[V1V2
]=[jL1 jMjM jL2
][I1I2
](transformer)
(11.27c)
[V1V2
]=[j(Lx+Lz) jLz
jLz j(Ly+Lz)
][I1I2
](T-equivalent circuit) (11.28)
Fawwaz T. Ulaby and Michel M. Maharbiz, Circuits c 2013 National Technology Press
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Lx = L1MLy = L2M
and
Lz =M
(transformer dots on same ends)
(11.29a)
(11.29b)
(11.29c)
Lx = L1+M
Ly = L2+M
and
Lz =M(transformer dots on opposite ends)
(11.30a)
(11.30b)
(11.30c)
La =L1L2M2L1M
Lb =L1L2M2L2M
and
Lc =L1L2M2
M(transformer with dots on same ends)
(11.31a)
(11.31b)
(11.31c)
M(max) =L1L2
(ideal transformer)
(11.34)
L2L1
=N22N21
= n2 (11.35)
V2V1
= n (ideal transformerwith dots on same side) (11.36)
I2I1=
1n
(ideal transformerdots on same ends) (11.39)
Fawwaz T. Ulaby and Michel M. Maharbiz, Circuits c 2013 National Technology Press
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V2V1
=N2N
=N2
N1+N2and
I2I1=
V1V2
=N1+N2
N2(step-down autotransformer)
(11.42)
V2V1
=NN2
=N1+N2
N2and
I2I1=
V1V2
=N2
N1+N2(step-up autotransformer)
(11.43)
VLsVLp
=ILpILs
= n (Y-Y and -) (11.44)
ST =
3VLIL (Y and ) (11.45)
VLsVLp
=ILpILs
=n3
(Y-)
and
VLsVLp
=ILpILs
=
3 n (-Y)
(11.46)
Fawwaz T. Ulaby and Michel M. Maharbiz, Circuits c 2013 National Technology Press
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f (t) = a0+
n=1
(an cosn0t+bn sinn0t)
(sine/cosine representation) (12.15)
a0 =1T
T0
f (t) dt
an =2T
T0
f (t) cosn0t dt
bn =2T
T0
f (t) sinn0t dt
(12.17a)
(12.17b)
(12.17c)
An =a2n+b2n
and
n =
tan1
(bnan
)an > 0
pi tan1(bnan
)an < 0
(12.26)
Ann = an jbn (12.27)
f (t) = a0+
n=1
An cos(n0t+n)
(amplitude/phase representation)
(12.28)
Even Symmetry: f (t) = f (t)
a0 =2T
T/20
f (t) dt,
an =4T
T/20
f (t) cos(n0t) dt, (12.31)
bn = 0,
An = |an|, and n ={
0 if an > 0180 if an < 0
Fawwaz T. Ulaby and Michel M. Maharbiz, Circuits c 2013 National Technology Press
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Odd Symmetry: f (t) = f (t)a0 = 0, an = 0,
bn =4T
T/20
f (t) sin(n0t) dt, (12.32)
An = |bn| and n ={90 if bn > 090 if bn < 0
Solution Procedure:Fourier Series Analysis Procedure
Step 1: Express s(t) in terms of an amplitude/phase Fourier series as
s(t) = a0+
n=1
An cos(n0t+n) (12.33)
with Ann = an jbn.
Step 2: Establish the generic transfer function ofthe circuit at frequency as
H() = Vout when s = 1cost. (12.34)
Step 3: Write down the time-domain solution as
out(t) = a0 H( = 0)
+
n=1
AnRe{H( = n0) e j(n0t+n)}.(12.35)
Pav =VdcIdc+12
n=1
VnIn cos(nin) (12.43)
Fawwaz T. Ulaby and Michel M. Maharbiz, Circuits c 2013 National Technology Press
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cn =an jbn
2and
cn =an+ jbn
2= cn
(12.47)
f (t) =
n=
cne jn0t
(exponential representation)
(12.48)
cn =1T
T/2T/2
f (t) e jn0t dt (12.50)
sinc(x) =sinxx
(12.54)
f (t) =
n=
cne jn0t (12.58a)
cn =1T
T/2T/2
f (t) e jn0t dt (12.58b)
F() =F [ f (t)] =
f (t) e jt dt (12.62a)
f (t) =F1[F()] =1
2pi
F() e jt d (12.62b)
K1 f1(t)+K2 f2(t) K1 F1()+K2 F2()
(linearity property) (12.65)
Fawwaz T. Ulaby and Michel M. Maharbiz, Circuits c 2013 National Technology Press
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(t t0) e jt0and
(t) 1
(12.67a)
(12.67b)
e j0t 2pi (0)and
1 2pi ()
(12.68a)
(12.68b)
e j0t f (t) F(0)(frequency-shift property)
(12.69)
f (t t0) e jt0 F()(time-shift property)
(12.70)
cos0t pi[ (0)+ (+0)] (12.71)
sin0t jpi[ (+0) (0)] (12.72)
Aeat u(t)A
a+ j, for a> 0 (12.73)
sgn(t) = u(t)u(t) (12.74)
u(t) pi ()+1j
(12.79)
f (t) j F() (12.81)
cos0t f (t)12[F(0)+F(+0)] (12.82)
Fawwaz T. Ulaby and Michel M. Maharbiz, Circuits c 2013 National Technology Press
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F() = F()(reversal property)
(12.85)
f 2(t) dt =1
2pi
|F()|2 d
(Parsevals theorem)
(12.86)
Fawwaz T. Ulaby and Michel M. Maharbiz, Circuits c 2013 National Technology Press
Chapter 1Chapter 2Chapter 3Chapter 4Chapter 5Chapter 6Chapter 7Chapter 8Chapter 9Chapter 10Chapter 11Chapter 12