Part Three
Copyrighted material - Taylor & Francis
Main formulae for Part 3 Advanced circuit theory and technology
Complex numbers
LR–C network
z = a + jb =r (cos θ + j sin θ ) =r ∠θ , √ where j 2 = −1 Modulus, r = |z| = (a 2 + b2 ) b Argument, θ = arg z = tan−1 a Addition: (a + jb) +(c + jb) = (a +c) + j (b +d)
1 fr = 2π
If z 1 =r1 ∠θ1 and z 2 =r2 ∠θ2 then Multiplication: z 1 z 2 =r1r2 ∠(θ1 + θ2 ) z 1 r1 and Division: = ∠(θ1 − θ2 ) z 2 r2
1 fr = √ 2π (LC)
General
Delta–star
V = R + j(X L − XC ) = |Z |∠φ I √ where |Z | = [R 2 + (X L − XC )2 ] and X L − XC φ = tan−1 R 1 I 1 X L = 2π f L XC = Y = = = G + jB 2π f C V Z Series: Z T = Z 1 + Z 2 + Z 3 + · · · 1 1 1 1 Parallel: = + + + ··· ZT Z1 Z2 Z3 P = VI cos φ or P = I R2 R S = VI Q = VI sin φ R Power factor = cosφ = Z If V = a + j b and I = c + j d then P = ac +bd Q = bc −ad S = VI∗ = P + jQ
Z1 =
R–L–C series circuit 1 fr = √ 2π (LC) √ fr = ( f 1 f2 )
ωr L 1 1 L Q= = = R ωr CR R C VL VC fr = = = V V f2 − f1
RD =
R 2L − L/C
L CR
Q=
ωr L IC = Ir R
RC2 − L/C
Determinants
[r ∠θ ]n =r n ∠nθ =r n (cosnθ + j sin nθ )
Z=
LR–CR network
a c a d g
De Moivre’s theorem:
1 R2 − 2 LC L
b = ad − bc d b c e e f = a h h j
d f d e f − b + c j g j g h
ZAZB , etc. Z A + Z B + ZC
Star–delta ZA=
Z1 Z2 + Z2 Z3 + Z3 Z1 , etc. Z2
Part 3
Subtraction: (a +jb) − (c + jd) =(a − c) + j (b −d) Complex equations: If a + jb = c +jd, then a =c and b =d
Impedance matching
|z| =
N1 N2
2
|Z L |
Complex waveforms
I=
2 + ··· I 2 + I2m I02 + 1m
i AV =
1 π
2
π
i d(ωt) 0
form factor =
r.m.s. mean
P = V0 I0 + V1 I1 cos φ1 + V2 I2 cos φ2 + · · · or P = I 2 R P power factor = VI 1 Harmonic resonance: nωL = nωC
Copyrighted material - Taylor & Francis
730 Electrical Circuit Theory and Technology Fourier series If f (x) is a periodic function of period 2π then its Fourier series is given by: f (x) = a 0 +
∞
(an cos nx + bn sin nx)
n=1
where, for the range −π to +π: π 1 f (x) dx a0 = 2π −π 1 π f (x) cos nx dx (n = 1, 2, 3, . . .) an = π −π 1 π f (x) sin nx dx (n = 1, 2, 3, . . .) bn = π −π If f (x) is a periodic function of period L then its Fourier series is given by: ∞
2πnx 2πnx f ( x) = a0 + + bn sin an cos L L
Part 3
n=1
L L where for the range − to + : 2 2 1 L/2 f (x) dx a0 = L −L/2 2πnx 2 L/2 an = f (x) cos dx (n = 1, 2, 3, . . .) L −L/2 L 2πnx 2 L/2 f (x) sin dx (n = 1, 2, 3, . . .) bn = L −L/2 L
Harmonic analysis 1 2 a0 ≈ yk a n ≈ yk cos nx k p p bn ≈
2 p
p
p
k=1
k=1
p
yk sin nx k
k=1
Hysteresis and eddy current Hysteresis loss/cycle = Aαβ J/m3 or hysteresis loss = kh v f (Bm )n W Eddy current loss/cycle = ke (Bm )2 f 2 t 3 W
Dielectric loss Series representation: tan δ = R S ωCS = 1/Q
Parallel representation: tan δ =
1 R p ωC p
Loss angle δ = (90◦ − φ) Power factor = cos φ ≈ tan δ Dielectric power loss = V 2 ωC tan δ
Field theory 2πε0 εr V F/m E = V/m b b ln r ln a a b μ0 μr 1 + ln H/m L= 2π 4 a πε0 εr Twin line: C = F/m D ln a D μ0 μr 1 + ln H/m L= π 4 a
Coaxial cable: C =
Energy stored: in a capacitor, W = 12 CV 2 J; in an inductor W = 21 LI 2 J in electric field per unit volume, D2 J/m3 ω f = 21 DE = 12 ε0 εr E 2 = 2ε0 εr in a non-magnetic medium, B2 ω f = 21 BH = 21 μ0 H 2 = J/m3 2μ0
Attenuators Logarithmic ratios: V2 I2 P2 = 20 lg = 20 lg in decibels =10 lg P1 V1 I1 P2 V2 I2 in nepers = 12 ln = ln = ln P1 V1 I1 Symmetrical T -attenuator: √ √ R0 = (R12 + 2R1 R2 ) = (R OC R SC ) N −1 2N R1 = R0 R2 = R0 N +1 N2 −1 Symmetrical π-attenuator: R1 R22 √ = (R OC R SC ) R0 = R1 + 2R2 2 N −1 N +1 R1 = R0 R2 = R0 2N N −1 √ L-section attenuator: R1 = [R OA (R OA − R OB )] R OA R 2OB R2 = R OA − R OB
Copyrighted material - Taylor & Francis Main formulae for Part 3
731
ABCD parameters Network
ABCD transmission matrix
Series impedance I1
I2
Z
V1
V2
1 Z 0 1
Shunt admittance I1
I2
Y
V1
V2
1 0 Y 1
L-network I1
I2 Z
V1
V2
Y
(1 + YZ) Z Y 1
T-network I1
I2 Z2
V1
V2
Y
(1 + YZ1 ) (Z 1 + Z 2 +YZ1 Z 2 ) Y (1 + YZ2 )
π -network I2
I1 Z V1
Y1
Y2
V2
Z (1 + Y2 Z ) (Y1 +Y2 + Y1 Y2 Z) (1 + Y1 Z )
(Continued)
Part 3
Z1
Copyrighted material - Taylor & Francis
732 Electrical Circuit Theory and Technology ABCD parameters (Continued ) Network
ABCD transmission matrix
Pure mutual inductance I1
I2
V1
⎛
0 ⎝ 1 jωM
V2
jωM 0
⎞ ⎠
M
Symmetrical lattice I1
I2 Z1 Z2
V1
⎛ Z2
⎞ Z1 + Z2 2Z 1 Z 2 ⎜ Z2 − Z1 Z2 − Z1 ⎟ ⎟ ⎜ ⎜ ⎟ ⎝ 2 Z1 + Z2 ⎠ Z2 − Z1 Z2 − Z1
V2
Z1
Characteristic impedance, Z0 =
Part 3
Filter networks
Low- and high-pass:
Z 0T Z 0π = Z 1 Z 2 = R02
1 L Low-pass T or π: f C = √ R0 = π (LC) C 1 R0 C= L= π R0 f C π fC ω 2 Z 0T = R0 1 − ωC Z 0π = 1− High-pass T or π: f C =
R0
ω ωC
I1 I2 I3 = = = eγ = eα+ jβ = eα ∠β I2 I3 I4 √ Phase angle β = ω (LC) √ time delay = (LC) m-derived filter sections:
2
1 R0 = √ 4π (LC)
B C
Low-pass m = L C
R0 1 L= C= 4π R0 f C 4π f C ω 2 C Z 0T = R0 1 − ω R0 Z 0π = ω 2 C 1− ω
1−
High-pass m =
1−
fC f∞ f∞ fC
2 2
Magnetically coupled circuits d I1 = ± j ωMI1 dt dφ2 dφ1 √ LA − L B M = N2 = N1 = k (L1 L2 ) = d I1 d I2 4
E 2 = −M
Copyrighted material - Taylor & Francis Main formulae for Part 3 Transmission lines √ Phase delay β = ω (LC)
wavelength λ =
velocity of propagation u = f λ = IR = IS e−nγ = IS e−nα ∠−nβ
ω β
2π β
VR = VS e−nγ = VS e−nα ∠−nβ √ R + j ωL Z 0 = (Z OC Z SC ) = G + j ωC √ γ = [(R + j ωL)(G + j ωC)] Ir ZO − ZR Vr = =− Ii ZO + ZR Vi Imax Ii + Ir 1 + |ρ| s= = = Imin Ii − Ir 1 − |ρ|
Standing-wave ratio,
s −1 s +1
2
Transients C−R circuit τ = CR Charging: v C = V (1 −e−(t /C R)) vr = Ve−(t /C R) i = Ie−(t /C R) Discharging: v c = v R = V e−(t /C R) L L−R circuit τ = R Current growth: vL = V e−(Rt /L)
i = Ie−(t /C R)
vR = V (1 − e−(Rt /L)) i = I (1 − e−(Rt /L)) Current decay: vL = vR = V e−(Rt /L) i = Ie−(Rt /L)
Part 3
Reflection coefficient, ρ =
Pr = Pt
733
These formulae are available for download at the website: www.routledge.com/cw/bird
Main formulae for Part 3 Advanced circuit theory and technology
Complex numbers
LR–C network
z = a + jb =r (cos θ + j sin θ ) =r ∠θ , √ where j 2 = −1 Modulus, r = |z| = (a 2 + b2 ) b Argument, θ = arg z = tan−1 a Addition: (a + jb) +(c + jb) = (a +c) + j (b +d)
1 fr = 2π
If z 1 =r1 ∠θ1 and z 2 =r2 ∠θ2 then Multiplication: z 1 z 2 =r1r2 ∠(θ1 + θ2 ) z 1 r1 and Division: = ∠(θ1 − θ2 ) z 2 r2
1 fr = √ 2π (LC)
General
Delta–star
V = R + j(X L − XC ) = |Z |∠φ I √ where |Z | = [R 2 + (X L − XC )2 ] and X L − XC φ = tan−1 R 1 I 1 X L = 2π f L XC = Y = = = G + jB 2π f C V Z Series: Z T = Z 1 + Z 2 + Z 3 + · · · 1 1 1 1 Parallel: = + + + ··· ZT Z1 Z2 Z3 P = VI cos φ or P = I R2 R S = VI Q = VI sin φ R Power factor = cosφ = Z If V = a + j b and I = c + j d then P = ac +bd Q = bc −ad S = VI∗ = P + jQ
Z1 =
R–L–C series circuit 1 fr = √ 2π (LC) √ fr = ( f 1 f2 )
ωr L 1 1 L Q= = = R ωr CR R C VL VC fr = = = V V f2 − f1
RD =
R 2L − L/C
L CR
Q=
ωr L IC = Ir R
RC2 − L/C
Determinants
[r ∠θ ]n =r n ∠nθ =r n (cosnθ + j sin nθ )
Z=
LR–CR network
a c a d g
De Moivre’s theorem:
1 R2 − 2 LC L
b = ad − bc d b c e e f = a h h j
d f d e f − b + c j g j g h
ZAZB , etc. Z A + Z B + ZC
Star–delta ZA=
Z1 Z2 + Z2 Z3 + Z3 Z1 , etc. Z2
Part 3
Subtraction: (a +jb) − (c + jd) =(a − c) + j (b −d) Complex equations: If a + jb = c +jd, then a =c and b =d
Impedance matching
|z| =
N1 N2
2
|Z L |
Complex waveforms
I=
2 + ··· I 2 + I2m I02 + 1m
i AV =
1 π
2
π
i d(ωt) 0
form factor =
r.m.s. mean
P = V0 I0 + V1 I1 cos φ1 + V2 I2 cos φ2 + · · · or P = I 2 R P power factor = VI 1 Harmonic resonance: nωL = nωC
Copyrighted material - Taylor & Francis
730 Electrical Circuit Theory and Technology Fourier series If f (x) is a periodic function of period 2π then its Fourier series is given by: f (x) = a 0 +
∞
(an cos nx + bn sin nx)
n=1
where, for the range −π to +π: π 1 f (x) dx a0 = 2π −π 1 π f (x) cos nx dx (n = 1, 2, 3, . . .) an = π −π 1 π f (x) sin nx dx (n = 1, 2, 3, . . .) bn = π −π If f (x) is a periodic function of period L then its Fourier series is given by: ∞
2πnx 2πnx f ( x) = a0 + + bn sin an cos L L
Part 3
n=1
L L where for the range − to + : 2 2 1 L/2 f (x) dx a0 = L −L/2 2πnx 2 L/2 an = f (x) cos dx (n = 1, 2, 3, . . .) L −L/2 L 2πnx 2 L/2 f (x) sin dx (n = 1, 2, 3, . . .) bn = L −L/2 L
Harmonic analysis 1 2 a0 ≈ yk a n ≈ yk cos nx k p p bn ≈
2 p
p
p
k=1
k=1
p
yk sin nx k
k=1
Hysteresis and eddy current Hysteresis loss/cycle = Aαβ J/m3 or hysteresis loss = kh v f (Bm )n W Eddy current loss/cycle = ke (Bm )2 f 2 t 3 W
Dielectric loss Series representation: tan δ = R S ωCS = 1/Q
Parallel representation: tan δ =
1 R p ωC p
Loss angle δ = (90◦ − φ) Power factor = cos φ ≈ tan δ Dielectric power loss = V 2 ωC tan δ
Field theory 2πε0 εr V F/m E = V/m b b ln r ln a a b μ0 μr 1 + ln H/m L= 2π 4 a πε0 εr Twin line: C = F/m D ln a D μ0 μr 1 + ln H/m L= π 4 a
Coaxial cable: C =
Energy stored: in a capacitor, W = 12 CV 2 J; in an inductor W = 21 LI 2 J in electric field per unit volume, D2 J/m3 ω f = 21 DE = 12 ε0 εr E 2 = 2ε0 εr in a non-magnetic medium, B2 ω f = 21 BH = 21 μ0 H 2 = J/m3 2μ0
Attenuators Logarithmic ratios: V2 I2 P2 = 20 lg = 20 lg in decibels =10 lg P1 V1 I1 P2 V2 I2 in nepers = 12 ln = ln = ln P1 V1 I1 Symmetrical T -attenuator: √ √ R0 = (R12 + 2R1 R2 ) = (R OC R SC ) N −1 2N R1 = R0 R2 = R0 N +1 N2 −1 Symmetrical π-attenuator: R1 R22 √ = (R OC R SC ) R0 = R1 + 2R2 2 N −1 N +1 R1 = R0 R2 = R0 2N N −1 √ L-section attenuator: R1 = [R OA (R OA − R OB )] R OA R 2OB R2 = R OA − R OB
Copyrighted material - Taylor & Francis Main formulae for Part 3
731
ABCD parameters Network
ABCD transmission matrix
Series impedance I1
I2
Z
V1
V2
1 Z 0 1
Shunt admittance I1
I2
Y
V1
V2
1 0 Y 1
L-network I1
I2 Z
V1
V2
Y
(1 + YZ) Z Y 1
T-network I1
I2 Z2
V1
V2
Y
(1 + YZ1 ) (Z 1 + Z 2 +YZ1 Z 2 ) Y (1 + YZ2 )
π -network I2
I1 Z V1
Y1
Y2
V2
Z (1 + Y2 Z ) (Y1 +Y2 + Y1 Y2 Z) (1 + Y1 Z )
(Continued)
Part 3
Z1
Copyrighted material - Taylor & Francis
732 Electrical Circuit Theory and Technology ABCD parameters (Continued ) Network
ABCD transmission matrix
Pure mutual inductance I1
I2
V1
⎛
0 ⎝ 1 jωM
V2
jωM 0
⎞ ⎠
M
Symmetrical lattice I1
I2 Z1 Z2
V1
⎛ Z2
⎞ Z1 + Z2 2Z 1 Z 2 ⎜ Z2 − Z1 Z2 − Z1 ⎟ ⎟ ⎜ ⎜ ⎟ ⎝ 2 Z1 + Z2 ⎠ Z2 − Z1 Z2 − Z1
V2
Z1
Characteristic impedance, Z0 =
Part 3
Filter networks
Low- and high-pass:
Z 0T Z 0π = Z 1 Z 2 = R02
1 L Low-pass T or π: f C = √ R0 = π (LC) C 1 R0 C= L= π R0 f C π fC ω 2 Z 0T = R0 1 − ωC Z 0π = 1− High-pass T or π: f C =
R0
ω ωC
I1 I2 I3 = = = eγ = eα+ jβ = eα ∠β I2 I3 I4 √ Phase angle β = ω (LC) √ time delay = (LC) m-derived filter sections:
2
1 R0 = √ 4π (LC)
B C
Low-pass m = L C
R0 1 L= C= 4π R0 f C 4π f C ω 2 C Z 0T = R0 1 − ω R0 Z 0π = ω 2 C 1− ω
1−
High-pass m =
1−
fC f∞ f∞ fC
2 2
Magnetically coupled circuits d I1 = ± j ωMI1 dt dφ2 dφ1 √ LA − L B M = N2 = N1 = k (L1 L2 ) = d I1 d I2 4
E 2 = −M
Copyrighted material - Taylor & Francis Main formulae for Part 3 Transmission lines √ Phase delay β = ω (LC)
wavelength λ =
velocity of propagation u = f λ = IR = IS e−nγ = IS e−nα ∠−nβ
ω β
2π β
VR = VS e−nγ = VS e−nα ∠−nβ √ R + j ωL Z 0 = (Z OC Z SC ) = G + j ωC √ γ = [(R + j ωL)(G + j ωC)] Ir ZO − ZR Vr = =− Ii ZO + ZR Vi Imax Ii + Ir 1 + |ρ| s= = = Imin Ii − Ir 1 − |ρ|
Standing-wave ratio,
s −1 s +1
2
Transients C−R circuit τ = CR Charging: v C = V (1 −e−(t /C R)) vr = Ve−(t /C R) i = Ie−(t /C R) Discharging: v c = v R = V e−(t /C R) L L−R circuit τ = R Current growth: vL = V e−(Rt /L)
i = Ie−(t /C R)
vR = V (1 − e−(Rt /L)) i = I (1 − e−(Rt /L)) Current decay: vL = vR = V e−(Rt /L) i = Ie−(Rt /L)
Part 3
Reflection coefficient, ρ =
Pr = Pt
733
These formulae are available for download at the website: www.routledge.com/cw/bird
