Formulae for further electrical and electronic principles AWS
Copyrighted material - Taylor & Francis
Formulae for further electrical and electronic principles
A.c. theory
I=
or
f =
1 T
1 fr = 2π
i 12 + i 22 + i 22 + · · · + i n2 n 2 Im or 0.637 Im π
For a sine wave: IAV = 1 I = √ Im or 0.707 Im 2 r.m.s. Form factor = average
maximum Peak factor = r.m.s.
General sinusoidal voltage: v = Vm sin (ωt ± φ)
1 R2 − 2 LC L
Ir =
VRC L RD = L CR
Q=
2π f r L IC = R Ir
P = VIcos φ or I 2 R
S = VI
power factor = cos φ =
R Z
Filter networks Low-pass T or π:
Single-phase circuits X L = 2πf L
XC =
1 2πf C
1 fC = √ π LC
R0 =
L C
1 π R0 f C
L=
R0 π fC
C=
V Z = = (R 2 + X 2 ) I
See Fig. F1. Series resonance: f r =
1 √
High-pass T or π:
2π LC
Q=
VL VC 2πf r L 1 1 or = = = V V R 2πf r CR R
Q=
fr fr or ( f 2 − f 1 ) = f2 − f 1 Q
L C
fC = C=
4π
1 √
LC
1 4π R0 f C
See Fig. F2.
R0 =
L C
L=
R0 4π f C
Q = VIsin φ
Section 2
1 f
T=
Parallel resonance (LR–C circuit):
Copyrighted material - Taylor & Francis
320 Electrical and Electronic Principles and Technology L 2
L
L 2
R0
R0
C
R0
C 2
R0
C 2
Figure F1
2C
2C
Section 2
R0
L
C
R0
R0
2L
2L
(a)
R0
(b)
Figure F2
D.c. transients
Operational amplifiers
C–R circuit τ = C R Charging: v C = V (1 − e−t /CR ) v r = Ve−t /CR
Inverter: A =
i = Ie−t /CR Discharging: v C = v R = Ve−t /CR i = I e−t /CR L–R circuit τ =
CMRR = 20 log10
L R
Current growth: v L = Ve−Rt/L
differential voltage gain dB common-mode gain
−Rf Vo = Vi Ri
Vo Rf =1+ Vi Ri V1 V2 V3 Summing: Vo = −Rf + + R1 R2 R3 Non-inverter: A =
Integrator: Vo = −
1 CR
Vi dt
Differential:
v R = V (1 − e−Rt/L )
Rf If V1 > V2 : Vo = (V1 − V2 ) − R1
i = I (1 − e−Rt/L ) Current decay: v L = v R = Ve−Rt/L i = Ie−Rt/L
If V2 > V1 : Vo = (V2 − V1 )
These formulae are available for downloading at the website: www.routledge.com/cw/bird
R3 R2 + R3
Rf 1+ R1
Formulae for further electrical and electronic principles
A.c. theory
I=
or
f =
1 T
1 fr = 2π
i 12 + i 22 + i 22 + · · · + i n2 n 2 Im or 0.637 Im π
For a sine wave: IAV = 1 I = √ Im or 0.707 Im 2 r.m.s. Form factor = average
maximum Peak factor = r.m.s.
General sinusoidal voltage: v = Vm sin (ωt ± φ)
1 R2 − 2 LC L
Ir =
VRC L RD = L CR
Q=
2π f r L IC = R Ir
P = VIcos φ or I 2 R
S = VI
power factor = cos φ =
R Z
Filter networks Low-pass T or π:
Single-phase circuits X L = 2πf L
XC =
1 2πf C
1 fC = √ π LC
R0 =
L C
1 π R0 f C
L=
R0 π fC
C=
V Z = = (R 2 + X 2 ) I
See Fig. F1. Series resonance: f r =
1 √
High-pass T or π:
2π LC
Q=
VL VC 2πf r L 1 1 or = = = V V R 2πf r CR R
Q=
fr fr or ( f 2 − f 1 ) = f2 − f 1 Q
L C
fC = C=
4π
1 √
LC
1 4π R0 f C
See Fig. F2.
R0 =
L C
L=
R0 4π f C
Q = VIsin φ
Section 2
1 f
T=
Parallel resonance (LR–C circuit):
Copyrighted material - Taylor & Francis
320 Electrical and Electronic Principles and Technology L 2
L
L 2
R0
R0
C
R0
C 2
R0
C 2
Figure F1
2C
2C
Section 2
R0
L
C
R0
R0
2L
2L
(a)
R0
(b)
Figure F2
D.c. transients
Operational amplifiers
C–R circuit τ = C R Charging: v C = V (1 − e−t /CR ) v r = Ve−t /CR
Inverter: A =
i = Ie−t /CR Discharging: v C = v R = Ve−t /CR i = I e−t /CR L–R circuit τ =
CMRR = 20 log10
L R
Current growth: v L = Ve−Rt/L
differential voltage gain dB common-mode gain
−Rf Vo = Vi Ri
Vo Rf =1+ Vi Ri V1 V2 V3 Summing: Vo = −Rf + + R1 R2 R3 Non-inverter: A =
Integrator: Vo = −
1 CR
Vi dt
Differential:
v R = V (1 − e−Rt/L )
Rf If V1 > V2 : Vo = (V1 − V2 ) − R1
i = I (1 − e−Rt/L ) Current decay: v L = v R = Ve−Rt/L i = Ie−Rt/L
If V2 > V1 : Vo = (V2 − V1 )
These formulae are available for downloading at the website: www.routledge.com/cw/bird
R3 R2 + R3
Rf 1+ R1
