ELECTRICAL CIRCUITS 104 – FORMULA SHEET
1.
DC CIRCUITS (Resistive)
(i)
Ohm’s Law: V = R.I or I = G.V
(ii)
Delta to Y Transformations (Ra, Rb, Rc) to (R1, R2, R3):
R1 =
(iii)
(iv)
3.
Rb Rc
R1 R2 + R2 R3 + R3 R1
….. Ra =
Ra + Rb + Rc
R1
Power: P = V .I = V G = I
Operational Amplifier Model:
2
2
SWITCHED CIRCUITS (First and Second Order Transient Responses)
(i)
First Order DE:
(ii)
Solution:
(iii)
R
(iv)
Overdamped Solution:
(v)
Critically Damped:
(vi)
Underdamped :
(ii)
A→∞
(iii)
(iv)
AC CIRCUITS (Phasors and Impedances)
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Complex Number:
a + jb = a 2 + b 2 ∠ tan −1
A cosθ + B sin θ =
sin θ = cos(θ − π2 )
A + B cos(θ − tan
2
2
b
a
−1 B
A
(v)
)
v(t ) = Vm cos(ω .t + φ ) ⇒ V = Vm ∠φ = Vm cos φ + jVm sin φ
Z = R + jX , Y = G + jB , Z L = jω .L , Z c = − j ω1.C
Mutual Inductance:
v1 = L
di1
1 dt
±M
di2
dt
and
k = M / L1 L2
⇒ V1 = ( jω .L1 ) I 1 ± ( jω .M ) I 2
(vii)
2
2
Pav = Vrms I rms cos φ = I rms
R = Vrms
G where Vrms = Vm / 2
*
S = V .I = Vrms I rms cos φ + jVrms I rms sin φ
(viii)
Complex Power:
(ix)
Energy stored: capacitor,
y + 2α
where A & B from boundary conditions
d
dt
y + ω o2 y = f (t ) where
Wc = 12 CV 2 ; inductor, WL = 12 LI 2 .
y (t ) = ( A.e −σ 1t + B.e −σ 2t ) + forced Response
y (t ) = ( A.e −α .t + B.t.e −α .t ) + forced Response
s = α ± jω d = α 2 + ω d2 ∠ tan −1
±ω d
α
= K∠ ± φ
y (t ) = A.e −α .t cos(ω d .t ) + B.e −σ .t sin(ω d .t ) + forced R
APPLICATIONS (frequency response, filters, 2-port networks)
(i)
2.
d2
dt 2
− Tt
ω o = 1 / LC (natural freq) and α is the damping factor
4.
Ro → 0
y + T1 y = f (t ) where T = RC or T = L/R
y (t ) = A + B.e
Second Order DE:
Solution :
where Ri → ∞
d
dt
V out
= H ( jω ) ∠H ( jω )
V in
The Magnitude Response can be expressed in dB = 20 log H ( jω )
Transfer Function:
H ( jω ) =
The half-power bandwidth is defined between 2 frequencies at which the
response drops down by -3dB (i.e. 0.707 of Maximum value).
Band-pass Filters are characterised by the centre frequency (ωο) and the halfpower bandwidth ( ∆ω ) . Quality factor, Q = ω o / ∆ω .
(viii)
(ix)
z-parameters are defined as:
V1 = z11 I1 + z12 I2 and V2 = z21 I1 + z22 I2
Transmission parameters are defined as:
V1 = a11 V2 - a12 I2 and I1 = a21 V2 - a22 I2
Hybrid parameters are defined as:
V1 = h11 I1 + h12 V2 and I2 = h21 I1 + h22 V2
Conditions for Reciprocity: z12 = z21, (a11.a22 - a12.a21) = 1, h12 = -h21.
Additional Conditions for Symmetry: z11 = z22, a11 = a22, (h11.h22 - h12.h21) = 1
(x)
Converting [Z] to [A]: a11
(vi)
(vii)
(xi)
z
z
∆
1
= 11 , a12 = z , a 21 =
, a 22 = 22
z 21
z 21
z 21
z 21
z
z
∆
1
Converting [Z] to [H]: h11 = z , h12 = 12 , h21 = − 21 , h22 =
z 22
z 22
z 22
z 22