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Electronic System Formulas

Contents

- Notation
- Impedance
- Admittance
- Reactance
- Resonance
- Disclaimer

Notation

The library uses the symbol font for some of the notation and formulas. If the symbols for the letters 'alpha beta delta' do not appear here [a b d] then the symbol font needs to be installed before all notation and formulas will be displayed correctly.

B
C
E
f
G
h
I
j
L
P
Q 		susceptance
capacitance
voltage source
frequency
conductance
h-operator
current
j-operator
inductance
active power
reactive power 		(Siemens)
(Farads)
(Volts)
(Hertz)
(Siemens)
(1Ð120°)
(Amps)
(1Ð90°)
(Henrys)
(Watts)
(VArs) 		R
S
t
V
W
X
Z
f
h
w
  		resistance
apparent power
time
voltage drop
energy
reactance
impedance
phase angle
efficiency
angular frequency
  		(Ohms)
(VA)
(seconds)
(Volts)
(Joules)
(Ohms)
(Ohms)
(degrees)
(per-unit)
(rad/sec)
 

Impedance

The impedance Z of a resistance R in series with a reactance X is:
Z = R + jX

Rectangular and polar forms of impedance Z:
Z = R + jX = (R2 + X2)½Ðtan-1(X / R) = |Z|Ðf = |Z|cosf + j|Z|sinf

Addition of impedances Z1 and Z2:
Z1 + Z2 = (R1 + jX1) + (R2 + jX2) = (R1 + R2) + j(X1 + X2)

Subtraction of impedances Z1 and Z2:
Z1 - Z2 = (R1 + jX1) - (R2 + jX2) = (R1 - R2) + j(X1 - X2)

Multiplication of impedances Z1 and Z2:
Z1 * Z2 = |Z1|Ðf1 * |Z2|Ðf2 = ( |Z1| * |Z2| )Ð(f1 + f2)

Division of impedances Z1 and Z2:
Z1 / Z2 = |Z1|Ðf1 / |Z2|Ðf2 = ( |Z1| / |Z2| )Ð(f1 - f2)

In summary:
- use the rectangular form for addition and subtraction,
- use the polar form for multiplication and division.

Admittance

An impedance Z comprising a resistance R in series with a reactance X can be converted to an admittance Y comprising a conductance G in parallel with a susceptance B:
Y = Z -1 = 1 / (R + jX) = (R - jX) / (R2 + X2) = R / (R2 + X2) - jX / (R2 + X2) = G - jB
G = R / (R2 + X2) = R / |Z|2
B = X / (R2 + X2) = X / |Z|2
Using the polar form of impedance Z:
Y = 1 / |Z|Ðf = |Z| -1Ð-f = |Y|Ð-f = |Y|cosf - j|Y|sinf

Conversely, an admittance Y comprising a conductance G in parallel with a susceptance B can be converted to an impedance Z comprising a resistance R in series with a reactance X:
Z = Y -1 = 1 / (G - jB) = (G + jB) / (G2 + B2) = G / (G2 + B2) + jB / (G2 + G2) = R + jX
R = G / (G2 + B2) = G / |Y|2
X = B / (G2 + B2) = B / |Y|2
Using the polar form of admittance Y:
Z = 1 / |Y|Ð-f = |Y| -1Ðf = |Z|Ðf = |Z|cosf + j|Z|sinf

The total impedance ZS of impedances Z1, Z2, Z3,... connected in series is:
ZS = Z1 + Z1 + Z1 +...
The total admittance YP of admittances Y1, Y2, Y3,... connected in parallel is:
YP = Y1 + Y1 + Y1 +...
In summary:
- use impedances when operating on series circuits,
- use admittances when operating on parallel circuits.

Reactance

Inductive Reactance
The inductive reactance XL of an inductance L at angular frequency w and frequency f is:
XL = wL = 2pfL

Capacitive Reactance
The capacitive reactance XC of a capacitance C at angular frequency w and frequency f is:
XC = 1 / wC = 1 / 2pfC

Resonance

Series Resonance
A series circuit comprising an inductance L, a resistance R and a capacitance C has an impedance ZS of:
ZS = R + j(XL - XC)
where XL = wL and XC = 1 / wC

At resonance:
XC = XL
ZSr = R
wr = (LC) -½ = 2pfr

Parallel resonance
A parallel circuit comprising an inductance L with a series resistance R connected in parallel with a capacitance C has an admittance YP of:
YP = 1 / (R + jXL) + 1 / (-jXC) = (R / (R2 + XL2)) - j(XL / (R2 + XL2) - 1 / XC)
where XL = wL and XC = 1 / wC

At resonance:
XC = (R2 + XL2) / XL = XL + R2 / XL
ZPr = YPr-1 = (R2 + XL2) / R = XLXC / R = wrL / wrCR = L / CR
wr = (1 / LC - R2 / L2)½ = 2pfr