Series
Parallel
Dissipation factor
Rs
±jXs
G
±jB
Capacitance
Cs=Cp(1+D
2
)
Cp=Cs/(1+D
2
)
D=Rs/Xs=ωCsRs
D=G/B=G/(ωCp)=1/(ωCpRp)
Inductance
Ls=Lp/(1+D
2
)
Lp=Ls(1+D
2
)
D=Rs/Xs=Rs/(ωLs)
D=G/B=ωLpG=ωLp/Rp
Resistance
Rs=RpD
2
/(1+D
2
) Rp=Rs(1+1/D
2
)
―
Q = Xs/Rs = 2πf Ls/Rs = ½πf CsRs
Q = B/G = Rp/│Xp│= Rp/2πf Lp = 2πf CpRp
― 20 ―
― 19 ―
4-15 MEASURING PRINCIPLES
4-15-1 What is impedance?
Impedance Z extends the concept of resistance to AC, which is
mathematically handled as a vector quantity on a complex plane.
As shown, the impedance vector consists of the real part (the
resistance R) and the imaginary part (the reactance X). Series
impedance Zs can be represented as Rs + jXs in Cartesian
form, and also can be represented as |Zs|
∠
θ (magnitude and
phase angle) in the polar form. The figure shows a mathematical
relationship between Rs, Xs, |Zs|, θ.
Zs = Rs + jXs or |Zs|
∠
θ
Rs = |Zs| cos
θ
Xs = |Zs| sin
θ
Xs/Rs = tan
θ
θ
= tan
-1
(Xs/Rs)
There are two types of reactance. One is inductive reactance X
L
,
and the other is capacitive reactance X
C
.
If
θ
> 0, the reactance is inductive. If
θ
< 0, the reactance is
capacitive.
The inductive reactance (XL) and the capacitive reactance (XC)
can be defined as follows.
X
L
= 2πƒL
X
C
= 1 / (2πƒC)
where:
L = Inductance
C = Capacitance
f = signal frequency
To understand the ratio of resistance and reactance, it is important
to consider Quality factor (Q) and Dissipation factor (D). Usually, Q
is used when measuring inductance and D is used when measuring
capacitance. Q is defined as the reciprocal of D.
Q = 1 / D = tan
θ
4-15-2 Impedance measurement
Impedance can be measured in series or in parallel.
In parallel mode, impedance can be represented as reciprocal of
admittance (Y).
The admittance can be defined as Y = G + jB.
where: G = Conductance
B = Susceptance
Imaginary axis (series mode)
Real axis
Zs=Rs+Xs
｜Zs｜
θ>0
Xs
Rs
θ<0
Z = Rs + jXs
Rs
jXs
Y = 1/Z = 1/Rp + 1/jXp = G+jB
Rp
jXp
Series impedance
Parallel admittance
Rs = Series resistance
Xs = Series reactance
Cs = Series capacitance
Ls = Series inductance
Rp = Parallel resistance
Xp = Parallel reactance
Cp = Parallel capacitance
Lp = Parallel inductance