Protek Z9216 - Chapter 1: Background on Components and Measurements; Definitions of Resistive and Reactive Parameters; Component Categories

Chapter
1
Background on Components and Measurements
Properties of Resistors, Inductors, and Capacitors
The measurements made by the Model Z9216, Digital LCR Meter are based on the definitions of
impedance and the properties of discrete components designed to provide impedances in electronic
circuits.
Definitions of Resistive and Reactive Parameters
Let the sinusoidal voltage and current in an electronic circuit at a particular frequency, f be
represented in the complex or phasor notation, given by
(
)
v
tVtV
θ
ω
+
= cos||)( (1a)
(
)
(
)
tj
jtj
eeVeV
vv
ω
θθω
|||| ==
+
(1b)
(
)
i
tItI
θ
ω
+
= cos||)( (2a)
(
)
(
)
tj
jtj
eeIeI
ii
ω
θθω
|||| ==
+
(2b)
where
1−=j , ω = 2π f, and
v
θ
and
i
θ
are symbols for phases of the voltage and current relative to
the frequency f. The impedance of a circuit component is defined as the complex number Z, in ohms,
that gives the ratio of the voltage across the component to the current in the component:
()
(
)
()
(
iv
i
v
j
tj
tj
e
I
V
eI
eV
tI
tV
ZZ
θθ
θω
θω
ω
−
+
+
====
||
||
||
||
)(
)(
)
(3)
Component Categories
From equation (3), we observe that if the phases of the voltage across the component and the current
in it are equal, then the impedance is a real number:
()
||
||
||
||
0
I
V
e
I
V
Z
j
== (4)
In this situation, the impedance is purely resistive, as an ideal resistor would be.
If the phase of the voltage is 90 degrees (π/2 radians) ahead of the phase of the current, then the
impedance is a positive imaginary number:
()
||
||
||
||
2/
I
V
je
I
V
Z
j
==
π
(5)
1