Impedance
Electrical Impedance (Z),
is the total opposition that a circuit presents to alternating
current. Impedance is measured in ohms and may include
resistance (R), inductive
reactance (X_{L}),
and capacitive
reactance (X_{C}).
However, the total impedance is not simply the algebraic sum of the
resistance, inductive reactance, and capacitive reactance. Since
the inductive reactance and capacitive reactance are 90^{o}
out of phase with the resistance and, therefore, their maximum
values occur at different times, vector addition must be used
to calculate impedance.
In the image below, a circuit diagram is shown that represents
an eddy current inspection system. The eddy current probe is a
coil of wire so it contains resistance and inductive reactance
when driven by alternating current. The capacitive reactance
can be dropped as most eddy current probes have little capacitive
reactance. The solid line in the graph below shows the circuit's
total current, which is affected by the total impedance of the
circuit. The two dashed lines represent the portion of the current
that is affected by the resistance and the inductive reactance
components individually. It can be seen that the resistance and
the inductive reactance lines are 90^{o} out of phase, so
when combined to produce the impedance line, the phase shift is
somewhere between zero and 90^{o}. The phase shift is always
relative to the resistance line since the resistance line is always
inphase with the voltage. If more resistance than inductive reactance
is present in the circuit, the impedance line will move toward
the resistance line and the phase shift will decrease. If more
inductive reactance is present in the circuit, the impedance
line will shift toward the inductive reactance line and the phase
shift will increase.
The relationship between impedance and its individual components
(resistance and inductive reactance) can be represented using a
vector as shown below. The amplitude of the resistance component
is shown by a vector along the xaxis and the amplitude of the
inductive reactance is shown by a vector along the yaxis. The
amplitude of the the impedance is shown by a vector that stretches
from zero to a point that represents both the resistance value
in the xdirection and the inductive reactance in the ydirection.
Eddy current instruments with impedance plane displays present
information in this format.
The impedance in a circuit with resistance and inductive reactance
can be calculated using the following equation. If capacitive
reactance was present in the circuit, its value would be added
to the inductance term before squaring.
The phase
angle of the circuit can also be calculated using some trigonometry. The phase angle is equal to the ratio between the inductance and the resistance in the circuit. With the probes and circuits used in nondestructive testing, capacitance can usually be ignored so only inductive reactance needs to be accounted for in the calculation. The phase angle can be calculated
using the equation below. If capacitive reactance was present
in the circuit, its value would simply be subtracted from the inductive
reactance term.
The applet below can be used to see how the variables
in the above equation are related on the the vector diagram (or
the impedance plane display). Values can be entered into the dialog
boxes or the arrow head on the vector diagram can be dragged to
a point representing the desired values. Note that the capacitive
reactance term has been included in the applet but as mentioned
before, in eddy current testing this value is small and can be
ignored.
Impedance and Ohm's Law
In previous pages, Ohm's Law was discussed for a purely resistive
circuit. When there is inductive reactance or capacitive reactance
also present in the circuit, Ohm's Law must be written to include
the total impedance in the circuit.
Therefore, Ohm's law becomes:
I
= V / Z
Ohm's law now simply states that the current (I),
in amperes, is proportional to the voltage (V),
in volts, divided by the impedance (Z),
in ohms.
The applet below can be used to see how the current
and voltage of a circuit are affected by impedance. The applet
allows the user to vary the inductance (L),
resistance (R), voltage (V)
and current (I). Voltage
and current are shown as they would be displayed on an oscilloscope.
Note that the resistance and/or the inductive reactance values
must be changed to change the impedance in the circuit.
Also note that when there is inductance in the circuit,
the voltage and current are out of phase. This is because the
voltage across the inductor will be a maximum when the rate of
change of the current is greatest. For a sinusoidal wave form like
AC, this is at the point where the actual current is zero. Thus
the voltage applied to an inductor reaches its maximum value a
quartercycle before the current does, and the voltage is said to
lead the current by 90^{o}.
