CALCULATING REACTANCE AND IMPEDANCE
Here are “scalar value” equations for these quantities; there are “vector value” equations which are
really neat, but they are not needed here.
Inductive reactance can be found with:
X L = 2 πfL where: XL is Inductive Reactance in Ohms.
π is Pi, a Greek letter which means “insert the number 3.1416”;
it is the ratio of the circumference to the diameter of a circle.
is the frequency in Hertz, or Cycles per Second.
is Inductance in Henries.
Here is an example: if f = 1MHz., and L = 10μH , X L = 62.8Ω
Capacitive reactance can be found with:
X = 1C π where: X 2 fC C
is Capacitive Reactance in Ohms.
is 3.1416 (as above)
is the frequency in Hertz, or Cycles per Second.
is Capacitance in Farads.
Here is an example: if f = 1MHz., and C = 100 pF , XC = 1592Ω
Impedance can be found with:
mined by the size and shape of the
conductors and the thickness and type
of material of the insulator between. A
common transmission line is co-axial
cable, such as RG-58, which has a
characteristic impedance of 50Ω.
What is characteristic impedance?
You do not measure it with an
ohmmeter. If a transmission line, no
matter how long, is terminated in its
characteristic impedance, the other
end will look like the characteristic
impedance to RF currents. It will be a
pure resistance if the termination is
the correct, pure resistance. This is a
very good arrangement because the
inductance and capacitance no longer
affect the frequency response.
Z = R 2 +X 2 where: Z
is Impedance in Ohms
is Resistance in Ohms
is Inductive Reactance, X L , or XC , Capacitive Reactance, in
Here is an example: if you connect a 10 ohm resistor and an inductor (whose reactance at a certain
frequency is 10 ohms) in series, the result is Z = 14.4Ω
between the two ground plane
surfaces suffers from the proximity
effect, capacitance to ground, and
inductance in the traces. Circuit board
traces have approximately 10 nH per
centimeter of inductance (1,000 nH =
1 μH) and, with the capacitance of the
ground plane, the traces can form a
low-pass filter which will produce a
falling frequency response, different
time delays for different frequencies,
and can produce ringing. As a result,
data pulses will be rounded and
timing may be wrong.
Circuit boards operating at frequencies greater than 1,000 MHz (this
includes high-speed data) must use a
different arrangement: a transmission
line. Transmission lines have a characteristic impedance (Ζo) which is deter-
Inductive reactance and capacitive With pure inductive reactance,
reactance are opposite: what one when the voltage is at a positive peak,
does, the other one does “upside the current is at zero! That means that
down.” Inductive reactance increases the current lags the voltage by up to 90°.
in direct proportion to the frequency; In a pure capacitive reactance, when
capacitive reactance decreases in the current is at a positive peak, the
inverse proportion to the frequency. voltage is at zero! That means that the
And, they cancel each other. If you current leads the voltage by up to 90°.
have equal quantities of inductive and I found this a difficult concept, but
capacitive reactance in series, you have there is an easy way to demonstrate it.
nothing left but a little bit of resistance. Read the sidebar “Seeing Is Believing.”
This is called resonance. ANYTHING that behaves with
In a resistance, the voltage and the characteristics shown in the
current are in phase. That is, when the comparison table (see the “Kinds of
voltage is at a positive peak, the current Opposition” sidebar) is a resistor,
is also at a positive peak. inductor, or capacitor.
The most dramatic experience
with wires that don’t behave as wires
that I have ever experienced happened
about a year ago with an instrument
called a vector voltmeter. Figure 1
shows the front panel. It’s a wonderful
old Hewlett-Packard instrument; it can
tell you things about a circuit that a
‘scope cannot. On my workbench, it is
indispensable, and now it is irreplaceable since it is about 35 years old.
So, when it stopped working
correctly, I had to fix it. It has an
indicator on the front panel marked
“APC unlocked” (Automatic Phase
Control; similar to a phase-locked
loop). When the APC is unlocked, the
input signal (up to 1 GHz) and the
internal phase-locked loop are out of
synchronization. An internal phase
comparator has to pull an internal
oscillator to the correct frequency and
phase, and keep it there even in the
presence of frequency modulation.
That’s a very demanding operation!
It was getting harder and harder
to lock the APC until, after a few
weeks, it wouldn’t lock at all. Without
a perfect phase lock, the instrument
Signal sampling up to 1 GHz is
accomplished with a 300 ps ( 3 x 10-10
second) pulse, adjusted to frequencies
between 0.98 MHz and 2.0 MHz,
producing a signal for measurement of
20 kHz. Such a short pulse contains
energy at 3 GHz. The pulses are pro-