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This is arguably the handiest tool anyone involved in electronic design
could wish for! It avoids the need to make impedance or reactance
calculations and there is no need to revise longforgotten formulas.
Reactance Chart
for easy RC, RL or
LC network design
W
ith this reactance chart, you
can easily check the 3dB
rolloff of a simple RC (resistorcapacitor) or RL (resistorinductor)
network or find the resonant frequency
of an LC (inductorcapacitor) network.
Why do we need such a tool? Sure,
you can easily Google to get a calculator for almost any purpose but typically such online calculators give
you a couple of fields to fill in with
the known values, say, resistance or
capacitance and frequency, and then
you click the “Calculate” button to get
the answer.
But this does not allow you to get an
overall picture of how passive components such as resistors, capacitors and
inductors interact to determine the frequency behaviour of circuits. For example, if you look at a typical amplifier circuit, it is not the active components such as op amps, transistors or
Mosfets which largely determine the
frequency response, it is the interaction of the above mentioned passive
components.
For example, in the very simplified
circuit of a complementary symmetry
amplifier in Fig.1, the low frequency
rolloff is determined by the interaction of resistor R1 and capacitor C1 in
the input circuit and also in the negative feedback network, by R2 and C2.
On the other hand, the high frequency performance is determined by the
interaction of inductors, resistors and
capacitors in the input and output of
the amplifier.
For example, ostensibly all that resistor R1 does is to provide input bias
current to transistor Q1. It also sets the
voltage at the output of the amplifier
(to 0V). But just as importantly, those
R1 and C1 values partly determine the
low frequency rolloff of the amplifier.
88 Silicon Chip
formulas but let us transfer the process to the reactance chart of Fig.2
(opposite), with a few examples. Say
you want to know the impedance of a
100nF (100 nanofarads or 0.1µF) capacitor at a frequency of 1kHz. We have
highlighted in red how you read the
values off the chart, in Fig.7.
The first step is to find the value of
100nF on the righthand vertical axis.
Then you trace down the line at 45° to
where it intersects the horizontal line
for 10kHz which again is marked on
the righthand vertical axis. You then
take a vertical (red) line down from
that “intersection” to the horizontal
axis. The value shown where the red
line intersects that horizontal axis is
about 1.6kΩ (the calculated impedance
is actually 1.592kΩ). So the three steps
in this process are shown as red lines
on the reduced chart of Fig.7.
Note that all the axes on this chart
are logarithmic and this means that
when you are interpolating values
between actual printed lines, the value you read off the respective axis is
always a bit of a guesstimate. That’s
By LEO SIMPSON
And capacitor C1 can also determine
the ultimate signaltoratio of the amplifier at very low frequencies, because
we need it to have a low impedance.
So there is more to these simple passive components than meets the eye.
So let’s look at how you can determine the impedance of any capacitor
or inductor from the wall chart. First,
the impedance of a capacitor at any
frequency can be calculated by the
formula
Z = 1/(2fC)
where Z is the impedance in Ohms;
i is the constant 3.1415926...;
f is the frequency in Hertz and
C is the capacitance in Farads.
Similarly, the impedance of an inductor at any frequency can be calculated by the formula
Z = 2fL
where L is the inductance in Henries
and f is the frequency in Hertz.
You can calculate impedances to
your heart’s content using the above
Fig.1: In this typical audio
amplifier, the overall frequency
response is mainly determined
by R1 & C1 at the input and R2
& C2 in the feedback network.
SIGNAL
INPUT
C1
B
R1
+VCC
B
E
E
C
C
+
–
C
E
B
OUTPUT
R2
B
C2
B
C
E
C
E
−VEE
siliconchip.com.au
SILICON CHIP
REACTANCE – INDUCTANCE – CAPACITANCE – FREQUENCY
1n
F
10
0p
F
10
pF
0.
1p
F
.0
1p
F
H
1
.0
H
1
0.
H
1
H
10
H
H
0
10
1m
1p
F
READY RECKONER
.COM.AU
H
m
10
10
nF
100MHz
10MHz
10
0n
F
H
0m
10
1H
1
F
1MHz
H
10
10
F
100kHz
0H
10
10
0
F
10kHz
H
10
1k
00
F
1kHz
10
kH
10
00
0
F
100Hz
siliconchip.com.au
1M
10
H
0k
10
00
0
0
F
10Hz
100k
10k
1k
100
January 2016 89
10
1Hz
1
L
C
OUT
OUT
R
Fig.4
HIGH PASS
F
F
pF
10
F
1p
H
1p
1m
.0
H
1m
1p
Fig.6
0.
H
1m
H
.0
C
green lines on Fig.7.)
You can use a similar process when
working with “high pass” filters and
in the simplest case, the positions of
the resistor and capacitor in the circuit
of Fig.3 are swapped to give the circuit in Fig.4. In this case, the circuit
passes high frequencies and progressively blocks lower frequencies due
to the impedance of the capacitor increasing as the frequency is reduced.
Feeling adventurous? Let’s take a
circuit example involving an inductor and resistor, an RL network set up
as a low pass filter. You will often see
examples of this sort of network at the
input of a preamplifier where we want
to block extremely high frequencies by
using a ferrite bead inductor.
In this case, if you look at the formula for the reactance of an inductor,
you will realise that it rises in a linear
fashion with increasing frequencies,
eg, a doubling a frequency will double the reactance.
By the way, for the purpose of using
this chart, the terms reactance and impedance mean the same thing. In fact,
some readers would regard the term
mH
10
0m
10
H
1m
F
OK though because if you had used
the formula to calculate the precise
value, you would always round it off
when selecting an actual component
value for a circuit. Which brings us to
the next example.
Say you need to come up with a
simple RC filter which will roll off frequencies above 20kHz (the 3dB point)
and then roll off at 6dB octave above
that point. This is the simplest possible “low pass” filter, meaning that it
passes low frequencies and attenuates
(rolls off) higher frequencies. The circuit is shown in Fig.3.
So if the resistor value R is known
to be 8kΩ and the wanted cutoff frequency is 20kHz, you take a vertical
line (green) up from the 8kΩ mark on
the horizontal axis until it meets the
horizontal line corresponding to a frequency of 20kHz on the righthand
vertical axis. You then take a line up
at 45° until it meets the top horizontal axis which corresponds to a value
of a whisker over 1nF.
(The calculated value is 992pF or almost exactly one nanofarard. We have
shown three steps in this process with
L
Fig.5
0.
Fig.3
LOW PASS
R
100MHz
10
10
H
m
nF
1n
C
IN
F
IN
0p
OUT
10
R
IN
10MHz
H
10
0m
0n
F
10
1H
1m
F
1MHz
H
10
10
10
10
0m
F
10kHz
0H
H
10
1k
00
mF
1kHz
10
10
00
0m
F
100Hz
kH
H
10
0k
00
10
00
mF
10Hz
1MW
90 Silicon Chip
mF
100kHz
Fig.7: the coloured
lines on this example
of the reactance chart
demonstrate examples
(see text) of how you
can find the impedance
of a capacitor or
inductor, the cutoff
frequency of a simple
RC or RL network or
the resonant frequency
of a series or paralleltuned LC circuit. Many
other impedance
calculation can by
done by a similar two
or 3step process.
“reactance” as being obsolete.
OK, so now we have a simple RL
low pass filter, as shown in the circuit
of Fig.5. Let’s say the value of the inductor is 500 microhenries (500µH).
You can find where the 500µH line on
the chart intersects the top horizontal
axis – it is marked in blue and is at an
angle of 45° (sloping up to the left) on
the chart of Fig.7.
In fact all the inductance lines slope
up to the left in the same way, just as
all the capacitance lines slope up the
to right. If we project that line down
to the horizontal line for a frequency
of 10MHz and then project down from
the intersection of those two lines
down to the bottom line of the chart
and the impedance can be read off as
just over 30kΩ (actually 31.4kΩ).
That’s fine, but what would be the
result if the circuit of Fig.5 used a
500µH inductor and a resistor value
of 1kΩ? What would be the cutoff frequency. In this case, we take the same
500µH sloping line and intersect it
with the 1kΩ vertical line. In this case,
the two lines intersect at a point corresponding to a frequency of just over
300kHz (actually, 318kHz).
Finally, let’s find the resonant frequency of a parallel LC network, as
shown in Fig.6. In this case, we will
use an inductor of 200 millihenries
(200mH) and a capacitor of 2 microfarads (2µF). In this case we need to
find the intersection of the sloping line
for a value of 200mH with the sloping
line for a value of 2µF. Both lines are
shown in pink and you will see that
if you project across to the right from
their intersection, you can read the
resonant frequency from the vertical
righthand axis as 250Hz (on Fig.7).
As you can see, this chart enables you
make many thousands of impedance,
resistance, capacitance or frequency
calculations, all without resorting to
SC
formulas or calculators.
100kW
10kW
1kW
100W
10W
GIANT A2 CHART NOW AVAILABLE!
The chart overleaf is great . . . but
imagine how much easier it would be
to use if it was larger! SILICON CHIP now
has available HUGE A2 (420 x 590mm)
charts, printed on heavy art paper, ready
for your lab, workshop or office!
Price is just $10.00 each inc GST + P&P,
mailed folded, or $20 each inc mailed
unfolded (in a protective tube).
Order from the SILICON CHIP Online Shop
(siliconchip.com.au/shop) or call SILICON
CHIP during office hours (94.30, NSW
Time MonFri) to obtain your copy.
1Hz
1W
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