August 1972 Popular Electronics
Table of Contents
Wax nostalgic about and learn from the history of early electronics. See articles
from
Popular Electronics,
published October 1954  April 1985. All copyrights are hereby acknowledged.

A lot of us still use older test
equipment at home and even in the company lab. As discussed in this 1972 article from Popular
Electronics magazine, the displayed rise time on an oscilloscope display is not necessarily
that true rise time of a signal  particularly when the speed approaches the rated bandwidth
of the equipment. In that case, it is necessary to mathematically compensate for the rise
times of each individual component used for making the measurement. Hooking the oscope probe
tip to the calibration point on the front of the instrument and adjusting the probe's trim
capacitor for a flat response is not always good enough. Most modern oscopes can calculate
and apply corrections automatically, negating the need for a manual correction. If your application
is not super critical from a timing standpoint, then you do not need to bother with correction,
but it is worth keeping rise time measurement inaccuracies in mind just in case you run into
an otherwise inexplicable problem that may turn out to be a timing issue.
RiseTime Measurements
By Raymond E. Herzog
What You See My Not be What You've Really Got
Suppose you feed the output of a pulse generator through a probe to an oscilloscope. You
adjust the scope time base for a fast sweep. (This is the typical setup for measurement of
rise time.) The result is the easily recognized risetime waveform shown in Fig. 1.
Fig. 1  Pulse rise time is measured between the 10% and 90% points on
the leading edge of oscilloscope trace. In this case, rise time is 100 nanoseconds.
Question. If the time base is set for a sweep of 50 nanoseconds per division,
what is the rise time of the generator's output pulse  ignoring any inherent scope inaccuracies.
"That's easy," you say, quickly multiplying 50 ns/div times the two divisions covered by
the pulse's leading edge. "100 ns is the rise time of the pulse. Right?"
Maybe yes; and maybe no!
Although 100 ns is the correct reading of the scope display, this value may not be the
true rise time of the pulse. So ignoring scope inaccuracies, what other reason is there for
suspecting that the measurement is not the real rise time?
Basically it is that, when the pulse passes through the probe, the scope amplifier, and
even the scope CRT, it suffers rise time deterioration. Thus, the displayed waveform can't
represent the true, original signal.
Although this is an unfortunate circumstance, it has one saving grace: it can be predicted.
So, if you erroneously said that the rise time in the example was 100 ns, it should be both
interesting and informative to learn how to determine what the error might be.
RiseTime Formula. True rise time can be determined by the use of the
tonguetwisting formula known as "the square root of the sum of the squares." In mathematical
terms:
where T_{RG} = true signal generator rise time
T_{RD} = displayed rise time
T_{RP} = probe rise time
T_{RO} = oscilloscope (amplifier and CRT) rise time
For a low to mediumpriced generalpurpose oscilloscope, a typical rise time is 35 ns.
For a standard probe, rise time is 5 or 10 ns. Putting these values into the formula, we
get
Thus, the actual rise time of the generator pulse is 93.1 ns and not the 100 ns displayed
on the scope; and the measurement was in error by 7.4%.
It is important, therefore, to keep in mind that the displayed rise time is greater than
the actual rise time. The amount of error is shown in Fig. 2, where per cent of error is plotted
against the ratio of the input signal's rise time to test equipment rise time. In our example,
this rise time ratio would be 93.1 divided by the square root of 35^{2} + 10^{2}
or 2.56.
Knowing that the measurement devices do introduce an error and with the aid of Fig. 2,
you can determine just what error to expect. Conversely, to make a measurement within a given
accuracy, Fig. 2 can be used to find what rise time response is needed in the test equipment.
Fig. 2  Measurement error is inversely proportional to the risetime ratio.
More About Rise Time. We often concern ourselves with only the frequency
response of an oscilloscope or an amplifier when, as we have seen, rise time is also important.
In using Fig. 1, we measured rise time between the 10% and 90% points of peak value on
the leading edge of the pulse. These two points are used as standards for waveform measurement
in the industry.
Looking at Fig. 1 again, you will note that the pulse rises linearly from the 10% voltage
level to the 90% level. Such a characteristic is known as a Gaussian response. When an ideal
unit step pulse (one with zero rise time) is applied to an amplifier (or cascaded amplifiers)
whose frequency response is RC limited, the response is Gaussian. For a Gaussian response,
there is a relationship between rise time and frequency (also called bandwidth):
0.35 = t_{r} X bw
where t_{r} = rise time in microseconds
bw = frequency bandwidth in MHz
Oscilloscope vertical amplifiers consist of cascaded RC stages so the above formula applies.
For instance, if a scope has a 10MHz bandwidth, its corresponding rise time would be 0.035
microseconds or 35 ns.
The 0.35 constant in the formula results from a combination of two factors:
RC rise time = t_{r} = 2.2RC
3 dB frequency = F_{C} = 1/ (2πRC) Consider first the
2.2RC factor and look at the universal time constant curve in Fig. 3. This curve will readily
be recognized as the capacitor charging voltage in a series RC network with an applied ideal
step pulse. Recalling that, in rise time measurements, the 10% and 90% points were used, we
find the corresponding RC value for these two points: namely 0.1 RC and 2.3 RC. Taking the
difference between these two values, we get 2.2RC  the time in which rise time is resolved.
Fig. 3  This is time constant curve for charging capacitor in series RC
circuit.
But wait just a moment! The curve in Fig. 3 is not Gaussian, so how can
it be used in our calculations? Well, irrespective of Gaussian  or whatever  the universal
time constant curve serves only as a reference for the 2.2 RC value. It is not the signal
pulse to be measured (as in Fig. 1) and thus does not have to be Gaussian.
Remember that what we did say about Gaussian response was that it pertained to an amplifier
stage whose frequency response is RC limited.
This brings us to the second factor in the 0.35 constant: the 3dB frequency, or the frequency
at which an amplifier's gain is down 3 dB from midfrequency gain. The influence of the stage's
resistance and capacitance is shown in the 3dB frequency formula: F_{C} = 1/ (2πRC).
Having defined these two factors, let's see how they determine the 0.35 constant. First,
transpose F_{C} = 1/(2πRC) into RC = 1/(2πF_{C}). Then from
t_{r} = 2.2RC, we get t_{r} = 2.2. [1/(2πF_{C})] = 0.35/F_{C}.
Finally, 0.35 = t_{r} X F_{C}. Or since frequency response is bandwidth, 0.35
= v X bw.
Posted July 18, 2017
