Oh this is a pure sine wave a sine wave is the simplest wave form that exists in nature it consists of a single fundamental frequency with no other components this sine wave is occurring at the rate of 1,000 cycles per second this is a square wave occurring at the same frequency both these wave forms are extremely useful in testing electronic apparatus in almost every case however the square wave is much more useful than the sine wave in the next few minutes we're going to discuss some reasons why this is true any wave shape other than a sine wave is known as a non sinusoidal or a complex wave form this is a complex wave form this is a complex wave this is a complex wave and this is a complex wave due to its basic appearance we call this wave form a square wave any repetitive complex waveform can be analyzed and broken down into its individual elements these elements are actually simple sine waves of different frequencies that have been added together in certain phase relationships and amplitudes before going any further let's discuss harmonics a harmonic is a waveform whose frequency is a whole number multiple of another frequency if we choose 1000 cycles as a fundamental frequency then it's second harmonic would be 2,000 cycles it's third harmonic would be 3,000 cycles the fourth 4,000 cycles and so on even harmonics are even whole number multiples such as the second harmonic the fourth sixth eighth and so on odd harmonics are the odd whole number multiples such as the third harmonic fifth seventh ninth of course the fundamental could be any frequency if it were four thousand forty four cycles the second harmonic would be eight thousand eighty eight cycles now let's analyze a square wave this square wave is occurring at the rate of one thousand cycles per second when it is broken down into its individual components we find it consists of the following a sine wave at the fundamental frequency of 1000 cycles per second a third harmonic at 3,000 cycles a fifth harmonic at 5,000 cycles a seventh harmonic a ninth and eleventh a thirteenth a fifteenth and so on notice that it consists of all odd harmonics a perfect square wave would consist of a sine wave at the fundamental frequency plus an infinite number of odd harmonics to obtain a square wave these harmonics must be added all in the exact same phase and in the proper amplitudes let's look at this graphically here is a 1000 cycle sine wave here is the third harmonic 3000 cycles at one third the amplitude if we combine these two signals that is add them algebraically the resultant waveform will be this if we now take the fifth harmonic 5,000 cycles at one fifth the amplitude of the fundamental and add it to the waveform the result will look like this with the fundamental and only two odd harmonics so far we can already see that this waveform is starting to take the shape of a square wave if we continue to add odd harmonics in phase and at the proper amplitudes the waveform will more and more approach that of a square wave adding harmonics of higher frequencies causes the leading edge of the wave to rise more rapidly and produce a sharper corner between the leading edge and the top of the wave it would require an infinite range of harmonics to produce a truly vertical edge and an actual sharp corner this situation is physically impossible to produce but waves can be generated that are very close to this ideal a wealth of information then is contained in a single square wave due to the harmonics involved this single wave form contains a wide range of frequencies information regarding the amplitudes and phase relationships of the higher harmonics is contained in the steepness of the leading edge and in the sharpness of the corner the flat top contains information regarding amplitudes and phases of the low frequency components that is the fundamental frequency and the first few harmonics the usefulness of the square wave in testing electronic apparatus becomes quite apparent now for a square wave to pass through an electronic circuit without any change or Distortion that circuit must be capable of passing a wide range of frequencies without altering their relative amplitudes or altering their phase relationships if the circuit is not capable of this the square wave will appear distorted at the output let's take some examples this little black box will represent an electronic circuit which we want to test it might be an amplifier it might be an attenuator at any rate for the purposes of demonstration we have designed this box so that we can introduce various nonlinearities to the square wave as it passes through an oscilloscope will show the distortions we might expect to see on a square wave first of all we need a square wave generator there are several methods of producing a square wave perhaps the simplest is by switching a DC voltage on and off however this method is cumbersome and inconsistent and it is difficult to attain a very high repetition rate this is a switch that is capable of producing an excellent square wave with a fast-rising leading edge it is a mercury pulsar another method of producing a square wave is to use a vacuum tube or a solid-state device such as a diode or transistor as an electronic switch such circuits make excellent square wave generators the calibrator in most laboratory type oscilloscopes is a satisfactory square wave generator this one operates at a fundamental frequency of approximately 1,000 cycles per second and we can accurately control the output voltage there are square wave generators of all shapes and sizes each design for various needs and requirements depending upon the intended use now with everything connected we find that the apparatus under test is passing the square wave correctly it's frequency response is fairly linear let us now introduce some nonlinearities and see the effect on the square wave let us assume our amplifier is not passing or amplifying the high frequencies as well as the low frequencies this would be the effect on the square wave notice that the leading edge does not rise as rapidly and that the corner is no longer sharp if the loss of high frequencies becomes greater the distortion increases let us now take an opposite effect this time we'll presume that our amplifier is amplifying the high frequencies more than the rest here is the result the leading edge rises very quickly and actually overshoots the top of the square wave this is called overshoot the flat top of the waveform contains the low frequency information here is an example of poor low frequency response here is an extreme case of poor low frequency response notice however that the rising front edge and the corner are still very sharp here is an example of excessive low frequency response this display usually indicates the presence of a resonant circuit it is known as ringing the upper frequency response of an amplifier usually can be increased consider by the use of resonant circuits in the amplifier the adjustment of these tuned circuits however can become complicated the use of the square wave as a test signal allows us to adjust such an amplifier for uniform frequency response without any ringing or peaking this is a simple voltage divider the total input resistance is 10,000 ohms the resistance at the output terminal is 1000 ohms 1/10 the total input resistance therefore this voltage divider will divide by a factor of 10 this can be demonstrated very quickly we apply 10 volts at the input if we measure the output we find it to be one volt this divider usually will feed into a circuit of some kind like the input of a vacuum tube as a result some capacitance is bound to appear across r1 this capacitance would be in the form of stray wiring capacitance and the input capacitance of the tube we will represent this capacitance by adding c1 to the circuit at DC and low frequency AC signals this capacitance as the frequency signal is increased however the reactance of a capacitor decreases and we find that c1 is bypassing part of the signal around r1 as a result we find that this voltage divider is no longer linear in its response to various frequencies with the input signal now at 20,000 cycles and still at 10 volts we find that the output has dropped below 1 volt and our divider is no longer dividing 10 to 1 the reason of course is that capacitor c1 at the higher frequency is shunting part of the signal around r1 thereby reducing the output voltage across R 1 the higher the frequency the greater will be the effect of c1 and the lower the output voltage there is a way to correct this situation let's add another capacitor to the circuit we will call this c2 now let's see what we've done capacitors c1 and c2 by themselves also form a voltage divider if their values are of the right ratio they can also divide the voltage by 10 to one a capacitor is reactance of course changes in relation to the applied frequency since both elements of this divider are capacitive they will both change in proportion to the frequency and the ratio will remain the same therefore they will continue to divide by ten to one even when the frequency is changed by combining this divider with the resistive divider we now have a frequency compensated voltage divider that will operate at a ten to one ratio up to very high frequencies in simplified form this is the input circuit of an oscilloscopes vertical amplifier resistor R represents the input resistance of the scope which is usually 1 megohm capacitor C represents the total stray wiring capacitance and the tube input capacitance on this scope this total input capacitance is 20 micro micro farad's in some applications even this high resistance and small capacitance can produce undesirable loading upon the circuit under test this loading can cause our oscilloscope presentations to be different from the actual waveform present if the oscilloscope were disconnected in this case we can minimize the loading effects by using a probe such as this this is a ten-to-one attenuating probe yet contains a 9 mega ohm resistor in series with the cable leading to the scope when the probe is connected to the scope this 9 mega ohm resistance is placed in series with the 1 Meg input resistance of the scope making the probes new total effective input resistance 10 mega ohms the resistive and capacitive loading on the circuit under investigation is reduced through the use of the probe notice that when the probe is connected to the scope a ten-to-one voltage divider is created just as in our earlier example notice also that there is a capacitance C which shunts resistor R this tends to reduce the high frequency response to counteract the effect of C and make this a frequency compensated voltage divider the probe also contains a small trimmer capacitor C P which is in parallel with the probe resistor R P capacitor C P in the probe is made adjustable so that when the probe is connected to a different scope it can be readjusted to compensate for differences of input capacitance how do we properly adjust this capacitance it is really quite simple again the square wave comes to our aid since the square wave contains a wide range of frequencies in a single wave form the voltage divider can easily be compensated using this one signal we feed the signal from the calibrator into the probe and then to the scope by adjusting CP in the probe we can see the effect it has on the square wave in this case the voltage divider is not correctly compensated and the high frequencies are being attenuated as evidenced by the rounding corner on the square wave in this case the divider is overcompensated and is accenting the high frequencies as indicated by the overshoot on the rising portion of the wave when the probe is properly adjusted the square wave is correctly displayed and we can rest assured that our voltage divider probe is responding equally to all frequencies the scope now we'll make a faithful reproduction of any signal we wish to display the rising portion of the square wave has become extremely important as a test signal in many cases we don't even care if the rest of the square wave is very square as long as the rising portion is just as straight and as sharp as it can be for example we can use the fast rising portion to check out a transmission line a good way of checking if the line is properly terminated is to feed in a fast rising wave while viewing the results with an oscilloscope when the line is properly terminated we will see only the applied signal if the line is not properly terminated we will have reflections which will appear on the scope with the line open the display will look like this if there is a short the display will appear this way this is an artificial delay line it is used to delay an electrical signal for a very small fraction of a second delay lines are used in television broadcast equipment radar equipment electronic computers and so on delay lines are also used in most laboratory type oscilloscopes to give the sweep time to start before the signal is actually displayed on the screen these delay lines are very tricky to tune and adjust proper adjustment can only be accomplished with a fast-rising square wave as we know the speed with which the rising portion of the square wave Rises depends on how many harmonics are present in the wave a perfect square wave would contain an infinite number of odd harmonics and the leading edge would rise from minimum to maximum in zero time such a wave of course is impossible to produce if we adjust this scope to expand the rising portion of this square wave we can see that it truly does not rise in zero time the time it does take to rise is called rise time to be a little more precise about it rise time is usually considered to be the amount of time it takes for the voltage to rise from 10% of its peak value to 90 percent of its peak value in other words it is the amount of time from here to here in this case the rise time is about 1 1 millionth of a second or 1 microsecond in today's modern world of electronic science it is becoming more and more important to measure things that happen in a fraction of a million or even a billionth of a second the use of the square wave especially the rising portion has become extremely important in checking a circuits ability to respond to high-speed signals to accurately measure a circuits response to a fast rise time it would be desirable if we could generate a perfect square wave a square wave which rises in zero time and if we could build a scope that could perfectly reproduce such a rise time of course these things are impossible nevertheless we continually strive to push ahead as far as possible this is a square wave generator that can produce a square wave with the rise time that is extremely fast we're not sure exactly how fast because we're limited as to how fast we can measure it this is one of the fastest precision laboratory or Silla scopes now available by connecting the signal from this generator into this scope and pushing the scope to its maximum speed we get this display the rise time of the waveform you are now viewing is less than a quarter of a billion of a second this means there are harmonics present which extend up close to one and a half billion cycles per second are we beginning to approach the generation of a perfect square wave with zero rise time just how close is a quarter of a billionth of a second to zero rise time well how far is it from a billion to infinity
