The VK-1,2 distortion measurement set is built according to the traditional design concept. An analogue oscillator produces the output signal which theoretically, as it follows from calculations, can be a pure sinusoid, its set parameters being amplitude and frequency.
     This sinusoid is applied to the input of the device under test, an amplifier for example, whose output is then analyzed to obtain its main AC characteristics. The output component of the same frequency as input, compared with this input in amplitude and phase, gives the device transfer characteristic at this frequency. The output components of other appearing frequencies (2, 3, 4, 5... times higher than the frequency of input sinusoid) give in their sum the signal distortion. Output of the device under test contains also a wide band component - noise which is generated within the device by its active and passive components.
     The task of removing the output component of the fundamental, originated from the oscillator frequency is performed by a notch filter being a key part of the classic distortion meter. The filter theoretically, according to calculations, can provide infinite suppression of the fundamental, an important requirement here is to leave untouched the residual harmonics of distortion and noise. To measure them directly in relative units (percents or dB), the notch filter input signal should be maintained constant, say at 1V level. Such normalization, manual or automatic, is justified only if the involved circuitry doesn't notably add its own distortion and noise to the ones to be measured.
     The filter output signal represents a mixture of distortion harmonics and noise, it can be monitored on the oscilloscope screen and measured by a RMS-millivoltmeter. To separate these harmonics from each other and from noise, further analysis is necessary. Possible solution is the use of a selective filter which automatically tunes itself to the desired harmonic of distortion, thus extracting it from the notch filter output. Spectrum analysis of this output is the most widespread method, and if earlier it required an expensive spectrum analyzer, now it can be done with the help of an ordinary computer and the proper software installed. The classic distortion meter usually contains also band limiting filters which reduce unwanted interferences and noise, for example below 20Hz and above 100kHz, thus increasing the accuracy of distortion measurement.
     The described above measurement scheme is ideal, as it considers only the distortion and noise produced by the device under test. In reality, however, the final measurement result includes also the distortion occurred within the oscillator, while the sinusoid is generated, within the notch filter and generally within all the circuitry the signal passes through. This nuisance component of the total measurement result must be at least one order lower than the distortion of the device under test we are interested in. The expected distortion contribution of the engaged oscillator and distortion meter can be roughly evaluated from the devices' declared characteristics. Its exact value at a certain frequency and level can be obtained by excluding the device under test from the measurement scheme and simply connecting the oscillator output to the distortion meter input.
     It's therefore very important to have a distortion measurement set in which both instruments, the oscillator and the meter, always and constantly can check each other. The VK-1,2 instruments from the very beginning were being designed and tested just as a set, this made it easier to bring them ultimately to the highest degree of perfection. Vital for the successful development was also the right choice of the instruments' topologies and principles of their operation, for the VK-1 it is a multistage phase-shifting oscillator, for the VK-2 - a twin-T notch network. The latter is an outstanding circuit which can provide any required suppression of the fundamental frequency, down to -150dB in practical realization. All depends on the performance of its system of automatic tuning to this frequency and generally on the level of designing other parts of the filter, including of course a proper choice of components.
     The used in the VK-2 distortion meter original tuning algorithm of the rejection filter and its optimized circuitry have allowed to achieve a reliable -135dB suppression of the fundamental frequency, the whole tuning process taking 3sec - 1,5sec of rough tuning and 1,5sec of fine tuning. In the tuned state, the twin-T notch network has at its output a 200nV signal of the fundamental frequency, left from the normalized 1V input (0,00002%). The participating in such huge signal attenuation are the all six elements of the network, three its R-arms and three its C-arms, so only they are responsible for the added distortion and noise. The subsequent +80dB amplification makes the filtered residual measurable by a standard millivoltmeter and visible on the screen of an ordinary oscilloscope. The amplifier isn't very critical to introducing distortion because its input signal is vanishingly small, the main requirement here is minimizing output noise to distinguish at its background as low distortion as possible. Helpful in this respect is the use of a third-order low-pass filter which effectively cuts off the frequencies beyond the range of interest (20Hz-100kHz). This problem, however, doesn't seem so actual when measuring distortion higher than 0,01%.
     The used in the VK-2 meter twin-T notch network is depicted in Fig.1 in its general form.

Fig.1. Twin-T notch network.

     Its transfer characteristic seems complicated, as it is represented in a general form too:

     The same expression after rearrangement:

     Infinite suppression of the input signal VIN , i.e. turning the network output VOUT to null will take place when both the real and imaginary parts of the numerator in expression (2) become equal to zero:

     Suppression frequency ω0 can be defined from the first equation of the system:

     As we see, it doesn't depend on the resistor R3 , but varying the value of R3 can help to meet the second condition (4) of tuning the network. Solution of this equation yields:

     To make calculations, component selection and tuning more simple and convenient, the values of adjacent R-arms (R1, R2) and C-arms (C1, C2) of the twin-T circuit are usually chosen equal, while the third C-arm has capacitance C3 of double value.

                                                  C1 = C2 = C;    R1 = R2 = R;   C3 =2C.

     Substitutions in (6) give resistance R3 of the third R-arm of the obtained symmetrical circuit:

    The transfer characteristic of the symmetrical twin-T notch network looks quite straightforward:

    We are interested in the magnitude (K) of the transfer function and the phase shift between the output and the input:

     The results of calculations are represented in table 1:

Table 1. Passive and active twin-T notch network - calculated data.

    As it follows from the obtained data, the main drawback of the passive twin-T filter is a wide frequency bandwidth of the input signal attenuation. This results in the output with a 9dB loss of the second harmonic and a 5dB loss of the third and makes it impossible to use such form of the network for accurate distortion measurements.
     Another problem concerns the notch depth which is very sensitive to a mismatch between the fundamental frequency of the input signal and the rejection frequency ω0 determined by the values of all six elements of the twin-T network. The network's 1% resistors guarantee themselves only -46dB suppression of the fundamental (see table), but there are also three capacitors which have greater tolerances of their values and are prone to temperature and time instability. Therefore, to ensure satisfactory operation of the network, for example as a 60-Hz notch filter, some sort of adjustment is needed.
     The network performance can be considerably improved by adding an amplifier and transforming the circuit to an active twin-T filter. The amplifier solves here a triple task. Firstly, it buffers the network to preserve it from excessive loading and to create optimum conditions for its operation. Secondly, it provides the necessary amplification for the residuals, this is particularly important in sensitive distortion measurements where these residuals are very small. And, at last, by injecting the positive feedback signal to the twin-T network, the amplifier sharpens its notch, thus leaving practically intact all that passes through the filter, except of cause the fundamental frequency. The notch depth at this frequency remains here the same, however, the reaching of its maximum becomes more difficult, particularly if an excessive amount of the positive feedback is applied.
     The active version of the twin-T filter is depicted in Fig.2.

Fig.2. Active twin-T filter.

     The circuit will be analyzed with the help of a nodal method. Considering the three node voltages V1, V2, VIN+ and remembering that the currents' algebraic sum in each node equals to zero, the following three equations can be written for the nodes 1, 2, 3:

     Potentials V1 and V2 can be expressed from the first and second equations of the system correspondingly:

     Their substitution into the third equation of the system and further rearrangements give:

     The expression for potential VIN+ can be obtained if we assume that the negative feedback signal applied to the inverting input is β2VOUT and both inputs of the op amp A have equal potentials. Therefore, VIN+= VIN-= β2VOUT. After replacement of VIN+ in the equation (10), the latter is easily solved relative to the filter output VOUT:

     As in the previous case, the suppression frequency ω0 = 1/CR.
    The magnitude (K) of the transfer function and the phase shift between the output and the input:

     For accurate distortion measurements, the optimal Q-factor value is Q=2, the calculations made with the help of relationship (12) yield a -0,5dB attenuation of the second harmonic at the filter output and only -0,2dB attenuation of the third. Such figures are quite tolerable, given that in sensitive measurements the noise affects the total measurement result much more seriously. Greater Q-factor values bring more trouble than benefit, they are particularly harmful for smooth operation of the system of automatic tuning, causing notable chaotic fluctuations of the tuned filter output.
     Calculations of the transfer function magnitude K and phase φ are made for the two versions of the twin-T notch network, passive and active, their results are placed in a common Table 1 for comparison.
     Graphically, the transfer characteristics of the circuit Fig.2 are obtained with the help of Multisim 10 software. Resistors R4 =240-Ohm and R5 =11,8kOhm set the amplifier gain A=1/β2=50, while the positive feedback elements R6 =560-Ohm and R7 =0 or 10-Ohm give the possibility to choose the passive, without feedback, or the active, with feedback, modes of the twin-T network operation, the feedback factor β1 being correspondingly 0 (Q=0,25) or 1/57 (Q=2).
     Before inputting to the program, the network values R, C were calculated for a given frequency of 1kHz with a great accuracy in hope to achieve a very deep notch. However, the results of simulation (see Fig.3) demonstrate a -75dB suppression of the fundamental for the filter without positive feedback (PFB) and only -45dB suppression for the filter with this feedback. Putting these limitations, the clever program indicates how difficult is to reach a maximum notch, relying only on good or even ideal component matching.

Fig.3. Transfer characteristics of the twin-T filter.

     Further analysis of the twin-T notch network will be done with reference to its application just in the distortion meter. Tuning the network to the fundamental frequency of the input signal is performed by varying the values of its components until reaching ω = ω0, where ω0 is the filter rejection frequency. As already shown,

     The assumptions C1 = C2 = C;    R1 = R2 = R;   C3 = 2C are usually made to simplify calculations, component selection and subsequent tuning, the rejection frequency being in this case ω0 = 1/CR. If the components are improperly matched, ω0 can be set by adjusting the value of a single element R2. Clearly, bigger tolerances of the component values require a broader range of adjustment, that leads to a complication of the tuning process. Therefore, a good distortion meter design should be oriented to the use of 1% network resistors and 2% selected capacitors.
     Also, it should be taken into account that differences in the component values can compensate each other, according to relationship (14). For example, if C1=0,95C; C2=1,05C; C3=2C; R1=1,03R and R2=0,97R, the rejection frequency ω0 =1,0034/CR is practically the same as in the ideal case (ω0 = 1/CR).
     Getting the equality ω = ω0 doesn't mean, however, the network complete tuning. This turns to zero only the quadrature (90°phase) component of the output. The second condition, nulling the in-phase component of the output, is fulfilled by choosing the value of R3 according to formula (6):

     In the symmetrical twin-T network, this condition might be satisfied automatically, if R3 was exactly equal to 0,5R, along with other exact components values. Unfortunately, for the real circuit the R3 adjustment is necessary and catching the filter output minimum by turning simultaneously two potentiometers becomes a tiresome procedure.
     Nevertheless, I conducted an experiment to learn what maximum notch of the rejection filter output can be reached at a certain level of accuracy the filter components are chosen with. The filter circuit used for its manual tuning is depicted in Fig.4.

Fig.4. Rejection filter tuning - simulation.

     It is an ordinary active rejection filter with the twin-T notch network composed of three basic resistors (R1, R2, R3) and three capacitors (C1, C2, C3). The amplifier gain K=50 and the filter Q-factor Q=2 are set with the help of resistors R5, R6 and R7, R8 correspondingly, the circuit is fed from generator V1 providing a 1V-RMS/994Hz sine-wave signal. To suppress this frequency, the calculated according to expressions (5),(6) filter parameters should be: R1=R2=10kohm, R3=5kohm, C1=C2=16nF, C3=32nF.
     To simplify the experiment, assume all the capacitors and resistor R1 have exact, the specified above values and the filter fine tuning will be carried out by choosing the values of resistors R2, R3 and by adjustment of potentiometers R12=20ohm, R13=10ohm, connected in series with them. Getting the filter's good minimum output requires at least a 0,1ohm accuracy of varying the R2, R3 values and 1% incremental adjustment of the R12, R13 values.
     The filter achieved tuned state is shown on the virtual oscilloscope screen, additional data are supplied by some measurement probes (see Fig.4). The final result of this interactive computer simulation is:
                          262uV output at 994Hz (1V input),
                          958mV output at 994x2=1988Hz (20mV input),
                          983mV output at 994x3=2982Hz (20mV input).
     Given that the amplifier gain is 50, the twin-T notch network provides
                          262uV/50/1V=-106dB signal attenuation at the fundamental frequency 994Hz,
                          958mV/50/20mV=-0.37dB at its second harmonic 1988Hz, and
                          983mV/50/20mV=-0.15dB at its third harmonic 2982Hz.
     In reality, however, the task of obtaining such considerable suppression of the fundamental isn't so feasible. Too much destabilizing factors exist, there are non-ideal electronic components, frequency fluctuations of the source oscillator, interferences coming from power supply and others.
     Usually, the whole process of tuning the twin-T notch network is divided into two stages - rough tuning and fine tuning. Firstly, the necessary frequency range of the input signal is being chosen by simultaneous switching the capacitors in the three C-arms of the network. Information about the signal fundamental frequency is supposed to be known, as the signal is derived from a concrete oscillator with the already made frequency settings, otherwise, the fundamental's value should be checked by a frequency meter. The usual number of the frequency ranges is three, so we have the decade frequency range selection which covers the whole 20Hz-20kHz audio band. The required for that 9 network capacitors are the most critical components of the circuit, they should be matched at least with a 2% value accuracy and be of high quality types, polystyrene for example, to ensure good stability and low distortion.
     The next is successive stepped switching of the resistors in the three R-arms of the twin-T network to bring the rejection frequency closer to the fundamental, when the network fine tuning becomes possible. For this purpose, the sections of 1% resistors are used in these R-arms, their commutations being accomplished by a common switch. In this case, the rejection frequency is varied by the so-called frequency steps and it's reasonable to keep them equal. The more the number of these steps dividing the frequency range, the easier is the subsequent fine tuning, but the more time should be consumed to locate the frequency step interval into which the fundamental frequency falls. This dilemma exists both in automatic and manual tuning systems and the VK-2 distortion meter just demonstrates how can it be resolved.
     The above manipulations prepare the twin-T network for the final tuning procedure - reducing both components of its output to a minimum achievable level. This fine tuning is carried out by smooth adjustment of the total resistances of the network R-arms, R2 and R3 for example, according to expressions (6), (14). The adjustment of R2 covers here the total required resistance variation of both adjacent R-arms (R1, R2) to tune the circuit exactly to the fundamental.
     There are three types of electronic components which behave as variable resistances and therefore are suitable for fine tuning of the twin-T network - potentiometers, FETs and resistive optocouplers. The potentiometers used in the circuit manual tuning shouldn't be grainy and noisy, their preferable type is high-quality wire-wound, so the achieved notch can be deep and stable. The potentiometers, along with constant metal film resistors, don't practically contribute distortion and this is their main merit.
     The FETs are cheap, easily and instantaneously controllable, even manufactured as matched pairs and therefore being able to reproduce any desired law of their resistance variation. Their disadvantage is increased distortion if the voltage applied to the FET's channel exceeds 100mV, this, however, doesn't exclude their use in low distortion designs. In the VK-1 audio oscillator, the FET is employed in the system of amplitude stabilization, but that isn't an obstacle in achieving ultra-low -130dB distortion of the output.
     The resistive optocouplers offer a huge dynamic range of the variable resistance (100-Ohm -10MOhm) and absolute isolation between the input LED and the output photocell resistance. Most important for the twin-T network fine tuning is a remarkable linearity the used optocoupler can provide. If the voltage across the photocell is kept below 100mV and its resistance doesn't exceed several kOhms, the network's produced distortion is less than -130dB. I tried the optocouplers of such brands as Silonex, Vactrol, Tesla and others, their performance may slightly differ, but the main characteristics are identical because the photocell material is the same - cadmium sulfide CdS. There is a small drawback inherent to this optocoupler type - an inevitable settling time of 10-200msec needed for the cell resistance to transit from one value to another. In the considered application, this effect may be neglected, but it should be taken into account when using the device in the circuits with fast signal processing.
     The optocouplers are key parts in my VK-1,2 instruments which were designed, built and tested some years ago. Recently, I have found at last a simulation model of the Silonex NSL32-SR3 optocoupler, that allowed me to conduct virtual testing of the instruments and to get full confirmation of the results earlier obtained in the course of real measurements and experiments. Here I would like to place the main characteristic of the NSL32-SR3 optocoupler, which can be useful in better understanding the work of the VK-2 rejection filter (Fig.5).

Fig.5. Silonex NSL32-SR3 optocoupler - characteristic.

     The characteristic isn't linear because the optocoupler by its nature is a semiconductor device. This explains the fact that photocell linearity begins notably deteriorate, if the applied voltage exceeds 1V. Distortion also depends on the photocell resistance, with the same voltage applied - the more resistance, the higher distortion.
     The described above manual process of the twin-T network tuning can be made automatic without altering the concept and sequence of operations - switching the capacitors, switching the resistors and smooth resistance variation. With this purpose, the mechanical switches should be replaced by relays, the adjustment potentiometers - by resistive optocouplers and also a system of automatic tuning must be introduced. It monitors the signals within the device and optimizes the tuning process which takes in this case some seconds. As a result, the network output exhibits the deepest possible notch and the tuned state is then continuously maintained. The automatic tuning system may be complicated, containing a processor, memory and other attributes of digital circuitry, but the core of this circuitry remains the same - an analogue filter which operates with analogue signals, so the main principle of any modernization is to do no harm.
     For example, in one's head may spark an idea to replace the mechanical switches and relays by their electronic counterparts, to make a distortion meter more compact, cheap and convenient. Indeed, the new distortion meter will work well, but only in the distortion range of 0,003-3%. Below a 0,003% limit the device will measure own distortion of the electronic switches and this distortion cannot be less than 0,001%, at signal levels of 1V and higher. Therefore, in ultra-low distortion applications (<0,001%THD), all commutations should be done only with the help of mechanical high-quality contacts.
     The system of automatic tuning of the VK-2 rejection filter was also upgraded, the aim was to reduce the time of tuning and distortion produced, the latter slightly varies with the used optocouplers' brand. The idea to build my own measurement instruments appeared first in the very beginning, when I was fiddling with commercial oscillators and distortion meters to test my early audio equipment, just at that time I understood what should be done and what shouldn't be done in designing such devices.
     My approach of the octave frequency range selection turned out to be successful both in the VK-1 oscillator and the VK-2 distortion meter. The VK-1 user can enjoy a fast and convenient procedure of obtaining the frequency response simply by consecutive selection of the octave spot frequencies with the help of a single rotary switch. These frequencies of exact, adopted in audio values are spread along the whole 20Hz-20kHz range evenly, with one- or third-octave intervals, and also, within each interval any desired frequency can be set with the help of a fine frequency control.
     In the VK-2 rejection filter with its system of automatic tuning, the user isn't interested to know into which range the fundamental frequency falls and may at all be unaware of this frequency value. All this is necessary for the system managing the tuning process, therefore, this system must perform some sort of frequency measurement. The octave range selection may seem here complicated and expensive, as it requires 30 high-quality capacitors being switched in the network C-arms (3 caps × 10 octave frequency ranges). However, these narrowed ranges are quite justified as they greatly simplify subsequent dividing of the range, by switching the resistors in the network R-arms.
     Pondering how to do this dividing best and how to optimize frequency monitoring, I came to the idea to combine these two processes. If each R-arm consists of binary-weighted resistors and the upper ends of the octave ranges are binary-weighted too, this well corresponds to binary frequency counting. Four switchable in parallel binary-weighted resistors of each R-arm of the twin-T network provide 16 equal intervals, or call them steps, within the octave frequency range. These steps of the rejection frequency ω0 are narrow enough to perform within the step the network fine tuning to the fundamental frequency, i.e. reaching the equality ω = ω0 . Localization of each step is determined by a combination of the switched on (1) and the switched off (0) resistors (4 in total ). But first of all, it's necessary to localize and switch on (1) the fundamental frequency octave range, one of a ten. Therefore, the total number of the unknown (0 or 1) switchings to be done, more correctly triple switchings, is 14.
     To solve this task at once, a 14-bit binary counter is used, its ten most significant outputs correspond to the ten octave ranges. In the first phase of the network rough tuning, call it a counting phase, the square-wave shaped pulses of the fundamental frequency are fed into this counter during a 0,762sec time window, the count result being a 14-bit output code of the counter. During the next 0,762sec of the so-called tuning phase, the counter outputs are step by step analyzed, starting with the most significant one. The first met "1" at the counter outputs, i.e. the most significant "1" of the output code, means that the octave frequency range, corresponding to this output, should be switched on, all other ranges are kept switched off. The next four less significant bits of the output code define the states of the four binary-weighted R-arm resistors, starting with the one of the lowest value (1 - switched on, 0 - switched off). The whole process of rough tuning lasts therefore 1,524sec, after that the network fine tuning becomes possible.
     A simplified circuit diagram of the VK-2 distortion meter is represented in Fig.6.

Fig.6. VK-2 distortion meter.

     The input (1-30)V signal is normalized by the regulated attenuator R10 - R11 and then buffered by a high quality amplifier A1. The amplifier delivers the normalized 1V signal to the input of the twin-T notch network and also to a square wave shaper 2 and to a quadrature phase shifter. The latter is the whole servo system comprising a phase shifting stage itself, a synchronous detector, an integrator, an optocoupler and also employing a square wave shaper 1. The target of such complication is to automatically maintain, within the whole 20Hz-20kHz range, a very accurate 90° phase shift between the outputs of the shapers 1, 2, these outputs being used in the high performance system of automatic tuning of the rejection filter. Both square wave shapers are built on discrete components and each of them produces two balanced, square wave, very symmetrical outputs with fast rising and falling edges and regulated swing.
     The amplifier A1 is all-discrete too, its configuration is just the same as that of the stages in the VK-1 audio oscillator, where distortion is guaranteed below -130dB at a 2V output on a 360-Ohm load. Here, in the VK-2 meter, the amplifier works in more comfortable conditions, its output is stable 1V and it sees an easy load in the form of the twin-T filter, therefore its distortion can not be registered even by the spectrum analysis. I suggest the distortion figure is less than -140dB.
     Returning to input normalization, it should be noted, that the level of the input signal may vary considerably, up to 30V, therefore the use of any kinds of electronic switches and electronically regulated resistors is absolutely unacceptable in the normalizing circuitry. This circuitry shouldn't be very complicated, containing numerous mechanical contacts, as that deteriorates the signal too, at last, such a block may appear to be expensive.
     An optimal solution in the sense of performance and economy is to use for normalization purpose a single high quality wire-wound potentiometer, thus shortening the signal path where distortion and noise can be picked up. The normalization block is a servo system which controls the process by producing the necessary indication (more - less) for manual adjustment and generating control signals for automatic adjustment, with the help of a motorized potentiometer. All that allows to set a 1V normalized level quickly and with 2% accuracy. The choice of a potentiometer value should satisfy two opposite requirements - this value should be low enough to minimize noise, at the same time it should be high enough to prevent excessive loading of the device under test. The compromised value is 2,2kOhm.
     The depicted in Fig.6 twin-T network configuration looks compact and straightforward. Being octave switchable, the capacitive elements C1= C2= 0,5C3= C take the values of a most typical binary sequence C = 1, 2, 4, 8, 16... 510nF. All capacitors are polystyrene, with at list 2% labeled tolerance, otherwise they should be selected or composed of two ones with a 2% accuracy. The network switchable metal film 1% resistors are only of five values based on R0=15kOhm, all their commutations being carried out by four high quality relays K1, K2, K3, K4 with triple contacts. Each of these relays may be replaced by two relays with twin contacts, their coils should be connected in parallel.
     The resistors of each R-arm are switched in parallel, therefore in calculations, it's reasonable to operate with their conductances which in parallel connection are simply summed. Considering the left adjacent R-arm, we see that its fixed conductance is G0=1/R0, the most significant switchable conductance is 1/2R0 and the least significant - 1/16R0. Designating 1/16R0=ΔG, we can obtain the total conductance of this R-arm:

     The general formula for the rejection frequency, expressed through conductance:

     In the same manner, Δω = ΔG/C, i.e. the step conductance determines the step in frequency within which the network fine tuning will be possible. This tuning is performed by smooth variation of the optocoupler resistance RV2 in the constantly connected branch of the second R-arm, which also includes a series resistor RS2 and parallel resistor RP2. Resistance and conductance of this branch:

     In an ideal case, R1=R2 and G1=G2, therefore the conductance of the constantly connected branch of the second R-arm is:

     When tuning the filter, the optocoupler resistance RV2 is varied between its minimum RV2min and maximum RV2max, the corresponding values of G02 being G02max and G02min. Smooth variation of G02 within the step conductance ΔG will be possible,
                          G02max - G02min > ΔG.      However, such variation of G2 is insufficient for varying the rejection frequency ω0

within the step Δω because conductance G1 isn't smoothly variable at all. Considering expression (15) and assuming G1= G2= G, we have:

     The above analysis shows that the doubled range of smooth conductance variation in the second R-arm can compensate the lack of such variation in the first R-arm and a slight difference between R1 and R2 is quite acceptable. Therefore, the final condition is:
                          G02max - G02min > 2ΔG.      Given that G02 is varied around G0= 1/R0 (see 16) and ΔG= 1/16R0,

the following two expressions, satisfying this condition, can be written, for further calculations:

     The set parameters are here: minimum optocoupler resistance RV2min, shunting resistance RP2 and base resistance R0. Substitution of their values into the first expression and its solving yield the value of series resistor RS2. Maximum optocoupler resistance RV2max is then defined from the second expression. A typical example: RV2min=1kOhm, RP2=4,3kOhm, R0=15kOhm. The results of calculations are RV2max>6,1kOhm and RS2<12,5kOhm.
     All relationships characterizing the third R-arm of the twin-T network are identical to those of the second, the difference being that R3= 0,5R2= 0,5R.
     Total conductance of the third R-arm:

     RV3 is the photocell resistance of the optocoupler U3 included in this R-arm, it is responsible for reaching the ultimate minimum of the filter output, given that preliminary fine tuning to the fundamental frequency is accomplished by the optocoupler U2. It seems both devices act simultaneously, because the whole process is fast (0,5sec) and precipitous. Computer simulation, however, runs this process in a slow time scale and it's clearly seen that a certain succession of the operations takes place - tuning to the fundamental is performed first and then both optocouplers jointly settle the output null. The degree of approximation to this null can be seen in Fig.3.
     Smooth RV3 variation from RV3min to RV3max causes the conductance G03 of the constantly connected branch of the third R-arm to be varied around its base value of 2G0= 2/R0. For successful tuning of the filter, the range of this variation should be wider than the step conductance 2ΔG=1/8R0, provided by stepped switching of the binary weighted resistors of this R-arm. Therefore:
                                       G03max - G03min > 2ΔG.      Final expressions for determining the components of R3:

     As in the earlier case, the set parameters are: minimum optocoupler resistance RV3min, shunting resistance RP3 and base resistance R0. The unknowns to be defined are series resistor RS3 and maximum optocoupler resistance RV3max. In the typical example, RV3min=1kOhm, RP3=2,7kOhm, R0=15kOhm. The calculations yield RV3max>3,4kOhm and RS3<6,3kOhm.
     The optocouplers U2 and U3 are the only source of distortion in the twin-T network itself. The photocell resistance is a sort of semiconductor, so it's prone to distortion by its nature. To minimize this distortion, the photocell resistance in both optocouplers should be maintained as low as possible, with the applied voltage not exceeding 100mV. Only in such strict conditions, distortion figures of -120dB and lower can be obtained. The signal level at the center points of both halves of the tuned twin-T network is the same and equals to 344mV, with the applied 1V input signal. At the same time, the measured voltages across the resistors RV2 , RV3 are correspondingly 55mV and 63mV, these figures being favorable to reducing distortion below -130dB. Obviously, this would be hardly achievable, if the optocoupler was used in the first R-arm, where the signal levels are still too high. Driving the optocouplers is not a problem, because a (1-9)kOhm range of the photocell resistance variation requires a mere (160-30)μA variation of the LED current (see Fig.5).
     Observing distortion below -120dB is possible only with the help of spectrum analysis of the VK-2 distortion meter output. The used for that SpectraLAB 4.32 software tests the combination of the VK-1,2 instruments, therefore the analysis result is total for both devices. All spectrum data, obtained for ten typical frequencies (31,5; 63; 125; 250; 500; 1k; 2k; 4k; 8k; 16k)Hz, are placed in Fig.11,12 of the VK-1 oscillator description, here only three graphs are adduced (Fig.7). These graphs represent the spectrums of the VK-1 + VK-2 combination output at the oscillation frequencies of 125Hz, 1kHz and 8kHz, the oscillator output set normally 1V (right vertical graph row) and reduced to 0,25V by pressing the noise reference button (left vertical row). As can be seen, neither distortion harmonics nor the fundamental exceed a -130dB level at all specified frequencies. In the second case, with a 0,25V oscillator output, the harmonics of distortion can not be revealed at all by the SpectraLAB 4.32 software, so the connected to the output RMS-millivolmeter measures and oscilloscope shows practically the pure total noise of the system.

Fig.7. Spectrums of the VK-1 oscillator + VK-2 distortion meter combination output, obtained with the help
of SpectraLAB 4.32 software: left - VK-1 output is set 0,25V, right - VK-1 output is set 1V.

     When inspecting the spectrum graphs, certain conclusions can be made. The sources of the third harmonic are probably the oscillator's amplifiers, deliberate loading of one of them leads to increasing this harmonic. The second harmonic is caused first of all by the optocouplers and its sharing between the oscillator and the meter is almost equal, may be in the VK-2 meter this harmonic is slightly higher, given that there are two optocouplers and their photocell resistances are incessantly varied.
     In the ultra-low distortion range (<0,001%), the direct measurement of total harmonic distortion (THD) becomes problematic, the main obstacle here being the noise component of the VK-2 output. This noise is a product of the whole circuitry of the oscillator and distortion meter. Noise contributed by the active components (transistors, ICs...) can be minimized by rational selection of their types, by optimization of their working conditions and simply by reducing their number. Noise generated within the passive components (the used metal film resistors) can be minimized only by reducing their values, according to the Nyquist formula:

     This is done both in the VK-1 oscillator, where the values of all gain-setting resistors in the amplifying stages are reduced to 3kOhm, and in the VK-2 distortion meter, where the network resistors R1, R2 are varied within (7,5-15)kOhm and R3 - within (3,8-7,5)kOhm. Further resistance reduction could create loading problems and lead to inevitable rise of distortion. To minimize noise, all signal amplifiers of the meter are configured as non-inverting, with the series feedback applied. Two of them are built on original discrete circuitry which can be best adapted to the requirements of each concrete application. The design of the input amplifier A1 was focused first of all on lowering distortion down to -150dB within 20Hz-20kHz, but its noise performance is excellent too (-120dB), that should be credited also to its use as a unity-gain buffer fed from a low-resistance source, in the input range of 1V to 30V.
     The filter amplifier A2 has an input stage whose parameters are well suited for work with the output resistance of the twin-T network. The unweighted input noise of the amplifier is 1,2μV in the measurement bandwidth of 100kHz, this equals -118dB relative to a 1V level. The set closed-loop gain is K2=50 (R4=240-Ohm, R5= 11,8kOhm), the chosen Q-factor Q=2 is established by resistors R6=560-Ohm, R7=10-Ohm, their values being calculated from (11).
     The rejection filter on A2 is followed by a third-order low-pass filter built around a standard fast op-amplifier. Its attenuation of the signal content above 100kHz is 18dB/octave, while in the pass-band, the filter provides an additional signal amplification KF=2. The chosen cut-off frequency is fixed, it allows to measure confidently the fifth harmonic of the highest audio frequency 20kHz. To prevent misunderstandings, I don't advocate making the cut-off frequency switchable.
     The last in the amplifying chain is a linear amplifier A3, the main requirements here being a stable closed-loop gain of K3=100 and satisfactory behavior of this amplifier in the conditions of deep output clipping, when the filter isn't still completely tuned. A standard fast op-amplifier is used, its noise doesn't practically affect the total noise figure.
     The overall amplification of the distortion products presented in the twin-T network output is 10000 (K2×KF×K3). Their magnitude of, say, 2μV could produce a decent 20mV reading (0,0002% distortion) of the RMS-millivoltmeter connected to the meter output and would be clearly seen on the oscilloscope screen. Unfortunately, total noise of about 2,5μV, referred to the A2 input, changes the picture considerably, and it completely swamps the distortion below 0,0001%. This -112dB noise, measured in the set 100kHz bandwidth, is generated predominantly within the network by its resistive elements. Fig.8 shows the spectral density of this noise monitored at the output of the rejection filter which includes the twin-T network and amplifier A2 (K2=50). Only smaller part of the noise is contributed by A2, its spectral density is rather monotonous along the whole 100kHz band.

Fig.8. Spectral density of the rejection filter output noise.

     The output of amplifier A3 is the main output the measurement result is taken from. In the distortion measurement ranges 0,1% and 1%, the gain of A3 is excessive and the measurement result is taken from the output of the rejection filter. The main output of the VK-2 distortion meter consists of two components, with a 90° phase shift between them, each of these components must be nulled at the fundamental frequency, according to expressions (3), (4).
     In the quadrature channel, the output signal is synchronously detected by an electronic switch U5 and then filtered and amplified by integrator A5. The integrator drives the LED of optocoupler U2, and the varied photocell resistance RV2 reduces the quadrature component of the output, thus closing this loop of automatic tuning. In the in-phase channel, the output signal is synchronously detected by an electronic switch U4 and then filtered and amplified by integrator A4. The varied photocell resistance RV3 of the second optocoupler reduces the in-phase component of the output and here the second loop of automatic tuning is acting.
     The detection accuracy is very high, because the used switches U4, U5 can handle equally well both microvolt and rail to rail clipped signals and they are controlled by precision, being in exact quadrature to each other, square-wave voltages of the fundamental frequency. Open-loop gain in each tuning channel is very high too, therefore reducing both fundamental frequency components of the output continues until reaching their -(130-135)dB level (see Fig.3,7).
     To optimize the tuning process dynamically, the time constants of integrators A4, A5 are changed with the help of electronically switchable resistors R12, R13. In the stationary conditions, when the input signal frequency remains fixed, the system of automatic tuning sets these time constants to the optimum needed for reliable maintaining of the filter tuned state. The system monitors the output voltage of integrator A5 and checks it to be within the set limits corresponding to the nominal range of varying the photocell resistance RV2.
     If the input signal frequency is suddenly changed, the rejection filter immediately responds to that by a sharp, up to clipping, rise of its output signal. The output voltage of integrator A5 goes out of the set limits and the system issues a command to reduce the resistances R12, R13 in order to speed up the recharging of capacitors C4, C5. During the nearest 1,524sec of the filter rough tuning, the input signal new frequency is measured and a new combination of the selected frequency range and the activated relays K1- K4 is established. After this preparation, the filter fine tuning commences and in its first 1,5sec it's performed with the still reduced time constants of integrators A4, A5. This time is quite enough to complete the processes of fine tuning and to reach a minimum at the distortion meter output. The ultimate settling of this output and its subsequent permanent holding are carried out with the increased, optimal time constants of the integrators A4, A5.
     A simplified circuit diagram of the system of automatic tuning is depicted in Fig.9.

Fig.9. System of rough automatic tuning of the rejection filter.

     The circuit operation is synchronized by a clock generator on NAND gates U10A, U10B, its frequency F0=21Hz being equal to the lowest frequency at which distortion measurements can be made. The 4-bit binary counter U12 divides this frequency by 16 and subsequent flip-flop U3 forms two time intervals T1= T2= 0,762sec, needed for the chosen two-phase operation of the system. During the so-called counting phase T1, the square-wave pulses of the input signal frequency pass through NAND gates U1B, U1A and come to the input of a 14-bit binary counter U2. Counting continues until the flip-flop U3 changes the state of its outputs, forcing gate U1C to block feeding the pulses to the counter.
     When analyzing the obtained 14-bit output code of the counter U2, it's not difficult to note that its most significant "1" appears to be at the same output for all frequencies within each octave range of the fundamental, therefore each range is strongly associated with one of the counter outputs, starting from the fifth output for a 20-40Hz range and finishing by the fourteenth output for a 10-20kHz range. The above "1" is followed by four less significant bits of the output code, these bits give more exact value of the measured fundamental frequency and therefore are indispensable for determining the state of relays K1- K4 which in the same binary manner, by the resistors' commutation, set more exact value of the rejection frequency. Only in this case absolute matching of the fundamental and rejection frequencies can be achieved.
     In the second, tuning phase T2 of the system operation, the low going not-Q output of flip-flop U3 activates a 14-to-1 multiplexer U13 which starts to sequentially connect the counter U2 outputs to the input of inverter U8A. The time of each connection is T0 and the multiplexer is controlled by the output code of clock counter U12. This code is also applied to the A0 - A3 inputs of a 16-bit decoder U15, making its outputs sequentially high, thus sending "1" in turn to the parallel inputs of a 10-bit register U16, starting from its first P0 input. Functioning of the multiplexer and the decoder is clock synchronized, each of them can be constituted of two components having lesser, standard bit capacity. The ten outputs of register U16 perform selection and indication of the octave frequency range of the distortion meter, only one of them is to be at high level.
     The first "1" appearing at the multiplexer output toggles the flip-flop built on NAND gates U8C, U8D, it is also delivered through inverters U8A, U8B to the serial data input of a 4-bit register U9 to start its serial loading. A monostable U11 is triggered by the rising edge of the U8D output and produces an output pulse which accomplishes parallel loading of register U16 with a delay, giving the clock synchronized data at the register inputs some time to settle. A single "1" is loaded and saved, the corresponding register output confidently indicates the octave range of the input signal frequency.
     During the next four steps of operation, the multiplexer transfers the bits carrying information about the state of relays K1- K4. The data are serially loaded to register U9 and appear at its four parallel outputs right after the first loaded "1". Loading is carried out on the high-to-low transition of the clock pulses which are inverted and gated by a NOR element U4B and then are sent to register U9. Only five pulses are permitted to be fed to the clock input of the register to shift the data arriving to its serial input, otherwise its outputs couldn't correctly represent the state of relays K1- K4.
     This requirement is fulfilled by using a 4-bit binary counter U6 which counts these pulses until its output code "101" changes the output of NAND gate U10D from high to low. This signal through inverter U10C inhibits further passing the clock pulses to the counter and register U9. The register outputs are coupled with the preset inputs of a binary counter U14, the information from these inputs being transferred to the counter corresponding outputs by a preset enable signal derived from the output of a NAND gate U7D. The gate emits a positive going pulse at the moment when the last output of the decoder U12 is activated. The output of U7D also resets the counter U2, preparing it for the next counting phase of operation.
     There is an analogue part of circuitry, comparators A6, A7, which monitors the output voltage of integrator A5 (Fig.6). In the tuned state of the rejection filter, this voltage lies within the set (0 - VC) range corresponding to a nominal range of the photocell resistance RV2 variation. The outputs of both comparators have a negative voltage level, that ensures the output of NOR gate U4D being high and the output of NAND gate U7C being low, the latter signal setting the optimal time constants of integrators A4, A5. If the rejection filter detuning occurs, one of the comparators toggles its output to positive voltage level and it leads to "1" at the output of gate U7C. This is a command to reduce the time constants in order to speed up the process of fine tuning in both channels of the optocouplers' control. When the output voltage of integrator A5 returns into the set interval, the negative transition of the comparator output through the gate U4D triggers a monostable U5. The monostable generates a low going output pulse keeping the state of gate U7C unchanged during additional 1,5sec. After that, with the restored optimal time constants of integrators A4, A5, the rejection filter reaches the ultimate both in notch depth and notch stability.
     The VK-1 oscillator and VK-2 distortion meter were designed and built more than ten years ago and since then these real instruments have been employed in numerous real measurements both when self-checking each other and when designing and testing various audio equipment. I'm very scrupulous about my electronics designs and consider them uncompleted and not suitable for public disclosure until their circuitry operation is successfully simulated by such a trustworthy program as Multisim 10. The last two years I've carried out strict and comprehensive virtual testing of the whole VK-2 distortion meter and its separate parts, all that has convinced me in remarkable properties of the Multisim 10 software and confirmed the validity of the distortion meter design and impeccability of its operation.
     Recently, I've developed the unprecedented in its reliability and accuracy method of measuring distortion with the help of my virtual VK-2 distortion meter which can successfully exist in parallel with its real counterpart. This instrument performs the fully transparent interactive distortion measurements of fantastic sensitivity - below -170dB (0.0000003%) within 20Hz-20kHz in less than 3sec.
     On the virtual oscilloscope screen you can see the extracted "live" distortion harmonics of an amplifier or oscillator whose circuit is entered to the simulation program along with the VK-2 distortion meter circuit. The exact RMS sum of these harmonics is measured confidently because it is free of any swamping noise and any added distortion being unavoidable in real distortion or spectrum analysis. The Multisim oscillator and the VK-2 virtual meter don't create distortion and noise by definition, this trick allows to investigate the linearity just of the device under test with the help of the classic, most right method - by removing the fundamental frequency from the analyzed signal. In principle, the use of automatic notch filter can provide infinite suppression of the fundamental, all depends on its performance, in the VK-2 device this performance is brought to the ultimate.
     The depicted in Fig.10 active rejection filter block consists of an input twin-T notch network with 10 selectable octave frequency ranges for the fundamental to be suppressed, a high-performance discrete amplifier (K=100), a 100kHz low-pass filter and at last the system of fine automatic tuning of the rejection filter (see Fig.13), its Q-factor is chosen Q=2, the achieved -175dB suppression of the fundamental frequency within 20Hz-20kHz is carried out in 3sec. The following then output amplifier (K=100) brings the total gain of the residuals to +80dB for measuring them by an ordinary RMS voltmeter, observing on the virtual oscilloscope screen and using them in the filter tuning.
     The accuracy of measuring the obtained distortion harmonics can be easily verified by applying their calibrated amounts, say -120dB, to the meter's input and analyzing its output, this accuracy being better than 0.5dB at all audio frequencies. An example of such verification is represented in Fig.10 where the sum of two signals, 1kHz and 2kHz, is applied to the rejection filter input, their values at this input being correspondingly 1V(0dB) and 1μV(-120dB), the latter is supposed to be the second distortion harmonic of the input 1kHz signal. The tuned to 1kHz filter notch network provides a huge -174dB signal attenuation at this frequency, while the 2kHz signal is passing through it practically intact (less than -0.5dB attenuation) and, being then amplified by 10000 times, appears at the VK-2 meter main output as the only its reading (9.65mV).

Fig.10. Virtual VK-2 distortion meter – the second harmonic verification.

     Applying the normalized 1V signal to the VK-2 meter input allows to interpret its output voltage reading directly in distortion percents - 1V means 0.01%(-80dB relative to 1V input), 100mV means 0.001%(-100dB), 10mV means 0.0001%(-120dB) and so on.
     And at last you can comfortably watch the whole process of the notch filter 3sec tuning in detail because the simulation is running in a slow time scale and usually takes 5-20min. Every nuance of extracting distortion harmonics from the set 1V RMS input signal is seen on the oscilloscope screen and saved in a file. The measurement probes placed at some points of the circuit monitor the simulation process at these points and give detailed time-varying information about its main parameters (voltage, current, frequency).
     The similar simulation procedure, but with the single 1V-1kHz input signal, results in its -174dB suppression provided by the rejection filter (see Fig.11), the test with the 1V-16kHz sine-wave input yields almost the same (-172dB) result (see Fig.12).

Fig.11. Virtual VK-2 distortion meter – the fundamental 1kHz harmonic verification.

Fig.12. Virtual VK-2 distortion meter – the fundamental 16kHz harmonic verification.

     The above two experiments show at the VK-2 meter’s main output the presence of only tiny, less than -170dB, pure sinusoids left from the input 1kHz and 16kHz sine-wave 1V test signals after passing them through the tuned rejection filter. This means that the VK-2 distortion meter doesn’t produce its own distortion and evaluates only the non-linearity of the device under test connected to its input.
     Of course, in reality the ubiquitous noise and other hindering factors reduce this notch potential of the VK-2 distortion meter, the level of suppression is also largely depends on the input parameters of op amps A4, A5 in Fig.6. When testing the real VK-2 distortion meter many years ago, the spectrum analysis data (see Fig.7) were obtained with the OP07 devices used as A4, A5 and their mediocre characteristics limited the fundamental frequency suppression at the level of -(130-140)dB.

Fig.13. System of fine automatic tuning of the rejection filter (1kHz setting).

     In conclusion, I would like to represent an example of the VK-2 distortion meter concrete application – checking the linearity of my discrete differential amplifier at 16kHz, the behavior at high audio frequencies is the most difficult examination for any amplifier and oscillator. The test scheme of Fig.14 contains the amplifier in its unity-gain inverting configuration, it is fed from the Multisim 10 non-distorting generator and its 2V output drives a 380ohm load. This voltage is then normalized at 1V level and applied to the input of the rejection filter which is set for operation in a 16kHz range.

Fig.14. Distortion measurement of the discrete differential amplifier at 16kHz.

     The whole process of measuring distortion takes 1.74sec, its result appears at the VK-2 meter output, and the measured distortion is shown on the screen and registered by an AC millivoltmeter, its RMS value being in this case 220uV/10000=22nV or -153dB relative to the 1V input. This distortion is mainly the second harmonic, it originates exclusively from the tested discrete amplifier because there aren’t simply other its sources (the input generator and rejection filter don’t create distortion by definition). The amplifier features also an open-loop gain of 100dB within 10Hz-100kHz and 0dB at 35MHz, its detailed description is placed in the application notes section of this site. By the way, it is just used in this distortion meter and in my VK-1 audio oscillator. The virtual tandem of this upgraded oscillator and the VK-2 meter easily exhibits -140dB distortion even at frequencies up to 20kHz.
     The computer simulation requires much more time than the real 1.74sec measurement process, with the average PC it usually continues 40-50min at 16kHz and 5-10min at 1kHz, all takes place as in the slowed video. This effect allows to study the measurement process in every detail.
     After stopping the simulation, all obtained measurement data and screenshots can be saved in a file. Screenshots of the extracted output distortion superimposed on the sine-wave input signal are very helpful in understanding the cause of this distortion. For example, narrow spikes on the distortion curve at the moments of the input sinusoid zero-crossing tell about the underbiased output transistors of a power amplifier, the third harmonic of distortion tells about clipping issues and so on.
     In general, the virtual VK-2 distortion meter is a powerful tool in designing and investigation of super-linear equipment, other methods of such demonstrative and accurate measurements of distortion below -130dB and direct exposing of its harmonics on the screen don’t still exist.
     The VK-2 meter full and complete circuit diagram is employed in simulation too, this time it measures distortion of the most linear 100W power amplifier ever created in the world. This VK-5P amplifier delivers a 100W power to a 4ohm load with 0.0000008% (-162dB) distortion at 1kHz and 0.0000085% (-141dB) at 16kHz. The corresponding schematics is depicted in Fig.15,16.

Fig.15. Distortion measurement of the 100W VK-5P power amplifier at 1kHz.

Fig.16. Distortion measurement of the 100W VK-5P power amplifier at 16kHz.

     The next Fig.17 represents the digital part of the meter. Its task is rough tuning of the twin-T notch filter by determining the octave frequency range to be chosen with the help of the filter capacitors and switching the filter resistors to make the suppression frequency maximally closer to the fundamental frequency of the input signal. Fine tuning of the notch filter will be then performed by the main, analogue part of the VK-2 distortion meter.

Fig.17. The digital part of the VK-2 distortion meter.

     Returning to the real built VK-2 distortion meter, it should be noted that it is a very compact and convenient device which hasn't been combined with the VK-1 oscillator within one case, although such temptation was at the early stage of development.

Fig.18. Front view of the VK-1 oscillator + VK-2 distortion meter set.

     The instruments can be used separately and as a set (Fig.18), dimensions of each instrument are 200×180×65mm, so both of them may be easily placed inside a briefcase and transported just to where you need to perform the test. The work with the VK-2 meter is childishly simple - by turning a normalization knob to find the state when both red LED indicators (less and more) don't light and then to press the octave frequency range button marked with the lighting green LED. In some seconds the measurement result appears at the output jack.
     The meter can be easily transformed to a fully automatic device, the only needed for that additional components are 20 high-quality relays with twin contacts and one motorized wire-wound potentiometer. However, this modernization would considerably raise the price and, what is most undesirable, would increase the device's physical size, breaking the VK-1,2 instruments' harmony.
     By publishing this material, I don't offer a DIY (do it yourself) project. I simply would like to share all information that concerns this type of distortion meter - its theoretical background, practical realization on the example of the VK-2 instrument and at last minimizing distortion and noise. There is no still commercial version of the VK-2 distortion meter, but I hope it might appear in the near future.
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