Digital
Analogical converter with Operational Amplifier |
Precision
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Created it, 06/09/09
Update it, 06/09/29
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2. 2. - OPERATIONAL AMPLIFIERS
The graphic symbol of the amplifier is given figure 10.
It is noticed that the food is not represented. However, on the circuits, it is obligatorily present. Generally, one uses a symmetrical food characterized by the presence of three terminals: for the positive tension, for the negative tension, for the mass.
The ideal operational amplifier shows the following characteristics:
Amplifier infinite :
amplification in tension which is the relationship between the output voltage VS
and the tension of Ve entry (figure 11) can
be considered practically infinite : 
Impedance of infinite entry : with an
impedance of infinite entry, the amplifier does not represent a load for the
preceding circuit ; in other words, it does not absorb any current.
Null output impedance : with a null ideal
output impedance, the operational amplifier can provide all the current required
by the load, without influence on the output voltage.
Zero output voltage for a tension
of null entry.
In this case, one says that “the Offset” is null. Offset is an English term, used to indicate the shift of the point of rest compared to the zero.
With the operational amplifier, it is enough to an extremely weak tension of entry to carry the output voltage to a very high value, very near to the supply voltage.
In the majority of the cases, it is necessary to decrease the profit of the whole by adding some components. As example, we will consider the assembly indicated figure 12.
In this circuit, one added two resistances R1 and R2. R2 ensures a negative feedback between the exit and the inverseuse entry of the operational amplifier.
In the ideal amplifier, the impedance of entry is infinite, therefore the current of entry is null.
The current of entry Ie also corresponds to the sum of the currents crossing R1 and R2.
I1 + I2 = Ie = 0
One deduces from it that :
I2 = - I1
The currents I2 and I1 can be also expressed in the following way :
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As the current of entry is null, one can consider that the Ve tension at the boundaries of the impedance of entry is also.
The preceding equation becomes :
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One thus deduces from it that amplification A of the assembly is equal to the report / ratio of resistances R2 and R1 and any more amplification of the operational amplifier does not depend. The latter was considered infinite. Actually, it is not it, but its value is so high (more than 105) that one can apply formula VS / VE = - R2 / R1 in all peace.
The sign “-” placed in front of the report/ratio of resistances R1 and R2, indicates that the output voltage VS is of opposite sign (or in opposition of phase) with VE.
Another use of the operational amplifier is illustrated figure 13.
The tension of entry VE is applied directly on the terminal + (entered noninverseuse). The network of reaction identical to the preceding circuit is consisted resistances R1 and R2.
In this type of configuration, the formula which expresses the output voltage east :
2. 3. - DIGITAL CONVERTER
ANALOGICAL A AMPLIFIER
OPERATIONAL
The principle of a digital/analogical converter with 3 bits using an operational amplifier is illustrated figure 14.
The contacts of the switches can be mechanical or electronic. When the bit is worth 0, the switch is opened, when it takes value 1, the switch is closed.
Let us see now what occurs with a binary number equal to 100. The first contact is closed, the two others are opened, as indicated figure 15.
By comparing this figure with figure 12, one notes that the two circuits are equivalent because two resistances R3 and R4 do not have any influence.
The output voltage VS is equal to :
It corresponds to half of the tension of entry. If one closes only the second switch (a number 010), one obtains a output voltage of :
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In this case, the output voltage corresponds to the quarter of the tension of entry.
So finally, one closes only the third switch (binary number 001), the output voltage becomes:
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That is to say a tension equalizes with the eighth of the tension of entry.
To obtain a tension VS equal to 1 / 16 of the tension of entry, it would be necessary to use a fourth switch and a resistance of 80 kW.
If several contacts are closed, the output voltage is obtained by adding the tensions corresponding to each taken switch separately.
Thus, for combination 101, one a :
VS = - (1 / 2 + 1 / 8) x VE = - 5 / 8 x 10V = - 6,25 Volts.
In practice, the circuit such as we have just described it, is not used. Indeed, if one wanted to work with 12 bits for example, the value of the last resistance would be equal to 20,480 MW.
It is rather difficult to carry out resistances of very great value with a good precision.
In addition, because of the great differences in values, the variations of resistances due to the temperature are not identical. The weight of each bit (1 / 2, 1 / 4, 1 / 8, etc…) is not exact any more and the precision of the system is bad.
2. 4. - RESISTANCE NETWORK (R - 2R)
The solution adopted to overcome the problems created by resistances of too different values is represented figure 16. It consists in using only two values of resistances : R and 2R.
In this circuit, the switches connect resistances 2R, either towards the reference voltage standard VR, or towards the mass, according to whether the bit corresponding is to 1 or 0.
The bit of strong weight (MSB) is located on the right resistance network R - 2R. When the switch corresponding to this bit is in position 1, the output voltage is equal to :
With traditional calculations on the tension divider bridges (which are not carried out here), one shows that the weight of each bit is 1 / 2, 1 / 4, 1 / 8, etc…
If bit 2 is for example to 1, the circuit becomes that indicated figure 17.
Resistance 2R, corresponding to bit MSB, does not have any influence, because it is connected between the mass and the entry of the operational amplifier which constitutes a virtual mass (potential very close to 0 V).
The resistance network located on the left of the feature into dotted can be summarized with that indicated figure 18-a.
According to the theorem of THEVENIN, the network located between point A and the mass can be replaced by a circuit made up of a generator in series with an equivalent resistance Req.
The generator has like tension the measured value with vacuum between point A
and the mass : here, VR / 2 is
obtained since point A is connected in the
middle of the chain of resistances connected at the boundaries of tension VR.
Equivalent resistance Req is equal to the
resistance seen between point A and the mass
when one replaces the generator of tension VR
by a short-circuit.
One obtains here two resistances of 2R in parallel, that is to say Req = R.
Finally, the assembly of figures 17 and 18-a are summarized with that of the figure 18-b.
Using this figure 18-b simplified, one notes that the output voltage is equal to :
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The weight of the bit N° 2 is thus 1 / 4 of VR.
An intermediate solution between the resistance network of figure 14 and that of figure 16 is represented figure 19.
In this circuit, one uses two groups of resistances of which each one is the double of the preceding one.
Between the two groups of resistances, one inserts a resistance of suitable value in order to cause is an attenuation of 1 / 16 if one carries out a digitize pure, that is to say an attenuation of 1 / 10 if one works in code BCD.
2. 5. - CONVERTERS “D / A” WITH INTEGRATED CIRCUITS
Converters D / A are currently available in the form of integrated circuits. The converters thus produced reach a precision of about 0,05 % to 0,0125 %.
On the market, one meets several types of converters D / A integrated, simplest is represented figure 20.
One recognizes the resistance network R - 2R and the ten switches which, of course, are produced with field-effect transistors.
Another more complex circuit is represented figure 21.
This converter uses a resistance network R - 2R whose each branch is fed by a generator of current (1 mA, 1 / 2 mA, 1 / 4 mA, 1 / 8 mA, etc…).
In both cases, the user must add the operational amplifier which is not built-in in the case. The choice of the amplifier will be according to the commutation rate necessary.
The exit of the resistance network can be connected to the entry “-” or the entry “+” of the amplifier and according to case's, one obtains at exit a negative or positive tension.
If one uses the converter represented figure 21, which delivers in fact a current I proportional to the binary number of entry, one can carry out two different connections (figure 22-a and 22-b).
Current I coming from the resistance network depends on the digital signal and the reference voltage standard VR.
Generally, one prefers the configuration of the figure 22-a, because it gets a highher degree of accuracy.
2. 6. - PRECISION OF THE CONVERTERS
The two principal characteristics of a converter D / A are : the resolution and precision.
As we saw previously, the resolution depends on the number of bits of entry which the circuit can treat. This number determines into how much levels the reference voltage standard VR can be divided.
Figure 23 gives the relation between the digital entry and the analogical exit of an ideal converter to 3 bits.
The resolution of this circuit corresponds to the increase in the analogical tension of exit, caused by the increase in a unit of the binary number of entry.
In this precise case, the resolution is 1 / 8 of VR. For a circuit with 4 bits, it would be 1 / 16 of VR.
With each bit configuration of entry, a output voltage corresponds. For example, the binary number 100 determines a output voltage equal to 0,5 VR. Actually, this value is slightly different, it can be of 0,49 VR or 0,51 VR. The difference between the ideal value and that really obtained (± 0,01 VR, i.e. 1 % per excess or defect) is called for degree of accuracy or simply greater detail.
One should not confuse resolution and precision. Indeed, one meets converters with weak resolution, for example with 3 bits, giving 8 levels, but with a very high degree of accuracy. Contrary, there are circuits with high resolution (10 to 12 bits of entry), but whose precision is poor.
The factors affecting the precision of the converters can be very diverse as we will see it now.
2. 6. 1. - OFFSET NOT NO ONE
When all the bits of entry are to 0, one should obtain 0 volt at exit. It is not always the case and one speaks then about error or shift of offset. This variation is constant and exists for all the binary values of entry, as shown in the figure 24.
2. 6. 2. - ERROR OF TRANSFER
This defect appears when the profit of the amplifier is excessive or too weak. One obtains then values of analogical tensions higher or lower than those envisaged. Figure 25 shows the shift between the actual values obtained and the ideal values. It is noted that the error is all the more large as the numerical value of entry is high.
2. 6. 3. - ERROR DUE TO NON-LINEARITY
An important cause of the inaccuracy is the bad linearity of the system. This one is due mainly to the resistance networks.
The precision of a converter depends on the absolute value of each resistance and of the relationship existing between the various resistances brought into service. It is very important that these reports/ratios are maintained in all the field of work.
Figure 26 shows the pace which could take the curve of transfer of a converter D / A to 3 bits of very bad quality.
It is quite obvious that such a converter is unusable.
It is inevitable that the value of resistances vary with the temperature. For this reason, one always uses resistance networks integrated in microcircuits; indeed, with this technology, resistances all are carried out with same material; moreover, they are very close and thus undergo the same variations in temperature.
In general terms, the precision reveals from how much a converter deviates from the theoretical behavior.
Generally, on the notes manufacturers, one meets two types of precision: absolute precision and relative precision.
The absolute precision is the difference between the analogical exit which one wishes when one applies a binary code in entry, and the exit really obtained.
To correct this variation, one can intervene on the profit of the operational amplifier or the reference voltage standard VR.
The relative precision is obtained by submitting the relationship between the variation and the theoretical value which one should obtain.
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