Analog Computers

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Anwendungsbeispiele für Analogrechner – Beispiel 6: Beschaltung von Parabelmultiplizier-Netzwerken

Complete English translation of the original German-language document (33 pages).

Application Examples for Analog Computers

Example 6 — 22 April 1966

CIRCUIT CONFIGURATIONS FOR PARABOLIC-MULTIPLIER NETWORKS

The following provides a comprehensive presentation of the circuit configurations of the parabolic-multiplier networks for multiplication, division, square-root extraction, and further special computing operations on the analog computers RAT 700, RAT 740, RA 741, RA 800, and RA 800 HYBRID.

Note: As a fundamental rule for all circuits using parabolic-multiplier networks, the input quantities Y_e of the network must in each case be applied directly as the output quantity of an amplifier, without the interposition of potentiometers or diodes.

Furthermore, for reasons of computing accuracy, it is always necessary to ensure a good drive level (Y_e approaching ±1) of the networks.

TABLE OF CONTENTS

  1. Multiplication — 3

    • 1.1 RAT 700 — 3
    • 1.2 RAT 700/2 — 5
    • 1.3 RAT 740 and RA 741 — 6
    • 1.4 RA 800 and RA 800 HYBRID — 9

  2. Division — 12

    • 2.1 RAT 700 and RAT 700/2 — 13
    • 2.2 RAT 740 and RA 741 — 14
    • 2.3 RA 800 and RA 800 HYBRID — 16

  3. Square-Root Extraction — 17

    • 3.1 RAT 700 and RAT 700/2 — 17
    • 3.2 RAT 740 and RA 741 — 18
    • 3.3 RA 800 and RA 800 HYBRID — 19

  4. Square Root of a Product — 20

    • 4.1 RAT 700, RAT 740 and RA 741 — 21
    • 4.2 RAT 700/2 — 22
    • 4.3 RA 800 and RA 800 HYBRID — 23

  5. Square Root of the Sum of Two Products — 23

    • 5.1 RAT 700, RAT 740 and RA 741 — 24
    • 5.2 RAT 700/2 — 25
    • 5.3 RA 800 and RA 800 HYBRID — 26

  6. Multiplication with Simultaneous Addition — 26

    • 6.1 RAT 700 — 27
    • 6.2 RAT 700/2 — 30
    • 6.3 RAT 740 and RA 741 — 30
    • 6.4 RA 800 and RA 800 HYBRID — 32

  7. Special Circuit Configurations for the Parabolic Multipliers — 32

CORRECTIONS to Application Example 6

Page 1, cell 6 must read: Input quantities Y_e

Page 7, Figure 5: The application example for the parabolic-multiplier network should read “links” [left] instead of “rechts” [right].

Page 11, Figure 9: Type “RA 2A, M2 A” to be equated with type “RA 56, 62 56” (Figure 16, bottom) — to be read: “to be substituted.”

Page 17, Equation (7) must read:

Y_a = +√x

Page 17: The arrow in Figure A.1 must point in the direction that the output of the central amplifier, which is located between output and input (summing point), does not change the sign of the function between G1 (8) (Figure 24) and G1 (9) (Figure 21).

Page 26, cell 3 must read: Instead of computing with the “Datrek Puker” RAT 700, RA 740, RA 800, RA 800 HYBRID, a series multiplier was used, as described. In it, the given parabolic-multiplier networks can be programmed in the same way as described in the present compilation for the same computing operations. The output point G is not a summing junction from which these alternatively-summed output points 5 can be formed by means of the amplifiers.

1. Multiplication

The multiplication of two variable quantities Y_e1 and Y_e2 is accomplished by means of a parabolic-multiplier network and one downstream (inverting) amplifier. Two interconnections are possible:

Y_a = +Y_e1 · Y_e2     (1)

Y_a = −Y_e1 · Y_e2     (2)

The output quantity Y_a appears at the output of the network with a negative sign inverted, that is, the sign of both input quantities determines the sign according to the polarity rule on the programming field. Without a downstream computing amplifier, the product is available as the summing-point quantity of the network’s downstream amplifier; in some circumstances, as negative of both input quantities. If one of the input quantities is negative, the sign inversion must be taken into account. If both input quantities are negative, the output quantity Y_a is formed as given in Eq. (1) (GI. (1)).

1.1 Multiplication on the RAT 700

On the plug-in board of type RAT 700, (PM 1) presents feedback resistors (e.g., R_o = 30, 20 kΩ) as non-switchable components; these are built into the network as plug-in elements. It is not possible to use these as an amplifier’s feedback, which is why for multiplication one amplifier is needed to produce the product. The downstream inverting amplifier inverts the product to produce (Y_a = +Y_e1 · Y_e2). The programming connection for multiplication according to GI. (1) is shown in Figure 1. The corresponding connections on the programming field of the RAT 700 are shown in Figure 2.

[page 4: figure only — Figure 1: Formation of the function according to GI. (1). The rectangle denotes the multiplier network without downstream amplifier. Figure 2: Connection on the programming field of the RAT 700 for forming the function according to GI. (1).]

1.2 Multiplication on the RAT 700/2

On the plug-in board of type RAT 700/2, there are two additional card slots for the reception of up to 4 multipliers, of which 1.1, referenced above, is one. It is possible to use the network’s output point as the output of the amplifier. The output point G is connected at the same node as the product Gl. (1) already specified above. A shorthand notation combines the network and the downstream amplifier into a single symbol according to Figure 3.

[page 5: figure only — Figure 3: Formation of the function according to GI. (1); simplified circuit diagram. Figure 4: Formation of the function according to GI. (2); simplified circuit diagram.]

The circuit configuration for forming the function according to GI. (2) is shown in Figure 4.

1.3 Multiplication on the RAT 740 and RA 741

Here there are two possibilities for programming a multiplier:

a) Programming by means of individual (discrete) parabolic-multiplier networks and the associated network cards.

b) Programming by means of parabolic-multiplier networks on function-amplifier cards and (normal) summing-amplifier cards.

Case a)

The individually-provided parabolic-multiplier networks are connected to one amplifier each. The network output is connected to the summing junction with the connections described in Section 1.1 above. The programming connections for Eq. (1) are shown in Figure 5. The corresponding connections on the programming field are shown in Figure 6.

[page 7: figure only — Figure 5: Formation of the function according to GI. (1). Figure 6: Connections on the programming field for forming the function according to GI. (1).]

In the circuit diagram, it is expedient to combine the two symbols for network and amplifier into a single symbol (Figure 3), to give the simplified circuit diagram shown in Figure 3. The circuit diagram for the function according to GI. (2) differs from that in Figure 3 only in the sign reversal of one input quantity.

To form the function according to GI. (1), the parabolic-multiplier networks are set according to the specifications in the function tables; then as shown in Figure 7, the two channels G_1 and G_2 are connected in parallel:

[page 8: figure only — Figure 7: Formation of the function according to GI. (1).]

The complete connections for a complete multiplier are shown in Figure 8; the representation in the circuit diagram is made as in Figure 3.

[page 8: figure only — Figure 8: Connections on the programming field for forming the function according to GI. (1).]

In the circuit diagram the two symbols for network and amplifier are conveniently combined into a single symbol (Figure 3) (simplified circuit diagram). The circuit diagram for the function according to GI. (2) differs from that in Figure 3 only in the sign reversal of one input quantity.

If it is necessary for programming-technical reasons to use a further network as an additional network next to one already used, the connections to the additional Rack Element are given to the operator (each according to the two-digit address of the element, from left to right).

Figure 2b shows the connection on the programming field of the RAT 700/2 for GI. (1).

[page 7: figure only — Figure 2b: Connection on the programming field of the RAT 700/2 for forming the function according to GI. (1).]

Case b)

The individual parabolic-multiplier-network cards are connected to one amplifier from the above; its output is connected at the summing junction at the same point as the GI. (1) output described in Section 1.1. It is possible to omit the downstream amplifier, because the feedback network (parabolic-multiplier network) is normalized to the same resistance level as the input resistance of the 10-fold input. This yields, for the positive divisor case, a circuit as in Figure 7.

Should it be necessary for programming-technical reasons to use the network as a separate network, and one wishes to connect it without the series resistor as in Figure 8, the assignment of the individual Rack elements is the same for the operator (from left to right, each pair of digits). The circuit configuration for the function according to GI. (2) is derived from this in an obvious way (see Fig. 4).

The complete connections for a complete multiplier are shown in Figure 9; the representation in the circuit diagram corresponds to that in Figure 3.

[page 9: figure only — Figure 9: Connections on the programming field for the formation of the function according to GI. (1).]

Should it be necessary for programming-technical reasons to use a further network for addition without an amplifier, the assignment of the individual Rack elements is the same for the operator (from left to right, each two-digit address). The circuit configuration for the function according to GI. (2) is derived in an obvious way (see Fig. 4).

1.4 Multiplication on the RA 800 and RA 800 HYBRID

If the parabolic-multiplier networks are plugged in in accordance with the instructions for the “Nichtlinear 800” insert, they can be programmed in the same way as the series multipliers. The circuit configurations for forming the two functions according to GI. (1) are shown in Figure 10a for the RA 800 and in Figure 10b for the RA 800 HYBRID.

*) The capacitor is omitted if amplifiers of type RA2N, RA2A, etc. are used.

[page 10: figure only — Figure 9: Formation of the second function according to GI. (2).]

The parabolic-multiplier-network input quantities in all programming fields are displayed repeatedly. In the items 1, the parabolic multiplier is identified as Canal A and B, the amplifier as Canal C and D, the summing amplifier as Canal E (vertical). It is equally valid to connect Canal X as a feedback path to create a feedback, so that previously-programmed polarity-inverting amplifiers are then connected via the Rack addresses +A⋅−A and +C⋅−C at the network output.

If no product is required, it becomes necessary either to connect the Canal A and B inputs in parallel or to connect Canal C with a polarity-reversal to B. A channel assignment is then straightforward because either the Canal A and B inputs are connected in parallel or Canal C with negative polarity (−) is brought to B.

If one wishes to use a product with negative polarity meanwhile, it is necessary to invert the polarity of the Canal B (−) inputs:

[page 11: figure only — Figure 10a: Connections on the programming field of the RA 800 for forming the second function according to GI. (1).]

[page 12: figure only — Figure 10b: Connections on the programming field of the RA 800 HYBRID for forming two functions according to GI. (1).]

*) The capacitor is omitted if the amplifier is of type RA 2a, 62 2A etc.

A simplified symbolic representation of the circuit description for forming two functions according to GI. (1), in which the two multipliers in each circuit need not be connected separately, is presented in Figure 11c. A simple assignment is possible in this way.

[page 12: figure only — Figure 11c: Formation of the second function according to GI. (1); simplified symbolic representation.]

2. Division

In multiplication, the computing amplifier receives as its input network the parabolic-multiplier network and as its feedback a resistor.

Division, as the inverse function of multiplication, is obtained by interchanging the input and feedback networks, i.e., the computing amplifier receives as its input network resistors and as its feedback the parabolic-multiplier network.

For division, three cases are to be distinguished:

a) Division with a positive divisor (GI. (3)) b) Division with a negative divisor (GI. (4)) c) Division with a divisor of changing sign (GI. (5))

The dividend may change its sign arbitrarily in each case.

Y_a = −(C₁/C₂) · (Z/N)     for 0 ≤ N ≤ +1     (3)

Y_a = +(C₁/C₂) · (Z/N)     for −1 ≤ N ≤ 0     (4)

Y_a = +(C₁/C₂) · (Z/N)     for −1 ≤ N ≤ +1     (5)

Since in Section 1 the programming of a multiplier for the various computer types has been treated in detail — both in symbolic representation and by specifying the necessary connections on the programming field — in what follows only the symbolic representation is given.

2.1 Division on the RAT 700 and RAT 700/2

Due to the resistance situation of the parabolic-multiplier network of the RAT 700, it is not practical to interchange input and feedback networks. For this reason, a downstream amplifier is required for both the input and the feedback, to create an open-loop amplifier. For stabilization of the computing circuit, a capacitor of at least 250 pF must be switched in parallel with the output and summing junction. This gives a circuit according to Figure 11 for GI. (3).

[page 14: figure only — Figure 11: Formation of the function according to GI. (3). Figure 12: Formation of the function according to GI. (4).]

The corresponding circuit for forming the function according to GI. (4) is shown in Figure 12.

A somewhat more complicated division circuit for a changing sign of the divisor (GI. (5)), using an additional absolute-value circuit and automatic switching, is shown in Figure 13:

[page 15: figure only — Figure 13: Formation of the function according to GI. (5).]

Constraints: ≥ 250 pF; Z is fed via capacitor C₁; N is applied to the absolute-value (|N| = 0) and sign-switching circuit (AR 32); Y_a = +(C₁/C₂)·(Z/N); −1 ≤ Z ≤ +1; −1 ≤ N ≤ +1.

2.2 Division on the RAT 740 and RA 741

As has already been seen in Section 1.2, the different resistance ratios of the parabolic-multiplier-network cards in conjunction with the resistance level of the computing resistors (R₁ = R_o = 200 kΩ, R₂ = 20 kΩ) allow a simplification of the programming. This means, for the case of division, that the amplifier behind the network can be omitted, since the feedback network (parabolic-multiplier network) is normalized to the same resistance level as the input resistance of a 10-fold input. For a positive divisor this gives a circuit according to Figure 14 (GI. (3)):

[page 16: figure only — Figure 14: Formation of the function according to GI. (3).]

Constraints: ≥ 250 pF; Z is applied with summing-capacitor S/C/G; Y_a = −(C/10)·(Z/N); −1 ≤ Z ≤ +1; 0 ≤ N ≤ +1; −Y_a.

The corresponding circuit for forming the function according to GI. (4) is shown in Figure 15:

[page 16: figure only — Figure 15: Formation of the function according to GI. (4).]

Constraints: ≥ 250 pF; Z applied via S/C/G; Y_a = +(C/10)·(Z/N); −1 ≤ Z ≤ +1; −1 ≤ N < 0; −Y_a.

The corresponding circuit for forming the function according to GI. (5) can be derived in a straightforward manner from Figures 13 and 14.

2.3 Division on the RA 800 and RA 800 HYBRID

Since the same resistance conditions as in the RAT 740 and RA 741 apply here, the programming of the division circuit proceeds analogously. The respective computing circuits are shown for GI. (3) in Figure 16 and for GI. (4) in Figure 17:

[page 17: figure only — Figure 16: Formation of the function according to GI. (3).]

Constraints: ≥ 250 pF*); Z input via S/C/G amplifier; Y_a = −(C/10)·(Z/N); −Y_a; −1 ≤ Z ≤ +1; 0 ≤ N ≤ +1. SM block with inputs +B, −B, +A, X−A; AR 32, N > 0.

*) The capacitor is omitted for amplifier type SRV 801.

[page 17: figure only — Figure 17: Formation of the function according to GI. (4).]

Constraints: ≥ 250 pF*); Z input via S/C/G; Y_a = +(C/10)·(Z/N); −Y_a; −1 ≤ Z ≤ +1; −1 ≤ N ≤ 0. SM block with inputs X+A, −A, +B, −B; AR 36, N < 0.

*) The capacitor is omitted for amplifier type SRV 801.

The division circuit for an arbitrary sign of the divisor is derived from Figures 16 and 13.

3. Square-Root Extraction

As a special case, the square-root extraction is treated as an inverse function of multiplication, in which a variable is multiplied by itself. The circuit configuration for square-root extraction is analogous to that for division, with the only difference that divisor and dividend are identical, i.e., both Divisor and Dividend must carry the same sign.

There are two cases to be distinguished:

a) Square root with positive radicand (GI. (6)) b) Square root with negative radicand (GI. (7))

Y_a = −√x     for 0 ≤ X ≤ +1     (6)

Y_a = +√x     for −1 ≤ X ≤ 0     (7)

Due to this, the open-loop amplifier cannot be omitted; a diode must be inserted between output and summing junction (to prevent sign reversal due to the function) and directed towards the amplifier. Since the computing circuit is prone to oscillation, it is necessary in this case also to connect a capacitor of at least 250 pF in parallel with the feedback.

For the case that the radicand changes sign during computation, the sign of Y_a must also change accordingly, which is why for the formation of this function it is necessary to use a circuit corresponding to that for division with a variable sign of the divisor, as in GI. (5).

3.1 Square-Root Extraction on the RAT 700 and RAT 700/2

This gives for positive radicand (GI. (6)) the circuit according to Figure 18, and for negative radicand (GI. (7)) Figure 19.

Page 19

Figure 18: Formation of the function according to Eq. (6)

The figure shows the circuit for x > 0, using amplifier AR 37, with a feedback capacitor ≥ 250 pF. The parabola-multiplier network receives inputs +X, +Y, −X, −Y. The output is:

Ya = −√(C₁/C₂) · √x

with the constraint −Ya, 0 ≤ x ≤ +1.

Figure 19: Formation of the function according to Eq. (7)

The figure shows the circuit for x < 0, using amplifier AR 38, with a feedback capacitor ≥ 250 pF. The parabola-multiplier network receives inputs +X, +Y, −X, −Y. The output is:

Ya = +√(C₁/C₂) · √(−x)

with the constraint −Ya, −1 ≤ x < 0.

3.2. Square Root with the RAT 740 and RA 741

Taking into account what was stated in sections 1.2. and 2.2., the simplified circuits for these two computer types are the circuits shown in Figure 20 for a positive radicand (Eq. (6)) and Figure 21 for a negative radicand (Eq. (7)).

Page 20

Figure 20: Formation of the function according to Eq. (7)

The figure shows the circuit for x > 0, using amplifier AR 39, with a feedback capacitor ≥ 250 pF. The amplifier inputs are labeled S, C, G (scale, coefficient, grid). The parabola-multiplier network receives inputs +X, +Y, −X, −Y and G. The output is:

Ya = −√(Cx)

with the constraint −Ya, 0 ≤ x ≤ +1.

Figure 21: Formation of the function according to Eq. (8)

The figure shows the circuit for x < 0, using amplifier AR 40, with a feedback capacitor ≥ 250 pF. The amplifier inputs are labeled S, C, G. The parabola-multiplier network receives inputs +X, +Y, −X, −Y and G. The output is:

Ya = +√(−C(x))

with the constraint −Ya, −1 ≤ x < 0.

3.3. Square Root with the RA 800 and RA 800 HYBRID

Taking into account the descriptions given in sections 1.3. and 2.3., the programming of the computing circuit for generating the square root according to Eqs. (6) and (7) follows from Figures 22 and 23.

Page 21

Figure 22: Formation of the function according to Eq. (6)

The figure shows the circuit for x > 0, using amplifier AR 41, with a feedback capacitor ≥ 250 pF. The servo-multiplier (SM) network receives inputs +A, −A, +B, −B. A footnote states: *) The capacitor is omitted for amplifier type SRV 801. The output is:

Ya = −√(Cx)

with the constraint −Ya, 0 ≤ x ≤ +1.

Figure 23: Formation of the function according to Eq. (7)

The figure shows the circuit for x < 0, using amplifier AR 42, with a feedback capacitor ≥ 250 pF*). The servo-multiplier (SM) network receives inputs +A, −A, +B, −B. The output is:

Ya = +√(−C(x))

with the constraint −Ya, −1 ≤ x < 0.

4. Square Root of a Product

The corresponding computing circuit results as a combination of the product formation (see section 1.) and the square-root circuit (see section 3.). The central computing amplifier serves both as an input amplifier and as a feedback network — a parabola-multiplier network.

Page 22

The realisable equations (8), (9) are:

Ya = +√(Ye1 · Ye2) = +√((−Ye1) · (−Ye2)) for 0 ≤ Ye1, Ye2 ≤ +1, or −1 ≤ Ye1, Ye2 ≤ 0 (8)

Ya = −√(Ye1 · (−Ye2)) = −√((−Ye1) · Ye2) for 0 ≤ Ye1 ≤ +1, −1 ≤ Ye2 ≤ 0 or −1 ≤ Ye1 ≤ 0, 0 ≤ Ye2 ≤ +1 (9)

4.1. Square Root of a Product with the RAT 700, RAT 740, and RA 741

Because the same network is present both in the feedback path and at the input of the central amplifier, circuits for forming the functions according to Eq. (8) (Figure 24) and Eq. (9) (Figure 25) result independently of the type of network. This means that the orange-coloured output jacks of the parabola-multiplier network on the RAT 700 remain unconnected.

Figure 24: Formation of the function according to Eq. (8)

The figure shows amplifier AR 43 with a feedback capacitor ≥ 250 pF. Two input amplifiers feed signals +Ye1 (−Ye1) and +Ye2 (−Ye2) respectively. The parabola-multiplier network has inputs +X, −X and G, with a second G output connected to an inverting amplifier. The output is:

Ya = +√(Ye1 · Ye2) (or +√((−Ye1)(−Ye2)))

with the secondary output −Ya.

Page 23

Figure 25: Formation of the function according to Eq. (9)

The figure shows the circuit arrangement for Eq. (9), with the same overall topology as Figure 24. Inputs +Ye1, −Ye1, +Ye2, −Ye2 are applied; the parabola-multiplier network and central amplifier produce:

Ya = −√(Ye1 · (−Ye2)) (or −√((−Ye1) · Ye2))

Both Ye1 and Ye2 may serve as input quantities to the amplifier (see section 1.). By interchanging the positive and negative input quantities at the multiplier network, a transition from the circuit of Figure 24 to the type of Figure 24 can be made to obtain Eq. (9). The same applies to Figure 25 in connection with Eq. (8).

4.2. Square Root of a Product with the RAT 700/2

Taking into account the descriptions in sections 1.2., 2.2., and 4.1., the circuit for forming the function according to Eq. (8) results as shown in Figure 26.

Figure 26: Formation of the function according to Eq. (8)

[Circuit diagram — figure only for lower portion of page]

Page 24

4.3. Square Root of a Product with the RA 800 and RA 800 HYBRID

Applying sections 4., 4.1., and 4.2. to the RA 800 bzw. RA 800 HYBRID, the circuit results according to Figure 27, in which the patch-panel of a servo-multiplier serves as the feedback network for both output jacks of a servo-multiplier network.

Figure 27: Formation of the function according to Eq. (8)

The figure shows the circuit with a feedback capacitor ≥ 250 pF*). Footnote: *) The capacitor is omitted for amplifier type SRV 801. The servo-multiplier network feeds back into the amplifier. The output is:

Ya = +√(Ye1 · Ye2)

with secondary output −Ya.

5. Square Root of the Sum of Two Products (Square Root of Two Squares)

It is generally possible, as with the ordinary square root, to make a summation before taking the root, by using a summing amplifier — this means adding with a second Eingangs-Parabelmultiplizier-Netzwerk (input parabola-multiplier network).

Thereby, the general circuit for the special case of the square root of the sum arises.

Page 25

a) the sum of two squares (Eq. (11)), as well as

b) the sum of the products of two quantities (Eq. (12)).

The special cases yield, each by means of a suitable sign arrangement for the input quantities made available, the following circuits that remain the same from a circuit-technology standpoint:

Ya = +√(Ye1² + Ye2² + Ye3² + Ye4²) (10)

Ya = +√(Ye1 · Ye2 + Ye3 · Ye4) (11)

or

Ya = +√(Ye1² + Ye2²) (12) (with only two squares)

Regardless of the sign with which the input quantities are applied, the circuit can be programmed for a positive output quantity, because through the inherent sign-reversal in the feedback, the output quantity appears at the multiplier input with the opposite sign. As previously pointed out, the circuit as a whole operates without any sign reversal when the output sign is positive directly from the circuit without requiring further inversion. A summing amplifier with the correct sign can reprogram the circuit so that the summed quantity appears directly, without an inverting amplifier, at the multiplier output.

5.1. Square Root of the Sum of Two Products with the RAT 700, RAT 740, and RA 741

For the same reasons as in section 4.1., for the three desktop-computer types the same circuit results, namely Figure 28, according to Eq. (10).

Page 26

Figure 28: Formation of the function according to Eq. (10)

The figure shows amplifier AR 47 with a feedback capacitor ≥ 250 pF. Four input signals +Ye1, +Ye2, +Ye3, +Ye4 are applied through two parabola-multiplier networks (each with inputs +X, +Y, −X, −Y and G). The output is:

Ya = +√(Ye1 · Ye2 + Ye3 · Ye4)

with secondary output −Ya.

5.2. Square Root of the Sum of Two Products with the RAT 700/2

On desktop computers (upgraded RAT 700 and RAT 700/2) whose networks are connected directly downstream of the amplifier, the circuit according to Eq. (10) is represented by Figure 29.

Figure 29: Formation of the function according to Eq. (10)

The figure shows amplifier AR 48 with a feedback capacitor ≥ 250 pF. Four input signals +Ye1, +Ye2, +Ye3, +Ye4 are applied through two parabola-multiplier networks. Each network has inputs +X, +Y, −X, −Y; the amplifier also has inputs G and 1. The output is:

Ya = +√(Ye1 · Ye2)

with secondary output −Ya.

Page 27

5.3. Square Root of the Sum of Two Products with the RA 800 and RA 800 HYBRID

For Eq. (10), the same circuit as in section 5.1. results here. It is realised with the networks on the panels of two servo-multipliers, as shown in Figure 30.

Figure 30: Formation of the function according to Eq. (10)

The figure shows amplifier AR 43 with a feedback capacitor ≥ 250 pF*). Footnote: *) The capacitor is omitted for amplifier type SRV 801. Two servo-multiplier (SM) networks are used; the first receives inputs +A X, −A, +B, −B and the second also +A X, −A, +B, −B. The second SM provides outputs SM C, +D, −D. The output is:

Ya = +√(Ye1 · Ye2 + Ye3 · Ye4)

with secondary output −Ya.

6. Multiplication with Simultaneous Addition

If in a computing circuit an operation of the form

Ya = Ye1 · Ye2 + α 0 ≤ Ya ≤ 1 (13)

                      0 ≤ α ≤ 1

is to be carried out, there are two possibilities for its implementation:

Page 28

  1. Formation of the product with the aid of a multiplier and subsequent addition of the product and α in a downstream summing amplifier. Requirements: 1 multiplier network and 2 amplifiers.

  2. Simultaneous addition of the product and α with the aid of the multiplier network itself. Requirements: 1 multiplier network and 1 amplifier.

The second method is generally known and therefore will not be explained further here. It is determined by the particular technical circumstances. Two cases must be distinguished:

a) α = const b) α = α(t)

Particularly in case b), and also in case a) when the product and α are positive, there are, from the standpoint of the computing-accuracy specification, reasons why the multiplier network may not be used as a summing network; this results in a preference for method 1).

6.1. Multiplication with Simultaneous Addition with the RAT 700

For the special feedback resistance R_x = 38.25 kΩ of the multiplier network (see section 3), a constant c_x is determined for the 1-unit inputs (R_0 = 500 kΩ, R_g = 50 kΩ). It is noted that the additional R_g in Eq. (13) gives the required value of c_x in the range of 1. For a 1-unit input with R_0 = 50 kΩ, the additional input…

Page 29

…is to be selected.

If R_x were exactly 38.25 kΩ, then c_x would be calculated as 0.765. However, for technical reasons R_x is subject to certain fluctuations and also has a temperature coefficient (TK = −8·10⁻⁴/°C ≈ 800 ppm). It is therefore recommended to apply method 2 as follows:

One programs a multiplier without input quantities and connects a 10-unit input of the amplifier with a positive or negative machine unit. The numerical value to be read off at the amplifier output corresponds to the sought amplification factor c_x. Values generally lie between 0.7 and 0.76. Thus equation (13) for the RAT 700 is written:

Ya = Ye1 · Ye2 + α/c_x (13a)

It should be noted that c_x via C_x = R_x/R_2 carries a TK = −8·10⁻⁴/°C, meaning that at changed room temperature c_x must be redetermined in order to guarantee maximum accuracy.

The necessary connections on the programming panel are shown in Figure 31, where it should be noted that without special measures (amplifying α(t) by 1/c_x), at the amplifier output only Ya = +Ye1 · Ye2 + c_x · α(t) appears. The corresponding computing circuit is shown in Figure 32.

Page 30

Figure 31: Connections on the programming panel of the RAT 700 for forming the function according to Eq. (13)

The figure shows the RAT 700 patch panel (amplifier AR 50). Input signals +Ye1 and +Ye2 are connected through the multiplier inputs (+Y, −X, −Y). An additional input of ±1 (or −1) feeds through coefficient α/c_x into the G (grid) input at position 10, and also at position 10 on the other row. Output Ya = +Ye1 · Ye2 ∓ α.

Figure 32: Formation of the function according to Eq. (13)

The figure shows the computing circuit (amplifier AR 51). Inputs +Ye1 and +Ye2 are applied. The multiplier network receives +X, +Y, −Y, and G. An additional input ±1 (or −1) feeds coefficient α/c_x. The output is:

Ya = +Ye1 · Ye2 ∓ α

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6.2. Multiplication with Simultaneous Addition with the RAT 700/2

Since the network itself does not have a 10-unit input available, this is obtained by plugging a 50 kΩ resistor plug into the freely accessible grid terminal G. The programming otherwise proceeds as in section 6.1. The corresponding computing circuit is shown in Figure 33.

Figure 33: Formation of the function according to Eq. (13)

The figure shows the computing circuit (amplifier AR 52). Inputs +Ye1 and +Ye2 are applied through the multiplier network (+X, +Y, −X, −Y, G). The 50 kΩ resistor provides the 10-unit input for the coefficient α/c_x with a ±1 (or −1) source. The output is:

Ya = +Ye1 · Ye2 ∓ α

6.3. Multiplication with Simultaneous Addition with the RAT 740 and RA 741

Here the two cases a) and b) mentioned in section 1.3. must again be distinguished.

In case a), α can be added via the 1-unit input of the downstream amplifier (see Figure 6).

In case b), a 10-unit input of the downstream amplifier acts as a 1-unit input (see Figure 8).

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The corresponding representations in the computing circuit are shown in Figures 34 and 35.

Figure 34: Computing circuit for case a)

The figure shows the computing circuit (amplifier AR 53). Inputs +Ye1 and +Ye2 are applied through the multiplier network (+X, +Y, −X, −Y, G) to amplifier S. A coefficient α with ±1 (or −1) is added. The output is:

Ya = +Ye1 · Ye2 ∓ α

Figure 35: Computing circuit for case b)

The figure shows the computing circuit (amplifier AR 54). Inputs +Ye1 and +Ye2 are applied through the multiplier network G_XY (+X, +Y, −X, −Y) to an amplifier with inputs 10, S, G, 10 and a feedback capacitor ≥ 250 pF. A coefficient α with ±1 (or −1) is added. The output is:

Ya = +Ye1 · Ye2 ∓ α

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6.4. Multiplication with Simultaneous Addition with the RA 800 and RA 800 HYBRID

From a programming standpoint, the same applies here as in section 6.3., case b) — there is also a 10-unit input available. Figures 35 and 36 show this on the programming panel and in the computing circuit.

Figure 36: Patching of a parabola multiplier according to Eqs. (14) and (15)

The figure shows the RA 800 patch panel with servo-multiplier network. The inputs are arranged for simultaneous multiplication and addition. The output is:

Ya = +(Ye1 · Ye2) ∓ α (or similar form)

7. Special Characteristics of the Parabola Multiplier

If one operates a parabola multiplier according to Figure 36, so there result the following computing operations:

Ya = +(x + y)²/4 = +(x² + 2xy + y²)/4 (14)

Ya = −(x − y)²/4 = −(x² − 2xy + y²)/4 (15)

A transfer to other computer types is possible with the aid of the corresponding figures for those types. Some special examples of parabola-multiplier networks in application examples will be found under section 2, example 2. For more details on the connections of the programming panel for the RAT 700, those connections are given; for other computer types, the programming can be read from the figures above without difficulty.