US 2697201 A Abstract available in Claims available in Description (OCR text may contain errors) Dec. 14, 1954 E. L. HARDER 2,697,201 ADJUSTABLE NONLINEAR RESISTANCE Filed sept. 27, 194s 2 Sheets-Sheet 1 l FIS-l. Pigna. I f Edwin L. Harder. M BY-fwm ATTO R N EY United States Patent O ADJUSTABLE NONLINEAR RESISTANCE Edwin L. Harder, Pittsburgh, Pa., assignor to Westinghouse Electric Corporation, East Pittsburgh, Pa., a corporation of Pennsylvania Application September 27, 1949, Serial No. 118,023 14 Claims. (Cl. 323-66) This invention relates to an electrical circuit for simulating an adjustable non-linear resistance and more particularly to a non-linear resistance circuit for an analog computer. In analog computors, it is frequently necessary to employ in an electrical analogy circuit constructed of circuit elements having electrical properties analogous to the properties of a physical system, a circuit element in the nature of a non-linear resistance to represent a nonlinear function of the physical system. Attempts to produce non-linear devices in the past have been chiefly of three types, namely, the type employing non-linear resistance materials, such as treated silicon carbide, the type employing grid-controlled circuits, and the type employing tapered potentiometers which are adjustable to effect non-linear resistances. As to combining the features of accuracy, adjustability, dependability and simplicity, applicant has invented an adjustable non-linear resistance device employing a plurality of parallel circuits, one or more of which is conductive depending upon the magnitude of the current flowing through the device, to produce any desired variation of voltage across the device. It is frequently necessary to have available such a readily adjustable non-linear resistance device for various applications of the analog computer with sufficient flexibility and accuracy for such analysis work. In some cases it is necessary to operate such a device at high speeds, such as up to several thousand cycles per second and to operate such device over a range of single valued functions which do not necessarily pass through the origin. It is accordingly an object of this invention to provide a circuit for effecting at its terminals a non-linear voltage-current characteristic. It is a further object of the invention to provide a circuit arrangement which can be made to produce any desired variation of voltage across the circuit arrangement as a function of the current therethrough. lt is a further object to provide a circuit for effecting a non-linear resistance curve comprising a plurality of parallel circuits, the resistance of any one of the parallel circuits being adjustable without affecting the conductivity of other parallel circuits. It is a further object of the invention to produce a circuit capable of effecting a resistance which increases or decreases with a change in the magnitude of the flow of current through the circuit. It is a further object of the invention to provide a method of simulating a desired non-linear resistance curve for an analog system. It is a further object to provide a method of simulating a desired non-linear resistance curve in an alternating current circuit. The foregoing objects are merely illustrative of the various aims and objects of the invention. Other objects and advantages will become apparent upon considering the following disclosure when considered in conjunction with the accompanying drawings, in which: Figure l is a diagrammatic showing of a non-linear resistance circuit capable of effecting a decrease in resistance with an increase in the flow of current through the circuit. Fig. la illustrates a typical voltage-current curve that may be effected by the circuit of Fig. l. Fig. 2 is a diagrammatic showing of a non-linear resistance circuit capable of effecting an increase in resistance with an increase in the flow of current through the 2,697,201 Patented Dec. 14, 1954 ice circuit. Fig. 2a illustrates a typical Voltage-current curve that may be effected by the circuit of Fig. 2. Fig. 3 is a diagrammatic showing of a non-linear resistance circuit employing a potentiometer instead of batteries to energize the parallel and other circuits in the non-linear resistance circuit and employing certain features of Figs. l and 2 collectively to effect an irregular voltage current curve. Fig. 3a illustrates an irregular voltage-current curve that may be effected by the circuit of Fig. 3. Fig. 4 illustrates a typical representation of a simplified physical or hydraulic system. Fig. 4a illustrates a pressure differential-flow curve representing the characteristics of a particular fluid they may be employed in the hydraulic system of Fig. 4. Fig. 5 is a diagrammatic showing of an electrical analog circuit employing non-linear resistors, as the electrical analog of the hydraulic problem illustrated in Fig. 4. Fig. 5a illustrates a voltage-current curve effected by the non-linear resistors in the diagram of Fig. 5 to closely approximate the curve of Fig. 4a. Fig. 6 is a diagrammatic showing of a versatile and adjustable non-linear resistance circuit which is adaptable to produce any one of a large variety of curves including the curves illustrated in Figs. la, 2a, 3a and 5a. Referring now to Fig. l, three branches of the nonlinear resistance device are connected between terminals 1 and 3. When current flows through the non-linear resistance device from terminal 1 to terminal 3, such flow is, for convenience in description, considered a positive direction of ow and the branch through which such current flows is considered the positive branch. When current flows from terminal 3 to 1, such flow is, for convenience in description, considered a negative direction of flow and the branch through which such current ow is considered the negative branch. The terms positive and negative are frequently employed to designate such distinction. The three branches are identified as a linear resistance branch 5, a positive non-linear branch 7, and a negative non-linear branch 9. The linear resistance branch 5 comprises, in series, a switch 11 which, for the purpose of the following description, is considered open, and an adjustable substantially linear resistance 13. The positive non-linear branch 7 comprises a crystal diode or rectifier 15 in series with a plurality of resistance circuits 17, 19, 21 and 23. The resistance circuit 17 comprises a substantially linear and adjustable resistor 25. The resistance circuit 19 comprises, in series, a unidirectional device or rectifier 27, a source of electrical energy, such as a battery 29, and an adjustable substantially linear resistor 31. The battery 29 has its polarity so arranged as to buck or oppose a positive flow of current from the terminal 1 through the parallel resistance circuit 19 to terminal 3, the positive terminal of the battery 29 being connected so that terminal 1 is, in effect, made positive with respect to terminal 3. However, the rectifier 15 has its polarity arranged so as to permit the positive flow of current from terminal 1 through the positive non-linear branch 7 to terminal 3 but to oppose the fiow of current in the opposite or negative direction through branch 7. In the resistance circuit 19, the polarity of the rectifier 27 is arranged so as to conduct a positive flow of current between terminals 1 and 3 and to oppose the flow of :urrent from the battery 29 through the resistance circuit 19. The resistance circuits 21 and 23 are similar to the resistance circuit 19 except that the batteries or iias in each of the resistance circuits 21 and 23 have 'ifferent potentials, and usually progressively higher otentials than the potential of the battery 29. Although it is to be observed that in Fig. l only four resistance circuits 17, 19, 21 and 23 are shown in the positive non-linear branch 7, a much larger number of resistance circuits similar to the resistance circuits 19, 21 and 23 may be employed to obtain greater accuracy and variation in approximating a desired non-linear current voltage curve. The negative non-linear branch 9 in Fig. l comprises a crystal diode or rectifier 32 in series with a plurality of parallel resistance circuits 33, 35, 37, and 39. The resistance circuit 33 comprises an adjustable substantiallylinear resistance 41. The resistance circuit 35 comprises, in series, a unidirectional device er rectitier 43, a source of electrical energy, such as a battery 45, and an adjustable substantially linear resistor 47. T he rectifier 32 has its polarity arranged so as to permit a negative iiow of current from the terminal 3 through the negative nonlinear branch 9 to the terminal 1, but to oppose the flow of current through branch 9 in the opposite or positive direction. In the resistance circuit 35, the battery 45 has its polarity arranged so that the battery 45 causes the terminal 3 to be positive with respect to the terminal 1. The rectifier 43 has its polarity arranged so as to conduct a negative flow of current from terminal 3 to terminal 1 and to oppose or buck any flow of current supplied by the battery 45. The resistance circuits 37 and 39 are similar to the resistance circuit 35 except that the potentials supplied by the batteries in the resistance circuits 35, 37, and 39 differ from each other. It is to be understood that the negative non-linear branch 9 may comprise still additional resistance circuits similar to the negative resistance circuits 35, 37 and 39 In considering the operation of the non-linear resistance device shown in Fig. l, let us tirst consider only the operation of the positive non-linear branch 7. We will also assume that the potentials supplied by the batteries in the resistance circuits 19, 21, and 23 are successively of greater magnitude, and that the greatest potential is supplied by the battery in the resistance circuit 23. When a direct current is caused to flow in a positive direction from the terminal 1 through the non-linear resistance device of Fig. l to the terminal 3, the polarity of the rectifier is arranged so as to permit the current iiow to iow through the positive non-linear branch 7. The effective conductance of the positive non-linear branch 7 is the conductance of the resistance circuit 17, namely Go, as long as the voltage drop between terminals 1 and 3 is less than the voltage supplied by any of the batteries in the resistance circuits 19, 21 and 23, respectively. It will be observed that current can not flow through the resistance circuit 19, for example, because the back voltage of the battery 29 exceeds the voltage applied to the circuit 19. Since progressively higher voltages are supplied by the batteries in the resistance circuits 21 and 23, such voltages also effectively block the circuits. If the voltage drop between the terminals 1 and 3 exceeds the voltage of the battery 29 and is less than the battery voltage in the resistance circuit 21, then the current iiowing in the positive direction from the terminal 1 through the positive non-linear branch 7 divides and fiows through both the resistance circuits 17 and 19, and the conductance is Go plus G1, the conductances of the resistance circuits 17 and 19. Again, if such voltage drop is greater than the voltages supplied by the batteries in the resistance circuits 19 and 21 and is less than the voltage of the battery in resistance circuit 23, current will divide and ow through resistance circuits 17, 19 and 21 but will not iiow through resistance circuit 23 because the voltage of the battery in resistance circuit 23 would be of such magnitude as to effectively block any fiow of current therethrough. When the current flows through the three resistance circuits 17, 19, and 21 the conductance is the sum of the conductances of the three circuits namely, Go, G1 and G2, respectively. When the current flow is of suiiicient magnitude to fiow through the four resistance circuits 17, 19, 21 and 23, the conductance is the sum of the conductances of the four circuits, namely Go, G1, G2 and G3, respectively. It will be observed that since the voltages of the batteries in the resistance circuits 19, 21, and 23 are successively higher, such circuits become progressively conductive as the voltage drop across the positive nonlinear branch 7 increases. By the circuit arrangement shown in Fig. l, the effective resistance between terminals 1 and 3 decreases as the flow of current between the terminals increases. Fig. la illustrates a typical current-voltage curve or non-linear resistance curve obtainable by the device shown in Fig. l. In Fig. la, E1, E2, and E3 represent the potentials supplied by the batteries in the resistance circuits 19, 21, and 23, respectively, and Go, G1, G2, and G3 represent the conductances of the resistance circuits 172 19, 21, and 23, respectively. In Fig. la, the horizontal ax1s represents the voltage-drop between the terminals 1 and 3 and the vertical axis represents the ow of current between the terminals 1 and 3. As well known in the art, the rectitiers, batteries and circuits have certain internal resistances which are negligible for most purposes; nevertheless, for accurate readings, it may be necessary to take such resistances into consideration along with the adjustable, substantially linear resistors in the circuits. If an alternating current is applied to the non-linear device shown in Fig. l, the positive portion of the current wave flows in the positive direction through the positive non-linear branch 7 and the negative portion of the current wave flows in the negative direction through the negative non-linear branch 9. The curve illustrated in Fig. la in its entirety represents a typical non-linear resistance curve obtainable when an alternating current is applied to the terminals 1 and 3 of the non-linear resistance device shown in Fig. l. The positive and negative portions of the curve are not and need not be symmetrical. Fig. 2 is similar to Fig. l except that the polarities of the rectifiers in the resistance circuits are reversed and that a source of electrical energy, such as batteries, are connected in series in each of the two non-linear branches similar to the positive and negative non-linear branches 7 and 9 of Fig. l. Referring to Fig. 2, two branches of the non-linear resistance device are connected between terminals 51 and 53, namely a positive non-linear branch 57 and a negative non-linear branch 59. The positive non-linear branch 57 comprises a rectifier 65 and a battery 66 in series with a plurality of resistance circuits 67, 69, 71 and 73. The resistance circuit 67 comprises an adjustable resistor 75. The resistance circuit 69 comprises, in series, a unidirectional device or rectifier 77, a source of electrical energy, such as a battery 79, and an adjustable resistor 81. The negative non-linear branch 59, when tracing the circuit from terminal 51 to terminal 53, comprises in series a rectifier 84, a source of direct current, such as a battery 82, and a plurality of resistance circuits 83, 85, 87 and 89. The resistance circuit'83 comprises an adjustable resistor 91. The resistance circuit comprises in series, a rectifier 93, a battery and an adjustable resistor 97. The resistance circuits 87 and 89 are similar to resistance circuit 85 except that the voltages of the batteries in the circuits 87 and 89 are progressively of higher potential. Similarly in the positive non-linear branch 57, the resistance circuits 71 and 73 are similar to the resistance circuit 69 except that the batteries in the circuits 71 and 73 are of progressively higher potentials. By comparing Figs. 1 and 2 it will be observed that Fig. 2 differs from Fig. 1 in only two respects, namely, the polarities of the rectiiiers in the resistance circuits 69, 71, 73, 85, 87 and 89 of Fig. 2 are reversed from the polarities of the rectifiers in the resistance circuits 19, 21, 23, 35, 37 and 39, respectively, of Fig. l, and batteries are added in series in the non-linear branches 57 and 59, the battery 66 being connected in series between the rectilier 65 and the parallel resistance circuits 67, 69, 71 and 73 and the battery 82 being connected in series between the rectifier 84 and the parallel resistance circuits 83, 85, 87 and 89. By way of explanation it may be pointed out that the non-linear resistance device of Fig. 2 does not contain a linear branch such as the linear branch 5 of Fig. 1. The linear branch 5 may be employed in or with any non-linear resistance device such as the device shown in Fig. 2. When the linear branch 5 is employed with a non-linear branch, the current flowing through the linear branch 5 adds to the current owing through the non-linear branch. In other words, the linear branch 5 adds to the flexibility of the non-linear resistance dev1ce. In the operation of the non-linear resistance device shown in Fig. 2, the rectiiers in each of the resistance circuits 69, 71, 73, 85, 87 and 89 have their polarities arranged so as to conduct current supplied by the batteries in such circuits, respectively, and to oppose a ow of current between the terminals 51 and 53 through such resistance circuits in the opposite direction. In the positive non-linear branch 57, the current supplied by the resistance circuits 69, 71 and 73 ilows through the resistance 75, when the voltage across the resistance circuit 6'7 is less than the lowest battery voltage, the voltage of the battery 79, such flow of current between the terminals 51 and 53 being etectively blocked by the rectifier 65. Similarly, all the current supplied by the batteries in the resistance circuits 85. 87 and 89 of the negative non-linear resistance branch 59 flows through the resistance 91 of the resistance circuit 83, the rectier 84 opposing any flow of such current between terminals 51 and 53. When no current ows from terminal 51 to terminal 53, and when the voltage across the resistance circuit 67 is less than the lowest battery voltage, the voltage of battery 79, the current supplied by the batteries in the resistance circuits 69, 71 and 73 of the positive non-linear branch 57, flows through the resistance circuit 67. As the flow of current from terminal 51 to terminal 53 increases in magnitude, the voltage between terminals 51 and 53 also increases in magnitude, and when the voltage equals or exceeds the potentials supplied by the batteries in the respective positive resistance circuits. 69, 71 and 73, such voltage bucks or opposes the voltage supplied by the batteries in the respective resistance circuits so that no current is supplied one or more of the batteries to flow through the resistance circuit 67. It will be readily observed that as the ow of current between the terminals 51 and 53 increases, the ettective resistance between such terminals also increases. Fig. 2a illustrates a typical curve in which the resistance increases with increased current flowing between terminals 51 and 53. If Go', G1. G2 and G3 are the conductance of resistance circuits 67, 69, 71 and 73, respectivelv, and Ei', E2 and E3 are the voltages, in order of magnitude. of the batteries in the resistance circuits 69, 71 and 73, respectively the resistance is at a minimum when the voltage across the positive non-linear branch 57 is less than the Voltage E1 the voltage of battery 79. When the voltage is less than E1 the conductance is the sum of the four conductances namely the sum of Go', G1', G2" and G3. When the voltage is greater than E1' and less than E2' the conductance is the sum of Go', G2', and G3. The conductances for the various voltages are illustrated in Fig. 2a. It may be observed that the batteries 66 and 82, can be adiusted so I that with no current flowing between terminals 51 and 53 the voltage of the battery 66 just balances the back voltage of the positive non-linear branch 57 and the voltage of the battery 82 just balances the back voltage of the negative non-linear branch 59 to cause the curve of Fig. 2a to pass through the origin, or with some sum of the voltages supplied by the batteries 66 and 82, the ratio can be altered to produce a bias so that the curve of Fig. 2a will not pass through the origin. Since a voltage-drop or voltage loss occurs, across the positive non-linear branch 57 even when no current flows between the terminals 51 and 53, the battery 66 is usually inserted to supply such voltage-drop or voltage loss, and to nullify any such voltage loss in the positive non-linear branch 57. Furthermore, the battery 66 may be of a selected value so as to have the curve cross the zero current axis at a desired value of voltage E. If it is desired to provide a non-linear resistance in an alternating current circuit to effect a resistance which increases with increased current then both the positive non-linear branch 57 and the negative non-linear branch 59 would be employed, the positive portion of the current owing through the positive branch 57 and the negative portion of the current flowing through the negative branch 59. The negative non-linear branch 59 performs in the same manner as the positive non-linear branch 57 except that it conducts the negative portion of the current ow, the same as the negative non-linear branch 9 in Fig. 1. In Fig. 3, a non-linear resistance device is shown with terminals 101 and 103 and two branches, namely a positive non-linear branch 107 and a negative non-linear branch 109, connected between the terminals. The positive non-linear branch 107 comprises a rectier 115 in series with a plurality of parallel resistance circuits 117, 119, 121, 123 and 124 which are connected through a potentiometer or voltage divider 111 to the terminal 103. Similarly, the non-linear branch 109, when tracing the circuit from terminal 101 to terminal 103, comprises a rectifier 132 in series with a plurality of parallel resistance circuits 133, 135, 137, 139 and 140 which are connected through a potentiometer or voltage divider 113 to the terminal 103. The resistance circuit 117 comprises an adjustable resistor 125. The resistance circuit 119 comprises a rectier 129 and an adjustable resistance 131 in series. In the negative non-linear branch 109, the resstance circuit 133 comprises an adjustable resistor 141, and the resistance circuit 135 comprises in series a rectier 143 and an adjustable resistance 147. Each of the remaining resistance circuits, namely circuits 121, 123, 124, 137, 139 and 140 comprise a rectifier and an adjustable resistance in series. The Voltage divider 111 has its end-terminals connected across a source of potential energy such as battery 149. Each of the resistance circuits 117, 119, 121, 123 and 124 of the positive non-linear resistance branch 107 is connected to slider-contacts on the voltage divider 111 so that each of the resistance circuits is energized by a progressively higher potential relative to the potential applied to the resistance circuit 117. The voltage divider 111 makes it possible to supply a voltage to the respective resistance circuits 119, 121, 123 and 124 without using batteries in such circuits. The terminal 103 is connected to a slider contact adjustable to an intermediate position on the voltage divider 111 so that a voltage, if desired, can be inserted in series in the positive nonlinear branch 107 of the non-linear resistance device of Fig. 3 to serve the same purpose as the battery 66 in the positive non-linear branch 57 of the non-linear resistance device of Fig. 2. Similarly in the negative non-linear branch 109, the resistance circuits 133, 135, 137, 139 and 140 are connected to slider contacts on the voltage divider 113 so that the circuits are respectively energized by a progressively higher potential, relative to the potential applied to the resistance circuit 133 the voltage divider 113 being energized at its end-terminals by a battery 151. In order to introduce a voltage in series in the negative non-linear branch 109, the terminal 103 is connected to a slider contact adjustable to an intermediate position on the voltage divider 113 to permit supplying a voltage to nullify any undesired voltage-drop across the resistance circuit 133. Furthermore, such a voltage of a desired value may be employed so as to have the resistance curve cross the zero current axis at a desired value of volta e E gIt will be observed that Fig. 3 is a variation of Figs. l and 2, one of the principal differences being the voltage dividers 111 and 113 in Fig. 3 which are employed to supply the various voltages required by the circuits. In the positive non-linear branch 107 the polarities of the rectiers in the resistance circuits 121 and 124 are the same as the polarities of the rectitiers in the resistance circuits 19, 21 and 23 of the positive non-linear branch 7 of Fig. 1 and the polarities of the rectitiers in resistance circuits 119 and 123 are the same as the polarities of` the rectitiers in the resistance circuits 69, 71 and 73 of the positive non-linear branch 57 of Fig. 2. It will also be observed in the negative non-linear branch 109 that the polarities of the rectiers in the resistance circuits 135, 137, 139 and 140 are not the same. By such an arrangement of the rectitiers it is possible to have multiple changes of curvature as illustrated in Fig. 3a. To explain the operation of the positive non-linear branch 107 we will consider that the voltages applied to the resistance circuits 117, 119, 121, 123 and 124 by the voltage divider 111 are zero, E1, E2, E3, and EA" respectively. We will also consider the conductances of the resistance circuits 117', 119, 121, 123, and 124 to be Go", G1, G2, and G3, and G4 respectively. It will be observed that as the positive voltage E across the positive non-linear branch 107 increases certain of the resistance circuits 119, 121, 123 and 124 cut in and other cut out producing any desired variation of conductance G with the positive voltage E or the current I through the positive non-linear branch 107. When current rst starts to flow across the branch 107, Go, G1" and G3" are active or conductive. When the voltage-drop across the branch 107 equals E1 then G1 cuts out, when the voltage-drop equals E2, G2 cuts in, when the voltagedrop equals E3, G3 cuts out, and when the voltage-drop equals E4, G4 cuts in. This operation is illustrated in Fig. 3a. By using various combinations of resistance circuits as shown in Figs. l, 2 and 3 and linear resistances, a wide variation of curves representing non-linear resistance curves are obtainable especially when wide variations are possible in the magnitude of the voltages applied to the resistance circuits and when wide variations in the magnitude of the resistances in the resistance circuits are possible. With this type of circuit the sign of the slope of the generated curve or function cannot change from positive to negative if it has an initial positive slope or cannot change from negative to positive if it has an initial negative slope. It is not possible to produce a curve having the same resistance twice for two different voltages or currents. In U. S. Patent No. 2,420,891, issued to G. D. Mc- Cann, Jr., et al. on March 20, 1947 and in U. S. application, Serial No. 700,130 tiled September 28, 1946, in the name of G. D. McCann, Jr., which are assigned to the Westinghouse Electric Corporation, the assignee of this invention, certain analyzers were disclosed including the mechanical transient analyzer and the analog computer. Such an analyzer oi' computer based upon analogies between physical systems such as mechanics and hydraulics and electrical systems, is able to solve almost all problems in the mechanical, hydraulic or thermal iields by reducing such problems to their electrical equivalent and solving such problems on the analyzers to engineering accuracy as an electrical problem. Such an analyzer makes it possible to readily solve numerous problems that previously could only be estimated by long mathematical commutations or solved by tedious and costly experiments. The non-electrical problem is set up and solved on the basis of the electrical equivalent. For example, to represent a mechanical system by one form of analogous electrical system, the following relationships may be employed: Table I Analogous electrical system Inductance (L.) Resistance (R.) Susceptance (l/C) Voltage (V) Mechanical .system Mass or inertia Velocity or damping Spring constant Force or torque Analogozls electrical system Voltage (V) Hydraulic system Pressure differential (P) Rate of flow (F) Current (I) Frictional resistance (r) Electrical resistance (R) For example, let us assume that we are confronted with an elementary hydraulic problem as illustrated 1n a simpliiied form in Fig. 4 in which, a pump 201 forces a fluid through a pipe or a line 203 in series with parallel lines 205 and 207, respectively. To determine the magnitude of iiow at different pressure differentials through each of the lines would require expensive measuring equipment which would be diicult to install. Furthermore, it might not be convenient or possible to supply the variations in pressure differentials necessary to make such tests. By means of the analog, the analysis, could be readily performed once the pressure diierential-ow relation for the particular fluid involved, is ascertained. The electrical analog of Fig. 4 is illustrated in Fig. 5. A potentiometer or voltage-divider 211 has a battery 213 connected across its terminals to supply a potential to the circuit of Fig. 5. A non-linear resistance 215 to represent the fluid resistance of the series pipe 203, is connected in a series circuit 214 between the voltage divider 211 and parallel circuits 217 and 219 which embody nonlinear resistors 221, and, 223 respectively, to represent the iiuid resistance of the lines 205 and 207, respectively. Each of the non-linear resistors 215, 221 and 223 is adjusted to produce a voltage current curve as illustrated in Fig. 5a which closely approximates the pressure differential-ow curve illustrated in Fig. 4a. It will be observed that the voltage-divider 211 may be adjusted to represent in the circuit of Fig. 5 the various pressure dilerentials to be supplied by the pump 201 in Fig. 4. By taking current readings in circuits 214, 217 and 219, one may readily determine the flow of the uid in the lines 203, 205 and 207, respectively and the distribution of flow between the lines 205 and 207. By measuring the voltage-drop across each of the non-linear resistors 215, 221 and 223, respectively, it is possible to determine the pressure losses in the lines 203, 205 and 207, respectively. Since pressure ditferential-ow of relationships are invariably non-linear in characteristic, the substantially linear resistors usually available are not suitable for such application, but with a non-linear resistor device as disclosed by applicant, it is possible to produce an electrical resistance having the desired characteristic to represent the relationships necessary to work the hydraulic problem by the electrical analog. Since the non-linear resistors 215, 221 and 223 are to eiect a curve in which the resistance increases with an increase in flow and such resistors are energized by a direct current, a circuit may be employed similar to the positive non-linear branch 57 of the non-linear resistance device shown in Fig. 2. ln Fig. 6 are shown two non-linear resistance circuits 251 and 253 connected to terminals 255 and 257. In the non-linear circuit 251 a plurality of parallel resistance circuits such as circuits 259, 261 and 263, are connected between conductors 265 and 267, conductor 265 being connected through a rectifier 269 by means of a reversing switch 271, and through a two pole switch 273, to terminal 257. The conductor 267 is connected through a switch 275, and the two-pole switch 273 to the terminal 255. rIhe resistance circuit 259 comprises, when traced from the conductor 265 to the conductor 267, a switch 277, an adjustable resistance element 279, a jack 231 to facilitate inserting additional voltage in the resistance circuit 259, and a voltage divider 283. A resistance measuring jack 285 is connected from conductor 267 across the adjustable resistance element 279 to facilitate measuring the resistance in the resistance circuit 259. The resistance circuit 261 comprises when traced from the conductor 265 to the conductor 267, a reversing switch 237, a crystal diode or rectilier 289, an adjustable resistance element 291, a voltage jack 292, a voltage divider 293 having its terminals energized by a battery 295, and a voltage divider 297. The voltage jack 292 is supplied to permit inserting additional voltage in the resistance circuit 261. A resistan-ce jack 299 is connected in parallel with the adjustable resistance element 291, the voltage jack 292, and the voltage divider 293 to permit measuring the resistance in the resistance circuit 261. It s frequently necessary to take into consideration and add he resistance of the rectifier 289 when considering the esistance ot a resistance circuit, such as resistance circuit 259. A battery 301 has its positive terminal connected through an adjustable resistance 303 and a double-throw double-pole switch 305 to the conductor 267, the negative terminal of the battery being connected to an endterminal of the voltage divider 283 the other end-terminal of the voltage divider 283 being connected to the conductor 267, and the resistance circuit 259 being connected to the slider arm of the voltage divider 283. The voltage divider 297 has its end-terminals connected to conductors 267 and 307 and its slider arm connected to an end-terminal of the voltage divider 293. A second battery 309 has its negative terminal connected to the positive terminal of the battery 301, and its positive terminal connected through an adjustable resistance 311, and through the double-throw double-pole switch 305 to the conductor 267 or 307. When the poles of the switch 305 are moved to the left, a voltage is supplied across the voltage divider 283 by the battery 301 and a voltage is ipgplied across the voltage divider 297 by the battery When the poles of the double-throw two-pole switch 305 are moved to the right the potentials of the batteries 301 and 309, in additive series, are applied across the voltage divider 283 and no voltage is applied by either of the batteries 301 or 309 between the conductors 267 and 307. Several additional resistance circuits similar to resistance circuit 261, such as resistance circuit 263 and others, may be connected between the conductor 265 and the conductors 267 and 307. It has been found desirable to employ as a unit, in parallel, a resistance circuit 259, four resistance circuits each similar to resistance circuit 261 but omitting the voltage jack 292 and the voltage divider 293 with its associated battery 295, and four additional resistance circuits, each similar to resistance circuit 261 employing either the voltage jack 292 or the voltage divider 293 and battery 295, but not both for supplying voltage to the associated resistance circuit. The variations therein, are optional with the designer and are governed largely by the requirements of the nonlinear unit. The non-linear resistance circuit 253 may be similar to the non-linear resistance circuit 251 and by reversing its connections to terminals 255 and 257 it will serve as the negative non-linear branch in an alternating current systern and the non-linear unit 251 will serve as the positive non-linear branch. In the unit 251, it will be observed that the polarity of the rectifier 269 may be reversed by the reversing switch 271. Similarly the polarity of the rectifier 289 in the resistance circuit 261 may be reversed by means of the reversing switch 287 so that the rectifier 289 may have the polarity of the rectifiers in the resistance circuits of the positive non-linear branch 7 of Fig. l or the polarity of the rectifier in the resistance circuits of the positive non-linear branch 57 of Fig. 2, as the particular situation may require. It will be observed that the non-linear circuit 251 of Fig. 6 employs the same principal of operation as the devices shown in Figs. l, 2 and 3, but embodies measuring jacks, reversing switches, additional voltage supplies and adjustments to permit the non-linear circuit 251 -to be more adaptable and useful. For instance, a voltage jack 313 is connected between terminals f 255 and 257 to permit taking voltage measurements, and a current jack 315 is inserted in the conductor connected to the -terminal 257 to permit measuring the fiow of current between terminals 255 and 257 through the nonlinear circuit. Numerous variations in the non-linear resistance devices shown in Figs. l, 2, 3 and 6 are possible such as the use of vacuum tube diodes, barrier-layer rectifiers or other unidirectional devices for the diode crystals or rectifiers employed in the devices described above. It has been found, however, that it is more difficult to provide power supplies for vacuum tube types of non-linear, resistance devices. For low impedances the barrier layer type of rectifier is preferable and for high impedances the vacuum tube type of rectifier is preferable. It is to be observed that the voltage-current curves produced by the non-linear resistance devices described above are a combination of straight line segments to approximate the desired curves. In actual practice it has been found that the segments do not intersect at an angle but that such intersections are rounded so that the segments more nearly approximate the desired curve. In the alternating -current devices the positive and negative portions of the voltage-current curve can be symmetrical or unsymmetrical as the occasion may require. It is also possible to add or subtract voltages and make adjustments so that the curve need not pass through the zero point or origin of the voltage-current axes. The above non-linear resistance devices have been described as employing substantially linear resistors in the non-linear resistors in the resistor circuits. The use of such non-linear resistors would facilitate obtaining certain extreme non-linear resistance curves, and the segments of the curves would not necessarily be straight lines. The foregoing disclosure and drawings are merely illustrative of the principles of this invention and are not to be interpreted in a limiting sense. The only limitations are to be determined from the scope of the appended claims. I claim as my invention: l. In an adjustable impedance unit, a first circuit cornprising, in series, a rectifier and an adjustable resistor, and at least one additional circuit shunting said resistor, said additional circuit comprising in series a rectifier, a source of voltage and an adjustable resistor. 2. In an adjustable impedance unit, a first circuit comprising in series a rectifier and an adjustable resistor, a resistor shunting said first circuit, and at least one additional circuit shunting said adjustable resistor, said additional circuit comprising in series a rectifier, a source of voltage and a resistor. 3. In an adjustable impedance unit, a first circuit connected between a pair of terminals, comprising in series a rectifier and a resistor, and at least one additional circuit shunting said resistor, said additional circuit comprising in series a rectifier, a source of voltage and a resistor, said rectifiers being oppositely poled relative to the termin'als. 4. In an adjustable impedance unit for effecting a nonlinear reisstance between a pair of terminals when energized by a unidirectional current from an external source, comprising a voltage divider having first, second and third connection points and a plurality of adjustable contacts thereon, the first connection point being connected to one of the terminals, a plurality of parallel resistance circuits connected between the adjustable contacts on the voltage divider and the other one of the terminals, and a first voltage source connected across the voltage divider between the second and the third connection points thereon, at least one of said parallel resistance circuits comprising in series a unidirectional element and an adjustable resistor and being connected to one of said adjustable contacts on the voltage divider to permit applying to such circuit a potential which will tend to oppose a flow of current from the external source in the circuit. 5. In an alternating-current adjustable resistance device for effecting a non-linear resistance between a pair of terminals comprising two branches connected in parallel between the terminals, each branch comprising a first resistor and at least one additional circuit shunting said first resistor, said additional circuit comprising in series a unidirectional element, a voltage supply and an adjustable resistor, the polarity of the voltage supply in one branch with respect to the terminals being opposite to the polarity of the voltage supply in the other branch. .6. 1n an alternating-current adjustable resistance device as claimed in claim 5 wherein at least one branch has a unidirectional element connected in series with the first resistor between said terminals, said unidirectional element having its polarity arranged so as to oppose a flow of current between said terminals from said voltage supply in the last-named branch. '7. In an alternating-current adjustable resistance device as claimed in claim 5 wherein at least one branch has a voltage source in series with said first resistor, the polarity of said voltage source with respect to said terminals being opposite to the polarity of said voltage supply in said branch. 8. An impedance device for simulating in an analog computer a non-linear impedance curve with sinuouslike portions in the slope thereof between a pair of terminals when a unidirectional current is supplied to such terminals from an external source comprising a first impedance circuit connected between the terminals arid at least two parallel circuits connected in shunt with .said first impedance circuit, each of said parallel circuits comprising a rectifier and an adjustable resistor and being biased by a voltage supply with its polarity so arranged as to tend to oppose the flow of current through such parallel circuit from the external source, the polarity of the rectifier in one parallel circuit with respect to the terminals being opposite to the polarilty of the rectifier in another of such parallel circuits. 9. In an impedance unit, a pair of terminals, rst and second circuits connected in parallel across such terminals, the first and second circuits comprising respectively, first and second resistors and first and second rectifiers, said rectifiers being poled to permit fi'ow of currents 1n opposite directions between the terminals, a plurality of first auxiliary circuits connected in parallel with the first resistor for energization only through the first rectifier, each of the first auxiliary circuits comprising a -resistor, a source of direct voltage and a rectifier in series, said sources establishing different voltages between the terminals for the first auxiliary circuits, and a plurality of second auxiliary circuits connected in parallel with the second resistor for energization only through the second rectifier, each of the second auxiliary circuits comprising a resistor, a source of direct voltage and' 'a rectifier in series, the number of the second auxiliary circuits that are conductive being determined by the magnitude of the current liowing through the second rectifier. l0. In an impedance unit, a pair of terminals, first and second circuits connected in parallel across such terminals, the first and second circuits comprising, respectively, first and second resistors and first and second rectifiers, said rectifiers being poled to permit fiow of currents in opposite directions between the terminals, a plurality of first auxiliary circuits connected in parallel with the first resistor for energization only through the first rectifier, each of the first auxiliary circuits comprising a resistor, a source of direct voltage and a rectifier in series, the rectifiers in the first auxiliary circuits being poled oppositely to the rst rectifier, and a plurality of second auxiliary circuits connected in parallel with the second resistor for energization only through the second rectifier, each of the second auxiliary circuits comprising a resistor, a source of direct voltage and a rectifier in series, the number of the second auxiliary circuits that are conductive being determined by the magnitude of the current flowing through the second rectifier. 11. ln an impedance unit, a pair of terminals, first and second circuits connected in parallel across such terminals, the first and second circuits comprising, respectively, rst and second resistors and first and second rectiers, said rectifiers being poled to permit flow of currents in opposite directions between the terminals, a plurality of first auxiliary circuits connected in parallel with the first resistor for energization only through the first rectifier, each of the first auxiliary circuits comprising a resistor, a source of direct voltage and a rectier in series, the rectifiers in the first auxiliary circuits being poled similarly to the first rectifier, and a plurality of second auxiliary circuits connected in parallel with the second resistor for energization only through the second rectifier, each of the second auxiliary circuits comprising a resistor, a source of direct voltage and a rectifier in series, the number of the second auxiliary circuits that are conductive being determined by the magnitude of the current owing through the second rectifier. 12. In an impedance unit, a pair of terminals, first and second circuits connected in parallel across such terminals, the first and second circuits comprising, respectively, first and second resistors and first and second rectifiers, said rectifiers being poled to permit flow of currents in opposite directions between the terminals, a plurality of first auxiliary circuits connected in parallel with the first resistor for energization only through the first rectifier, each of the first auxiliary circuits comprising, a resistor, a source of direct voltage and a rectifier in series, the rectifiers in a portion of the first auxiliary circuits being poled similarly to the first rectifier and the rectifiers in the remaining portion of the first auxiliary circuits being poled oppositely to the first rectifier, and a plurality of second auxiliary circuits connected in parallel with the second resistor for energization only through the second rectifier, each of the second auxiliary circuits comprising, a resistor, a source of direct voltage and a rectifier in series, the number of the second auxiliary circuits that are conductive being determined by the magnitude of the current owing through the second rectier. 13. In an impedance unit, a pair of terminals, a first rectifier, a first source of direct voltage and a first resistor in series between such terminals, said first rectifier being poled with respect to the source of direct Voltage to conduct current supplied thereby, a plurality of auxtiliary circuits connected in parallel with said first resistor, each of the auxiliary circuits comprising a resistor, a second source of direct voltage and a rectifier in series, the rectifier and the second source of direct voltage, respectively, in at least one of the auxiliary circuits being poled oppositely to the first rectifier and the first source of direct voltage. 14. In an adjustable impedance unit comprising a pair of terminals between which a non-linear impedance is simulated when current from an external source fiows in a predetermined direction between the terminals, a first impedance circuit connected between the terminals, and at least one additional impedance circuit shunting said first impedance circuit, such additional impedance circuit comprisilng in series a voltage supply, a rectifier and an adjustable substantially linear resistor, the polarities of said voltage supply and said rectifier being arranged so as to oppose a iiow of current from the external source through the additional impedance circuit, a unidirectional element connected in series with the impedance circuits between the terminals, the polarity of the unidirectional element being arranged so as to permit current from the external source to iiow between the terminals and to oppose a flow of current from said voltage supply between the terminals. References Cited in the file of this patent UNITED STATES PATENTS Number Name Date 2,104,336 Tuttle Jan. 4, 1938 2,248,563 Wolfi July 8, 1941 2,401,404 Bedford June 4, 1946 2,434,155 Haynes Jan. 6, 1948 2,548,913 Schreiner et al. Apr. 17, 1951 2,581,124 Moe Jan. l, 1952 Patent Citations
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