< 262-7538 
 
 i>TAT£ WATER SURVEY DIVISION 
 LIBRARY CORY 
 
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 An Electric Analog 
 
 For Computing Direct Surface Runoff 
 
 April 1966 ARS 41-118 
 
 Agricultural Research Service 
 
 UNITED STATES DEPARTMENT OF AGRICULTURE ■ 
 
CONTENTS 
 
 Page 
 
 INTRODUCTION. c 1 
 
 TERMINOLOGY . X 
 
 ANALOG DEVICES. 2 
 
 WATERSHED ANALOGY. 3 
 
 TEST MODEL OF ELECTRIC ANALOG . 8 
 
 Electric Analogy . 9 
 
 Calibration of Analog . X7 
 
 Input to Analog. 19 
 
 Operation of Analog .. 22 
 
 EXAMPLES OF HYDROGRAPHS . 23 
 
 CONCLUSIONS. 23 
 
 ACKNOWLEDGMENT. 27 
 
 LITERATURE CITED. 01 
 
1 / 
 
 AN ELECTRIC ANALOG FOR COMPUTING DIRECT SURFACE RUNOFF 
 
 By 
 
 2 / 
 
 J. Marvin Rosa 
 
 INTRODUCTION 
 
 The purpose of this study was to demonstrate the feasibility of an 
 electric analog device for routing runoff to generate hydrographs on experi¬ 
 mental watersheds,, A small, inexpensive analog model was desired for quickly 
 routing runoff from hourly records of precipitation on headwater areas,, Even 
 the most simple routing through multiple stages, as if the watershed and 
 channels act like a series of reservoirs, is adequate if time intervals are 
 very short. The use of many increments of time and storage would complicate 
 any long-hand or graphical computation previously known, but this is no 
 problem with fast electronic devices. A rapid solution could be obtained by 
 utilizing the analogy between the flow of electricity in a passive network of 
 resistors and capacitors as used in oil-reservoir simulations and soil- 
 moisture movement studies. 
 
 A simplified, schematic explanation of the hydrology of watersheds is 
 first outlined to show that they function like a series of reservoirs. After 
 derivation of the analogy to a reservoir in electrical units, a description 
 of some construction and of calibration details completes the discussion of 
 electronic equipment. Finally, the operation of this analog model is shown 
 to simulate the time-consuming routing computations previously required. 
 
 Some actual hydrograph computations illustrate the applicability of this 
 model at a cost of about $100. The term analog "model" is more descriptive 
 of this network for solving the storage function by the use of common radio 
 parts than is the term "computer." 
 
 TERMINOLOGY 
 
 Symbol Unit 
 
 Electrical 
 
 I , g , = Current (ma = milliamperes)--- amperes 
 
 L^ , g , 3 = Inductance--- Henrys 
 
 R , , = Resistance (K = 1,000 ohms) - ohms 
 
 V , 2 , 3 = Voltage- volts 
 
 C , 2 , 3 = Capacitance (/if. = microfarads) - farads 
 
 Qc = Charge on capacitors - coulombs 
 
 T s = Time of storage-*•- seconds 
 
 1 / Soil and Water Conservation Research Division, Agricultural Research 
 Service, USDA, in cooperation with the Idaho Agricultural'Experiment Station. 
 
 _2/ Research hydraulic engineer. Soil and Water Conservation Research 
 Division, Agricultural Research Service, USDA, Moscow, Idaho. 
 
 1 
 
Symbol 
 
 TERMINOLOGY--Continued. 
 
 Unit 
 
 Hydraulic 
 
 Ii 
 
 Io 
 
 Qo 
 
 Qt 
 
 t 
 
 e 
 
 X 
 
 k 
 
 = Inflow - 
 
 = Outflow - 
 
 = Initial discharge --- 
 
 = Discharge at time - 
 
 - Time - 
 
 = 2.7183 
 
 = Relative effect of inflow and outflow, when 
 = "Muskingum" routing coefficient 
 
 Hydrologic 
 
 = Direct storm runoff - 
 
 = Accumulated runoff for periods - 
 
 = Increment of runoff --— 
 
 = Mass direct runoff - 
 
 = Storm rainfall - 
 
 = Accumulated rainfall for periods 
 
 = Rainfall for periods - 
 
 = Mass precipitation - 
 
 = Storage - 
 
 = Recession constant 
 
 c « f . s • 
 
 C . f . S . 
 
 c.f. s. 
 c.f.s. 
 
 seconds 
 
 inches 
 
 inches 
 
 inches 
 
 inches 
 
 inches 
 
 inches 
 
 inches 
 
 inches 
 
 inches 
 
 ANALOG DEVICES 
 
 Analog devices depend on--and are named for--the use of analogies, which 
 are similarities of properties of relationships. As stated by Rogers and 
 Connolly (13 )' simulation results whenever a physical model is represented 
 in such a way that its characteristics, parameters, and behavior can be easily 
 identified, adjusted, and studied as accomplished in the use of analog models. 
 The term "analog computer" refers to operational amplifiers, which are as¬ 
 sembled into a computer for general purposes. Such a computer must be capable 
 of (1) multiplying constants, (2) adding variables, (3) multiplying variables, 
 (4) integrating variables with time, or (5) generating a function of a 
 variable. 
 
 All analogs could be called computers, but all computers do not operate 
 as analog models. One major type of computer is called digital--which deals 
 in numbers according to an ordered sequence of simple arithmetic operations. 
 
 It computes by repeatedly refining an approximation so that its accuracy is 
 potentially unlimited. Another type of computer is some sort of physical 
 model that is set in operation to generate a solution. This analog unfortu¬ 
 nately is limited in accuracy by the physical elements of the system. The 
 most direct analog "model" (not usually called a computer) performs like a 
 scale model under experimentation in a hydraulic laboratory. It warrants 
 being in a class by itself--the passive network. 
 
 3 / Underscored numbers in parentheses refer to Literature Cited at the 
 end of this report. 
 
 2 
 
As utilized by Harder and others (6), electric analogs are models of a 
 dynamic system where flow networks can be integrated by instantaneous electri¬ 
 cal currents. Such electric analog models have been used by the petroleum 
 industry, according to Scheidegger (14), and should have a similar application 
 to ground-water problems. Problems involving ground-water behavior, as 
 studied by Skibitzke and Brown,—' can be solved by using electric analogs 
 where solutions are unattainable in any other way. There is no limit to the 
 application of such electric analogs to hydrologic problems, according to 
 Tribus (15), since it is possible to construct analogous electrical networks 
 for every dynamic system. 
 
 Electric analogs for routing streamflow to generate hydrographs on small 
 watersheds are a recent development. Linsley, Foskett, and Kohler (9) in 1948 
 first utilized an electronic routing machine to route floods from basins of up 
 to 15,000 square miles. In 1952 Paynter (11) demonstrated the analogy between 
 flood routing by admittance methods and transients in electrical transmission 
 lines. Glover (4) in 1953 presented an electronic circuit for an analog model 
 which provided for the square-law resistance. And in 1956 Rockwood and 
 Hildebrand (12 ) described an electrical circuit for multiple-stage storage 
 routing on large rivers. 
 
 Messerle (10 ) first reported an electronic high-speed simulator developed 
 at the University of Melbourne in 1953. Japanese developments were described 
 in 1955 and 1957 by Ishihara and Ishihara (7) and Ishihara (8). At about this 
 same time, foreign articles by Dziatlik (3) and Halek (5) mentioned analog 
 apparatus. By 1958 electric analogs were mentioned in the literature of most 
 countries. 
 
 WATERSHED ANALOGY 
 
 Basically, the watershed analogy can be represented as an exchange of 
 water between storage and flow or losses, the time pattern of precipitation 
 is a variable input following the trace of a recording gage. The first losses 
 are intercepted by vegetation, which is shown as part of evapotranspiration 
 in figure 1, 
 
 The first increment of runoff could be called channel interception or the 
 rain falling directly on the wet surface area of the open channel. Direct 
 evaporation from water surfaces could be extracted here and combined with other 
 losses. Other increments of overland flow follow each other down the channel 
 after separation at the soil surface by infiltration. All increments of runoff 
 to this point are routed through storage; First through detention on the rough 
 land surfaces or humus (A 0 ) layer; then as channel storage, including bank 
 storage, less other riparian losses downstream. These losses and storages are 
 shown diagrammatically in figure 1. 
 
 Moisture entering the topsoil (A ) is available for percolation or is 
 eventually extracted by the process oi evapotranspiration in proportion to 
 temperature or other controlling factors. As far as immediate direct runoff 
 
 4/ Skibitzke, H. E., and Brown, R. H. Analysis of hydrologic systems. 
 U.S. Geol. Survey unpublished paper, 10 pp. 1961. 
 
 3 
 
is concerned, the characteristics of the detention volume in the soil profile 
 determine the percolation rate of free water into the next incremental volume 
 of storage, shown as subsoil (B 1 ). Here again, retention is trapped and re¬ 
 leased gradually either to direct runoff laterally if there is a limiting 
 layer (Rg), or to percolation downward into the rock mantle (C ), as shown 
 in figure 1. 1 
 
 Rainfall 
 
 c 
 
 o 
 
 <h 
 
 
 
 Surface 
 
 Detention 
 
 
 Topsoil A 
 
 AM> 
 
 Overland 
 
 Detention a, 
 
 \ 
 
 Flow 
 
 
 
 \ 
 
 
 
 
 ^ 1 hour 
 
 
 <h 
 
 © 
 
 3 
 
 o 
 
 I- 
 
 I 
 
 I 
 
 Subsoil 
 Detention g ( 
 
 
 
 Channel 
 
 Routing 
 
 (limiting 
 layer) q 2 
 
 
 
 Analog 
 
 Percolation 
 
 Mantle 
 
 
 Watershed 
 
 Recharge 
 
 Quick 
 
 Seepage 
 
 ft 
 
 2 days 
 
 Boseflow 
 
 Ground 
 
 Water 
 
 Routing 
 
 Analog 
 
 20 days 
 
 Figure 1.--Schematic diagram of watershed analog,, 
 
 4 
 
A shallow profile might contribute quick seepage at this point after 
 saturation, as represented by a limited storage-discharge relation. A deep 
 porous mantle would continue to receive watershed recharge, which drains out 
 much later through an aquifer, as represented by a great volume of ground 
 water storage. Wherever the increments of outflow drain into the channel from 
 each increment of storage, there would be a division of the flow (for routing 
 purposes) between the limited storage in channels relatively far above the 
 point gaged and the greater storage in underground sources leading eventually 
 to the gaging station. The runoff component of overland flow and quick 
 seepage (usually called direct runoff), as shown in figure 1, is quickly 
 routed through a channel analog representing the drainage patterns. Base flow 
 or ground water is similarly routed through a ground water analog capable of a 
 much greater time of storage. 
 
 The multiple-stage storage routing method is outlined in figure 2 as it 
 applies to a complex channel system. This represents a watershed that must be 
 divided into 8 to 10 subareas in order to evaluate the runoff from different 
 rainfall patterns and complex soil-cover conditions. Thus, at many points 
 down the main channel either tributary or local inflow must be added. 
 
 In order to express variable channel cross sections, slopes, and rough¬ 
 nesses, which could be computed to give different storage volumes, some four 
 or five measured sections are assumed for each reach. Thus, there are perhaps 
 five stages through which to route the water down each reach of the main 
 channel between subareas. To get from one cross section to the next, the flow 
 must be routed through the storage, which (between closely spaced cross 
 sections) can be assumed to act like a series of reservoirs. 
 
 Figure 3 has been adapted from Bruce (1) to represent the behavior of 
 underground reservoirs--in this case, the ground-water table. Recharge at any 
 point would be distributed through a series of storage units until discharged 
 at a distant point. Subsurface flow would respond to disturbances of the 
 pressure surface represented by water table levels. Unit recharge on a water¬ 
 shed slope must move, toward an outlet, through a network of pores or minor 
 underground channels of constantly changing diameter. This can be represented 
 simply as a series of tiny tubes, end-to-end, with storage changing between 
 sections. 
 
 A watershed could be modeled after figure 3 as a three-dimensional network 
 of resistors and capacitors and stacked to the shape of the water table, since 
 the governing fluid equations are identical in form to those representing the 
 behavior of such an electric circuit. Scheidegger (14) has shown that such 
 analogs will solve complex flow equations and that the proper size of capaci¬ 
 tors and resistors may be selected by trial and error to match a known past 
 history of behavior. 
 
 Figure 4 shows the analog model for several stages of storage which could 
 represent either the channel reaches for surface runoff, as previously 
 discussed and represented in figure 2, or the units of storage available in 
 the soil or rock mantle for the subsurface or ground water flow shown in 
 figure 3. The schematic diagram in figure 4 is drawn simply as if the hourly 
 rainfall entered through a funnel into a system of pipes and tanks. Each tank 
 
 5 
 
Figure 2.--Multiple-stage storage routing. 
 
 6 
 
Inflow 
 
 Figure 3.--Part of a three-dimensional electrical representation Q). 
 
 H ourly_Roinfol I 
 
 7 
 
is analogous to storage in charging a capacitor (C), and flow between tanks 
 is controlled by a valve that is analogous to a variable resistor (R). Thus, 
 the time it takes for a tank to empty would be a function of capacity and 
 rate of outflow. The final outflow past the last stage would be a hydrograph 
 integrating the effect of all prior stages of storage and resistance. 
 
 TEST MODEL OF ELECTRIC ANALOG 
 
 The analogy between an electric current in a capacitor-resistor circuit 
 and water flowing through storage can be derived for the simple case of a 
 reservoir. If several reservoirs are connected, then the equations become 
 too complex for easy solution. Therefore, a simple model of the network is 
 the answer. 
 
 A circuit diagram for an electronic flood-routing device developed by 
 the U. S. Weather Bureau in 1948 and built commercially for each of the River 
 Forecasting Centers is shown in figure 5. Resistances and capacitors, 
 
 8 
 
cross connected, are analogous to the "Muskingum" method of routing, where S 
 is storage, k is the time of storage, and X is the relative effect of inflow 
 and outflow in the equation: 
 
 S = k [XI + (l-X)o] 
 
 where, 
 
 k = 2C, (R +R) and X =- ^- . 
 
 1 2 ( R i + k e ) 
 
 This flood-routing device represents only one stage so that outflow has 
 to be recorded and fed back into the input in order to duplicate multiple 
 stages of flood routing,, When X = 0.0, storage is related to discharge alone 
 as in a reservoir and when X = 0.5, a channel is represented with inflow and 
 outflow equally important in determining storage within a river reach. 
 
 Another flood-routing analog, using two stages of resistors and capaci¬ 
 tors, was built for the U.S. Corps of Engineers at Portland, Oreg., in 1956. 
 The circuit diagram of this analog, as described by Rockwood and Hildebrand 
 (12), is shown in figure 6 C It was a small battery-powered test model, only 
 about 12 inches long, having variable rheostats across each bank of capaci¬ 
 tors so that leakage currents could be increased. Thus, an exponential re¬ 
 lation of storage and discharge could be approximated by slowly reducing the 
 resistance as outflow increased. Such an operation releases tremendous surges 
 from storage and would be difficult to link mechanically to a variable output. 
 Leakage across the capacitors is analogous to actual losses of input, so that 
 output would have to be adjusted to unity. Limited use was found for this 
 experimental development because flood-routing problems at about that same 
 time were programmed to digital computers made available to U.S. Corps of 
 Engineer offices. 
 
 Electric Analogy 
 
 The electric analogy to reservoir-type storage, upon which the multiple- 
 stage storage routing analog is based, has been described by Rockwood and 
 Hildebrand (12). Inflow is related to outflow by the continuity equation 
 
 dS 
 
 1 = 0 + Zt 
 
 and for reservoir-type storage the assumption is made that 
 
 S = T s 0 
 
 (2) 
 
 where, 
 
 I = inflow rate, 
 
 0 s= outflow rate, 
 
 S = storage volume, and 
 T s = a time constant. 
 
 9 
 
oc 
 
 o 
 
 o 
 
 10 
 
 Figure 6.--Circuit diagram of a test model analog (12) 
 
The electric counterpart is achieved by placing a resistor (R) and capacitor 
 (C) in parallel and introducing a current (1^) into the network at the junction 
 as shown in figure 7. From elementary electricity the following equations 
 hold: 
 
 v r 
 
 giving 
 
 Q = (RC)I 3 (3) 
 
 where, 
 
 R = resistance, 
 
 C r= capacitance, 
 
 V r c = voltage across resistor or capacitor, 
 
 Q = charge on capacitor, and 
 
 I , g , 3 = current in branches. 
 
 The comparison may now be made between the storage routing equations and 
 the electric analogy: 
 
 Storage routing Electric circuit 
 
 i=o + £! 1=1+42 
 
 rtr 1 2 3 4 5 “ dt 
 
 = \ + h 
 
 = Z 3 + — 
 
 3 dt 
 
 = V c , V r = 3gR, V c 
 
 S = T s 0 Q = (8 C) 
 
 Therefore, the product of (RC) is equivalent to the time constant T s or 
 T s = RC. Watershed storage and river reaches can be considered as several 
 successive reservoir-type stages of storage. This can be achieved analogously 
 by connecting several (R C) networks so that Iq for the first network becomes 
 for the second, etc. 
 
 Briefly, the project involved building a similar but more complex electric 
 analog than that developed by the Corps of Engineers at Portland (12 ) by means 
 of the following modifications: 
 
 1. The number of stages was increased from two to five, and switches pro¬ 
 vided so that any number may be used from one to five. 
 
 2. The time constant is adjusted by a variable resistor with capacitors 
 chosen to keep leakage to an insignificant level. 
 
 3. In addition to provision for expensive recorders to monitor inflow and 
 outflow, two meters were added for direct reading. 
 
 4. Batteries were replaced by a power supply incorporated into the chassis. 
 
 5. An automatic input or "memory" unit was built to store 24 preset values. 
 
 11 
 
(a) Single Stage 
 
 6 
 
 Input 
 
 O 
 
 -AAAAA- 
 
 
 T Z 
 
 i5_yvWW- 
 
 u 
 
 r 3 
 
 -^-MAAA 
 
 I 
 
 I, 
 
 t 
 
 (b) Multiple Stages 
 
 Figure 7.—Basic circuit diagrams* 
 
 12 
 
A list of parts for the multiple-stage, storage-routing analog, as built 
 by Beyers^' and reported by Chow (2) is shown below. Capacitors were selected 
 so that they would operate linearly within their rating and duplicate the 
 simple storage equation, equation 2, where T s = RC. 
 
 LIST OF PARTS FOR ANALOG MODEL 
 
 V 2 -6X4 vacuum tube--------- $ 1.65 
 
 V 2 -VR 105 gas regulator tube- 1.25 
 
 -6BA6 vacuum tube -- 2.00 
 
 T x -Pri.= 115 V, sec = 6.3V, 350 V.C.T., 50 ma. transformer- 7.35 
 
 M x -0 - 1 ma. milliammeter- 10.40 
 
 Ms-0-1 ma. milliammeter- 10.40 
 
 -Dpdt switch--— .40 
 
 4r-"'-Spdt switches- 1.00 
 
 K —- 48 m h inductance- 1.25 
 
 -9.6 m h inductance- 1.00 
 
 C x , Cg - 200 /if, 250 WVDC capacitor- 8.00 
 
 Cs- 300 /if, 200 WVDC capacitor. 2.80 
 
 £3 ,C 4 ,C6> C7 -**- 
 s C 9 - 
 
 . 
 
 500 /if, 200 WVDC capacitor 
 40 /if, 600 WVDC capacitor 
 
 25 K ohm potentiometer - 
 
 10 K ohm potentiometer - 
 
 43.20 
 
 2.50 
 
 2.50 
 
 3.75 
 
 ,<R( 5 „-5 K ohm potentiometer - 
 
 Ry-— 100 K ohm potentiometer 
 
 '- RgyRq-50 K ohm resistor- 
 
 1.25 
 
 1.25 
 
 .30 
 
 ---20 K ohm resistor--- 
 
 , R^---180 K ohm resistor-- 
 
 R -2 ---10 K ohm, 2-watt resistor 
 
 .15 
 
 .15 
 
 .85 
 
 Total 
 
 $103.40 
 
 Figure 8 shows photographs of the assembled analog model. In closeup it 
 can be seen as desk-calculator size, 10 inches high by 18 inches long and 8 
 inches deep, weighing about 20 lb. On top is mounted a separate input device 
 which feeds 24 preset values (on resistances) into the analog at 2.3-second 
 intervals for automatic operation. The large knob on the front panel also 
 serves the same purpose for hand operation where observations are noted on the 
 panel-mounted milliammeters. Along the base of the front panel are the con¬ 
 trols for setting various resistances and switches for connecting stages. 
 
 5/ Beyers, L. A. The electronic analogy to spring snow melt on a 
 watershed. M.S. Thesis, University of Idaho, Moscow. [Unpublished.] 1962. 
 
 13 
 
Front view Rear view 
 
 Figure 8.—Analog test model as viewed from front and rear. 
 
 The equipment design, as shown in figure 9, was based on a maximum inflow 
 of 1 milliampere. This inflow figure was used to determine all ratings by- 
 inserting milliammeters in the inflow and outflow circuits. These provide 
 continuous measurements from which data points can be read to reproduce the 
 outflow hydrograph. In order to permit the operator to take a minimum of 20 
 data points for each run, an RC time of 10 seconds was chosen for each of the 
 5 stages desired. 
 
 The RC values of the components in each of the stages could be filled out 
 naturally from the two choices already made. The ready accessibility of capac¬ 
 itors provided values of 200, 300, and 500 microfarads with a minimum working 
 volts rating of 200. Since the capacitance per stage must increase in each 
 succeeding stage and since leakage should be kept low, the section of series 
 200 /if. - 300 /if. - 500 /if. - 700 /Lif. - 1000 /if. was chosen. This selection 
 forced the values of resistance for each stage. Consulting the schematic, 
 figure 9, it can be found for RC = 10 sec. in each stage that: 
 
 (1) c x 
 
 (2) C 2 
 
 (3) C 3 
 
 (4) C 4 
 
 (5) C 5 
 
 200x10” 6 farads, Rs^+Rg+I^+R^^. =50xl0 3 ohms; 
 300x10”^ farads, Rrd^+I^+R^Rg =33x10 ohms; 
 500x10” ^ farads, R=^+R 4 +R g =20xlCT 3 ohms; 
 700x10”^ farads, R=R 4 +R 5 =14xlCP ohms; and 
 1000x10"^ farads, R=R 5 =10xlC)P ohms. 
 
 14 
 
Input Output 
 
 \ 
 
 n 
 
 CM 
 
 PC 
 
 rv 
 
 O 
 
 vO 
 
 o 
 
 L_ 
 
 
 11° 
 
 
 
 
 
 1 
 
 < 
 
 o 
 
 l 
 
 t 
 
 
 1 
 
 II , 
 
 
 II V 
 
 c-» r 
 
 o L 
 
 
 
 r: 
 
 r i 
 
 S t' 
 
 £ o 
 
 1 v 
 
 < l 
 
 Figure 9.--Analog circuit, multiple-stage storage routing. 
 
 15 
 
The plate current from a 6BA6 tube was chosen to provide an independently 
 controlled inflow. This tube was designed for currents in this range and was 
 selected because a pentode provides plate current independent of plate voltage. 
 A VR105 tube permits a constant voltage source for the independent grid control 
 of the plate current. 
 
 The design of this analog model was based on an overall precision of not 
 less than 95 percent. The capacitor characteristics are shown in figure 10. 
 
 Leakage currents in the electrolytic capacitors introduce calibration and 
 error considerations. The maximum voltages across the capacitors are: 
 
 C 1 = 50 volts, C 2 = 33 volts, C 3 = 20 volts, C 4 = 14 volts, and C 5 = 10 volts. 
 Leakage curves were plotted and approximated with straight lines for calibra¬ 
 tion. The maximum error through leakage is the sum of the deviations from the 
 straight lines and is insignificant in this equipment. 
 
 16 
 
Calibration of Analog 
 
 In calibration tests of discharging storage, points observed along the 
 recession curve were found to have a constant depletion ratio. The time of 
 storage (storage divided by outflow) in seconds was linearly related to the 
 product of capacitance times resistance or: 
 
 T s = R C (4) 
 
 where, 
 
 R = ohms x 10 3 , 
 
 C = farads x 10"^, and 
 
 T s = seconds. 
 
 Using this relation, the depletion of reservoir storage may be represented by 
 the usual expression in the form 
 
 Q t = Qoe‘ t/RC (5) 
 
 or alternately in the general form 
 
 Q t = QoK 1 (6) 
 
 where, 
 
 Qo = initial discharge, 
 
 Qt = discharge, (t) time units after Q 0 , 
 
 K = recession constant having a value less than one, and 
 e = Napierian base. 
 
 From a typical depletion curve, figure 11, K can be computed by reading 
 Q 0 and Q t separated by (t) units of time. Equations 4, 5, and 6 may be 
 combined, assuming t = 1, to obtain the following equations: 
 
 K = e -1 /^ (7) 
 
 or 
 
 Ts = (8) 
 
 Ln K 
 
 Calibration of the analog is for the purpose of verifying the linear 
 operating characteristics of the selected capacitors. To have a constant time 
 of storage, leakage of current across each capacitor should be at a minimum. 
 
 The circuitry of the analog was designed so that voltages would not exceed the 
 rating at which leakage becomes significant. Observations of the depletion of 
 storage, for one or all stages of the completed RC-network, show that depletion 
 ratios between 5-second current readings are essentially uniform (fig. 11). 
 
 17 
 
DISCHARGE — u AMPS. 
 
 Figure 11.--Calibration of analog by storage depletion. 
 
 18 
 
The time of storage is determined from values of capacitance multiplied 
 by the variable resistance. Assuming that 5-second time intervals have been 
 selected to represent 1-hour input, for example in one stage: 
 
 h 
 
 3.0, C , 5 =2.7, RC = 8 .1 second, equivalent T s 
 
 8.1 
 
 5 
 
 1.6 hrs. 
 
 Input to Analog 
 
 An analog computer has been designed by Robinson and Beyers^ to calcu¬ 
 late runoff increments from hourly rainfall rates (fig. 12) as input to the 
 multiple-stage storage routing analog. The computer is based on the equation 
 suggested by the Soil Conservation Service (16 ) and is stated as: 
 
 Q = —-— (9) 
 
 ^ P + S K ' 
 
 where 
 
 Q = direct storm runoff, 
 
 P = storm rainfall, and 
 
 S = potential storage or maximum difference between P and Q. 
 
 The theory of the runoff analog computer is based on three elementary 
 electrical relations: Ohm's Law and two formulas giving the equivalent re¬ 
 sistance for series and parallel resistors. These three relations in equation 
 
 form are as follows: 
 
 V = R I (Ohm's Law) (10) 
 
 R e q =R l +R s +R 3 +... (Resistors in series) (11) 
 
 Req = fis (Two resistors in parallel) (12) 
 
 Equation 9 may be written as 
 
 Q = P - . s . J L (13) 
 
 P + S 
 
 where the second term on the right in this equation is the same as equation 
 12, if S and P are interpreted as resistances. This form of the runoff 
 equation was used to derive a formula for a special case of runoff increment, 
 which may be easily generalized to apply to any increment. The derivation 
 proceeded by letting P 2 = p + pg and P 3 = p x + p 2 + p^ , where p x , p 2 , and p 3 are 
 the average rainfall for the first three time periods of the rain following 
 the initiation of runoff. Therefore, P g and P 3 represent the storm rainfall 
 (accumulated rainfall) after the second and third time periods, respectively. 
 
 j6/ Robinson, A. D., and Beyers, L. A. An analog computer for calculating 
 runoff increments from average rainfall rates. U.S. Soil and Water Cons. Res. 
 Div. Unpublished Report, 10 pp„ 1962. [Mimeographed.] 
 
 19 
 
Figure 12,--Schematic diagram of computer: Q = ..- , 
 
 20 
 
The time periods used depend upon the desires of the computer operator or the 
 available data. Using equation 13: 
 
 % 
 
 S P 3 
 P 2 + S 
 
 Pi + P 2 “ 
 
 s ( p i+ P 2 ) 
 (?!+ P 3 ) + S 
 
 (14) 
 
 and 
 
 Qs 
 
 s p 3 
 
 p 3 + s 
 
 Pl + p 2 + p 3 " 
 
 S ^ P l + P 2 + P 3 ) 
 
 (Pi + p 2 + p 3 > + s 
 
 (15) 
 
 where Q 2 and are the total runoffs (accumulated runoff) after the second and 
 third time periods, respectively. The runoff increment for the third time 
 period is given by: 
 
 p + p + 
 
 r- s (Pi + p s +p 3 ) 
 
 1 - r P + P . s <Pi + p s > i 
 
 (16) 
 
 
 J (P x + P s + p 3 ) + s 
 
 J L 1 s ( Pl + P 3 ) + s J 
 
 reduced 
 
 to: 
 
 
 
 % = p 3 
 
 - s (p,+ P2 +p 3 ) + 
 
 S(p x + P 2 ) 
 
 (IV) 
 
 (P! + P 2 + P 3 ) + S 
 
 (p x + p 3 ) + s 
 
 In developing an electrical analogy to equation 17, the assumption was 
 made that the currents labeled I in Branch 1 and Branch 2 of figure 12 are 
 equal, constant, and independent of any changes of resistance, power supply 
 voltages, or brush positions. The reason for the constant current devices is 
 to insure this. Another assumption was that both resistances labeled = £p 
 are zero. Referring to the circuit in figure 12 and using equations 11 and 12, 
 the equivalent resistance between points A and D is found to be 
 
 R eqAD 
 
 R + (%+ Re) R 
 
 (R 1 + Bg) + R 
 
 and between points B and D to be 
 
 ^BD 
 
 (%+ Re+ Rj) R 
 
 («!+ V + R 
 
 (18) 
 
 (19) 
 
 Applying equations 10 and 18, the potential difference between points A and D 
 is found to be 
 
 Vt = I [~IL + (Ri + Rg) R 1 
 L 3 (R 1+ Rg) + R J 
 
 and between points B and D to be 
 
 v 2 . i r (R ^ + w R i 
 
 L (R l + Rg+ ft,) + R J 
 
 21 
 
As can be seen from equations 10 and 19, the potential difference between 
 points A and B, which is one of the output voltages of the computer, is 
 
 V x - V Q = 
 
 = i [% 
 
 + («1+ 1*8) R 
 
 (%+ Bg) + R 
 
 ]- X [- 
 
 (Ri+ Be+ %) R 
 
 (R x + FgH- Rj) + R 
 
 ] 
 
 (20) 
 
 This can be written as 
 
 (8!+ Rg) R 
 («!+ Eg) + R 
 
 (R x + Rg+ R;) R 
 (R 1 + 1^+ Rj) + R 
 
 ( 21 ) 
 
 If the pairs of resistors labeled R^^, Rg, and , and R are chosen such that 
 R = Pi» "Pg* ^ = p 3 , and R = S, equation 21 may be written as 
 
 = 1 [P 3 + 
 
 (Pi + P 3 > S 
 (P!+ P 2 ) + S 
 
 . (Pi+ p s + p 3 ) s -| 
 
 ^ P l + P S + p 3 ) + S 
 
 ( 22 ) 
 
 An examination of equations 17 and 22 shows that V 1 - V 2 is proportional to 
 - Q 2 . This fact is the basis for the design of the computer. 
 
 The above derivation is for the brush positions shown in figure'12. If, 
 for example, the brushes labeled C and A are on commutator segments 5 and 6, 
 respectively, in Branch 1, and if at the same time the brush labeled B is on 
 commutator segment 6 in Branch 2, then the output voltage is proportional to 
 Q6"Q5® Sweeping the brushes at a uniform rate will, therefore, produce a 
 voltage output which is a step function. The height of the individual step is 
 the runoff increment, and the width of the step is the time unit. This is the 
 same as the time unit over which the inputs are averaged. 
 
 In deriving equation 22 it was assumed that the resistances labeled 
 Rjl = £p in figure 12 were zero. This would be the case if the analog were 
 being programmed to start at the beginning of a rainfall. When the computer 
 is programmed to start at a time other than the beginning of the rainfall, the 
 two resistances Rjj are set equal to the accumulated P at this time. This 
 feature also extends the total rainfall time for which the computer can be 
 us'ed. 
 
 Operation of Analog 
 
 Operation of the analog model is very simple for reproducing recorded 
 hydrographs from small watersheds. Given the time distribution of rainfall 
 for routing of excess rainfall into a downstream hydrograph, output can be 
 computed from input by the proper choice of only three values: 
 
 1. The potential storage for solution of equation 9. 
 
 22 
 
2. The time of storage , T s , to determine the RC value so that resistances 
 can be set on the analog. 
 
 3. The number of stages or the lag time which may be estimated from the 
 length of channels, slope, area, etc. 
 
 By plugging a recorder in the output jacks of the analog, the computed 
 hydrograph is quickly obtained and after a few trial-and-error solutions the 
 best choice of arbitrary values is soon indicated. Only one choice of RC 
 values, number of stages, and potential storage will best duplicate the avail¬ 
 able hydrographs in general shape and time of occurrence. These values can be 
 said to have scaled the analog so that it is now a model of that particular 
 watershed. Thus, any other rainfall record could be converted into a hydro¬ 
 graph from that watershed or any other ungaged drainage with similar soil, 
 cover, geology, drainage density, slope, area, etc. 
 
 EXAMPLES OF HYDROGRAPHS 
 
 The practical application of this analog model can be illustrated by a 
 sample computation (table 1) and by typical hydrographs from the South Fork of 
 the Palouse River from Moscow, Idaho, to Pullman, Wash., as shown by figures 
 13 and 14* Hourly rainfall values from recording gages on the watershed were 
 first tabulated, then converted to runoff (i.e. excess precipitation) by use 
 of equation 9, in the form of EQ = EI4-S, after estimating the potential storage 
 (S). Finally, the increments of runoff (AQ) were entered into the analog after 
 conversion to milliamperes and routed through a number of stages and time of 
 storage so that the final downstream generated hydrograph agrees with the 
 actual. For reproduction of hydrographs on tributaries of the Palouse River, 
 a value of T s ranging from 5 to 10 hours and the use of 5 to 10 stages are 
 required for drainage areas of 34 to 84 square miles. The physical signifi¬ 
 cance of the time of storage, T s , may be characteristic of land form, soil 
 mantle, and cover type. The number of stages indicates the length of channel-- 
 that is, the number of routing reaches. 
 
 CONCLUSIONS 
 
 The application of electric analog methods to the solution of a complex 
 network of channels and reservoirs is relatively new in the United States. 
 Because the future of these devices seems promising for both ground water and 
 surface routing, a test model of multiple stages of linear reservoirs was 
 assembled simply and inexpensively. Recommendations resulting from testing 
 this model analog are: 
 
 1. The number of stages should be increased to 10 or more. 
 
 2. Leakage currents could be decreased by greater capacitance or by 
 more expensive capacitors. 
 
 3. The time of storage should be varied exponentially. 
 
 23 
 
TABLE 1.— Analog hydrograph computations on Palouse River , 
 Fourmile Creek at Shawnee, Wash., April 20, 1937 
 
 
 Watershed 
 
 Analog 
 
 Hydrograph 
 
 Time 
 
 t 
 
 Rain 
 
 Runoff-^ 
 
 Input—/ 
 
 Output—^ 
 
 Computed^/ 
 
 
 EP 
 
 2Q • 
 
 
 Z\Q lX 50 
 
 
 AQg -f 50 
 
 
 In. 
 
 In. 
 
 In. 
 
 Ma. 
 
 Ma. 
 
 In./hr. 
 
 
 
 
 0.00002 
 
 0.00 
 
 0.00 
 
 0.0000 
 
 1000 
 
 0.01 
 
 0.00002 
 
 .00017 
 
 .01 
 
 .00 
 
 .0000 
 
 1100 
 
 .03 
 
 .00019 
 
 
 
 
 
 
 
 
 .00034 
 
 .02 
 
 .00 
 
 .0000 
 
 1200 
 
 .05 
 
 .00053 
 
 .00082 
 
 .04 
 
 .00 
 
 .0000 
 
 1300 
 
 .08 
 
 .00135 
 
 .00165 
 
 .08 
 
 .00 
 
 .0005 
 
 1400 
 
 .12 
 
 .0030 
 
 
 
 
 
 1500 
 
 .17 
 
 .0060 
 
 .0030 
 
 .15 
 
 .01 
 
 .0007 
 
 
 s 
 
 
 .0022 
 
 .11 
 
 .02 
 
 .0009 
 
 1600 
 
 .20 
 
 .0082 
 
 
 
 
 
 
 S) 
 
 
 .0066 
 
 .33 
 
 .03 
 
 .0011 
 
 1700 
 
 .27 
 
 .0148 
 
 
 
 
 
 
 
 
 .0166 
 
 .83 
 
 .04 
 
 .0013 
 
 1800 
 
 .40 
 
 .0314 
 
 .0149 
 
 .79 
 
 .07 
 
 .0019 
 
 1900 
 
 .49 
 
 .51 
 
 .0463 
 
 .0037 
 
 .19 
 
 .11 
 
 .0026 
 
 2000 
 
 .0500 
 
 .0020 
 
 .10 
 
 .18 
 
 .0041 
 
 
 2100 
 
 .52 
 
 .0520 
 
 —— 
 
 .. 
 
 .24 
 
 .0053 
 
 2200 
 
 •• •“ 
 
 
 m mm 
 
 .. 
 
 .27 
 
 .0059 
 
 2300 
 
 * •" 
 
 
 mm mm 
 
 
 .27 
 
 .0059 
 
 2400 
 
 
 
 -- 
 
 -- 
 
 .24 
 
 .0053 
 
 1/ EQ 
 
 EF 5 
 
 " EP + 4.68 
 
 2/ Factor for full-scale deflection of one milli-amp. = 50. 
 3/ 10 stages at 2.5 seconds = 25 seconds, RC = 2.5 seconds. 
 
 4/ AQg/50 + 0.0005 in./hour. 
 
 T s = 5 hours. 
 
 24 
 
Discharge -Whr. Ppt . . ln . /hr> 
 
 Figure 13.—Hydrographs: Fourmile Creek at Shawnee, Wash. (71.6 sq. mi.) 
 
 25 
 
Discharge - in./hrs. Ppt. - in./hr 
 
 Figure 14.—Hydrographs: South Fork Palouse River at Pullman, Wash. 
 •(84.4 sq. mi.) 
 
 26 
 
The analog should be thought of, not as a mathematical computer, but as 
 a "tool to think with" in terms of the physical system. The operation of the 
 analog model is not based on man-devised algebra, as is that of the digital 
 computer, but is governed by the physical laws of the electrical circuits. 
 
 An outstanding advantage of the analog model is that it can be con¬ 
 structed of standard radio parts and can be operated by personnel with only 
 routine engineering training. The use of the analog model is a never-ending 
 game; there must always be feedback between the user and the system. 
 
 Some advantages of this multiple-stage, storage-routing analog are: 
 
 1. Flexible operation to meet variable storage-discharge relations. 
 
 2. Application to all routing where analytical or graphical solutions 
 were previously required. 
 
 3. Success in routing through channels or basin storage for rain or 
 snowmelt. 
 
 4. A saving in operating time by replacing tedious graphical methods, 
 thus making possible many trial routings that otherwise would not 
 be attempted. 
 
 5. Simplicity and cost of about $100 for parts. 
 
 An improved analog model would be particularly useful for: 
 
 1. Calibrating experimental watersheds before new rainfall or runoff 
 data are available. 
 
 2. Forecasting water supplies or estimating design floods. 
 
 3. Demonstrating routing procedures to local groups. 
 
 ACKNOWLEDGMENT 
 
 The author wishes to express his appreciation to Professor A. D. Robinson 
 and L. A. Beyers, Physical Sciences Department, University of Idaho, Moscow, 
 for their assistance in development of the model analog. 
 
 LITERATURE CITED 
 
 (1) Bruce, W. A. 
 
 1943. An electrical device for analyzing oil-reservoir behavior. Amer. 
 Inst. Mining and Metall. Engin. Trans. 151: 73-85. 
 
 (2) Chow, Ven Te [ed.]. 
 
 1964. Handbook of applied hydrology. McGraw-Hill, New York. Chapt. 29, 
 pp. 10 and 11. 
 
 (3) Dziatlik, H. 
 
 1955. Electrical model for a river. Przeglad Elektrotechniczny 31(7): 
 447-455. 
 
 (4) Glover, R. E. 
 
 1953. Electrical analogies and electronic computers. Amer. Soc. Civ. 
 Engin. Trans. 118: 1010-1027. 
 
 27 
 
(5) Halek, V. 
 
 1958. Hydraulic problems using electric analogy. Vodni Hospodarstvi, 
 No. 10, pp. 297-305. 
 
 (6) Harder, J. A., Mockros, Lyle, and Nishizaki, Ray. 
 
 1960. Flood control analogs. Water Resources Center, Cont. No. 24, 
 University of California, Berkeley, 40 pp. 
 
 (7) Ishihara, T., and Ishihara, Y. 
 
 1955. On the electronic analog computer flood routing. Japan Soc. 
 
 Civ. Engin. Trans., No. 24: 44-57. 
 
 (8) Ishihara, Y. 
 
 1957. On the application of an electronic analog computer for flood 
 routing to actual rivers. Japan Soc. Civ. Engin. Trans., No. 
 43:43-47. 
 
 (9) Linsley, R. K., Foskett, L. W., and Kohler, M. A. 
 
 1948. Electronic device speeds flood forecasting. Engin. News Record 
 141(26):64 and 66. 
 
 (10) Messerle, H. K. 
 
 1953. Electronic high speed simulation of hydraulics. Inst. Engin. 
 Jour. 25(3):35-41. 
 
 (11) Paynter, H. M. 
 
 1952. Methods and results, M.I.T. studies in unsteady flow. Boston 
 Soc. Civ. Engin. Proc. 39(2):120-165. 
 
 (12) Rockwood, D. M., and Hildebrand, C. E. 
 
 1956. An electronic analog for multiple stage reservoir type storage 
 routing. U.S. Corps of Engin. Tech. Bui. No. 18, 12 pp. 
 
 (13) Rogers, A. E., and Connolly, T. W. 
 
 1960. Analog computation in engineering design. McGraw-Hill, New York, 
 450 pp. 
 
 (14) Scheidegger, A. E. 
 
 1960. The physics of flow through porous media. Macmillan, New York, 
 313 pp. 
 
 (15) Tribus, N. 
 
 1958. The use of analogs and analog computers in heat transfer. 
 
 Okla. Engin. Expt. Sta. Pub. 100, p. 2. 
 
 (16) U.S. Soil Conservation Service. 
 
 1964. National engineering handbook. Sec. 4, Hydrology, Part 1, 
 
 Watershed Planning, U.S. Soil Conservation Service, p. 10.3. 
 
 28 
 
UNIVERSITY OF ILLINOIS-URBANA 
 
 3 0112 099059351