# Resistor Network Solver for the HP-67 Programmable Calculator

November 18, 2008

I acquired my HP-67 at the beginning of 2008, and it sat on the shelf until November when I finally found the time to rebuild the battery pack and overhaul the card reader. I then set out to enter a program I’d written for the HP-67 earlier in the year (and debugged on an HP-41C, after hand-translating it).

I wrote this program because I often need to determine voltages at the nodes of simple resistor networks (for example, while designing my USB Powered AA Battery Charger). This program can find node voltages in networks of one or two nodes, with up to five resistors and/or constant current sources per node. The following is an example of such a network:

The program uses Kirchoff’s Current Law together with Ohm’s Law to compute the voltage at the node(s). Kirchoff’s law states that the sum of all the currents flowing into a node must equal zero. In other words, for every Amp of current that flows in, an Amp of current must flow out. Ohm’s Law states that the current flowing through a resistor equals the voltage across the resistor divided by the resistance.

For the single node case, finding the node voltage for a network with `N` resistors and `M` current sources is a matter of solving the following equation for `V _{node}`:

Fortunately, the solution is rather simple, and is well suited to being computed by a programmable calculator such as the HP-67:

Solving a problem with two nodes is an iterative process of alternately solving two problems with one node each. First we solve the problem for one of the nodes, treating the the other node as a voltage source and guessing its voltage (any reasonable guess will do). This gives an estimate of the voltage of the first node. We can then solve the second node the same way, treating the first node as a voltage source having the voltage we just estimated, and giving an estimate for the voltage of the second node. Now we can go back and solve the first node again and we’ll get a better estimate of its voltage. We can continue like this, going back and forth, getting ever closer estimates of the node voltages. When the estimates stop changing at the level of precision we need, we’re done.

## Using the Program

First type in the program and save it, or read it from a previously recorded magnetic card. The card should be labelled as follows:

RESISTOR NETWORKS OF 1 OR 2 NODES | ||||
---|---|---|---|---|

DROP | 1↔2 | CLEAR | i→I_{i} |
i→P_{i} |

V,_{i}R_{i} |
I_{i} |
→V_{node} |
→I,_{1}I,…_{2} |
→P,_{1}P,…_{2} |

### Single Node Problems

Consider a single node network consisting of two resistors and a constant current source as shown:

Solve the problem using the following steps:

Description | Keystrokes | Display |
---|---|---|

Select engineering notation | h ENG DSP 2 |
0.00 00 |

Clear any existing data | f c | 0.00 00 |

Enter V and _{1}R_{1} |
5 ENTER 56 EEX 3 A |
1.00 00 |

Enter V and _{2}R_{2} |
0.3 ENTER 27 EEX 3 A |
2.00 00 |

Enter I_{3} |
0.1 EEX 3 CHS B |
3.00 00 |

Compute V (Volts)_{node} |
C | 3.65 00 |

Compute I (Amps)_{1} |
D | 24.1 -06 |

Compute I_{2} |
R/S | -124. -06 |

Compute (recall) I_{3} |
R/S | 100. -06 |

Compute P (Watts)_{1} |
E | 32.5 -06 |

Compute P_{2} |
R/S | 416. -06 |

Compute P (not applicable)_{3} |
R/S | 0.00 00 |

Look up I again_{2} |
2 f d |
-124. -06 |

Look up P again_{2} |
2 f e |
416. -06 |

Notice that the computed value of `I _{2}` is negative, meaning that current if flowing out of the node through

`R`.

_{2}Since `I _{3}` was entered as a fixed current as opposed to a voltage or resistor, computing

`I`just returns the entered value, and computing

_{3}`P`returns zero.

_{3}If you make a mistake when entering an input to a node (a voltage/resistance pair or a current), you can remove the most recently entered input using the DROP command (press `f a`

).

### Two Node Problems

Consider the following two node network:

Each node has four connections. The leftmost node (`V _{node1}`) is connected by resistors

`R`,

_{1ab}`R`,

_{2a}`R`, and

_{3a}`R`to voltages

_{4a}`V`,

_{node2}`V`,

_{2a}`V`, and ground respectively.

_{3a}The rightmost node also has four connections, three resistors and a constant current input of 0.1mA. The resistors `R _{1ab}`,

`R`, and

_{2b}`R`connect to voltages

_{3b}`V`,

_{node1}`V`,

_{2b}`V`respectively.

_{3b}To solve this problem, first consider the leftmost node in isolation, and partially solve it using the following steps:

Description | Keystrokes | Display |
---|---|---|

Clear any existing data | f c | 0.00 00 |

Enter guess for V, and _{node2}R_{1ab} |
0 ENTER 68 EEX 3 A |
1.00 00 |

Enter V and _{2a}R_{2a} |
5 ENTER 56 EEX 3 A |
2.00 00 |

Enter V and _{3a}R_{3a} |
0.3 ENTER 27 EEX 3 A |
3.00 00 |

Enter V (ground = 0V) and _{4a}R_{4a} |
0 ENTER 47 EEX 3 A |
4.00 00 |

Estimate V (Volts) assuming _{node1}V = 0_{node2} |
C | 1.10 00 |

Now solve the rightmost node, using the estimated voltage of the left node:

Description | Keystrokes | Display |
---|---|---|

Switch to node 2. Display of 1.00 indicates that R is already entered._{1} |
f b | 1.00 00 |

Enter V and _{2b}R_{2b} |
6 ENTER 33 EEX 3 A |
2.00 00 |

Enter V and _{3b}R_{3b} |
3 ENTER 22 EEX 3 A |
3.00 00 |

Enter I_{4b} |
0.1 EEX 3 CHS B |
4.00 00 |

Estimate V assuming _{node2}V = 1.10_{node1} |
C | 4.80 00 |

Finally, alternate between the two nodes, recomputing `V _{node}` until the values stop changing:

Description | Keystrokes | Display |
---|---|---|

Switch to node 1 | f b | 4.00 00 |

Estimate V (Volts) assuming _{node1}V = 4.80_{node2} |
C | 1.88 00 |

Switch to node 2 | f b | 4.00 00 |

Estimate V assuming _{node2}V = 1.88_{node1} |
C | 4.93 00 |

Switch to node 1 | f b | 4.00 00 |

Estimate V (Volts) assuming _{node1}V = 4.93_{node2} |
C | 1.90 00 |

Switch to node 2 | f b | 4.00 00 |

Estimate V assuming _{node2}V = 1.90_{node1} |
C | 4.93 00 |

Switch to node 1 | f b | 4.00 00 |

Estimate V (Volts) assuming _{node1}V = 4.93_{node2} |
C | 1.90 00 |

The two `V _{node}` values stop changing when

`V`= 1.90V and

_{node1}`V`= 4.93V, so that is the final solution.

_{node2}Just as with the single-node solution, you can view and review the currents and power dissipation through each resistor connected to the currently selected node.

## Program Listing

Line | Instruction | Comments |
---|---|---|

001♦ | LBL a | Back up by one input |

002 | RC I | |

003 | 2 | |

004 | x>y? | Don’t back up if we’re already at the beginning |

005 | RTN | |

006 | − | |

007 | ST I | |

008 | SF 1 | Must recompute V_{node} |

009 | GTO 1 | Display number of inputs entered |

010♦ | LBL B | Enter a current as V=_{i}I, _{i}R=0_{i} |

011 | 0 | (0 I)_{i} |

012♦ | LBL A | Enter a voltage,resistance pair, with X=R, Y=_{i}V_{i} |

013 | RC I | Test that we’re not out of space |

014 | 2 | |

015 | 0 | |

016 | − | |

017 | x=0? | |

018 | 1/x | Force an error if we’re out of space |

019 | R↓ | (R _{i}V)_{i} |

020 | x↔y | (V _{i}R)_{i} |

021 | STO (i) | Store V in R0, R2, R4, …_{i} |

022 | ISZ I | |

023 | x↔y | (R _{i}V)_{i} |

024 | STO (i) | Store R in R1, R3, R5, …_{i} |

025 | ISZ I | |

026 | SF 1 | Must recompute V_{node} |

027♦ | LBL 1 | Display number of inputs entered |

028 | RC I | |

029 | 2 | |

030 | ÷ | |

031 | RTN | |

032♦ | LBL C | Compute (if necessary) and display V_{node} |

033 | F? 1 | Need to recompute? |

034 | GTO 5 | |

035♦ | LBL 3 | Compute numerator / denominator |

036 | RCL B | Σ V/_{i}R + Σ _{i}I_{i} |

037 | RCL C | Σ 1/R_{i} |

038 | ÷ | V_{node} |

039 | STO D | Save for later use |

040 | CF 1 | Record that we’ve computed it |

041 | RTN | |

042♦ | LBL 5 | Compute numerator and denominator |

043 | RC I | |

044 | STO A | Save I for re-use |

045 | CLx | Clear numerator and denominator |

046 | STO B | |

047 | STO C | |

048 | ST I | Clear I (original value was saved in A) |

049♦ | LBL 2 | Loop to sum numerator and denominator |

050 | RC I | |

051 | RCL A | |

052 | x=y? | Are we done? |

053 | GTO 3 | |

054 | RCL (i) | (V)_{i} |

055 | ISZ I | |

056 | RCL (i) | (R _{i}V)_{i} |

057 | ISZ I | |

058 | x≠0? | Is this a voltage,resistance pair? |

059 | GTO 4 | |

060 | R↓ | X=0 means a current is in Y |

061 | RCL B | Add current to numerator |

062 | + | HP-67/97 can’t do STO+ B |

063 | STO B | |

064 | GTO 2 | Next input |

065♦ | LBL 4 | Process voltage,resistance pair |

066 | ENTER | (R _{i}R _{i}V)_{i} |

067 | 1/x | (1/R _{i}R _{i}V)_{i} |

068 | RCL C | Add 1/R to denominator_{i} |

069 | + | |

070 | STO C | |

071 | R↓ | (R _{i}V)_{i} |

072 | ÷ | (V/_{i}R)_{i} |

073 | RCL B | Add V/_{i}R to numerator_{i} |

074 | + | |

075 | STO B | |

076 | GTO 2 | Next input |

077♦ | LBL D | Display I … _{1}I_{N} |

078 | SF 0 | Indicates we want I, not _{i}P_{i} |

079 | CLx | |

080 | STO E | We’re using E as an index in this loop |

081 | GTO 0 | Use same loop as subroutine E |

082♦ | LBL E | Display P .. _{1}P_{N} |

083 | CF 0 | Indicates we want P, not _{i}I_{i} |

084 | CLx | |

085 | STO E | |

086 | LBL 0 | Entry point for subroutine E to call us |

087 | RCL E | |

088 | RC I | |

089 | x=y? | Exit if we’re done |

090 | GTO 1 | Display number of inputs |

091 | F? 0 | If called from subroutine D, set flag 2 |

092 | SF 2 | |

093 | GSB 8 | Compute I (or _{i}P) for input specified in register E_{i} |

094 | R/S | Stop to display result |

095 | RCL E | Point to next voltage,resistance pair |

096 | 2 | |

097 | + | |

098 | STO E | |

099 | GTO 0 | Process next pair |

100♦ | LBL d | Compute I for i=X_{i} |

101 | SF 2 | |

102♦ | LBL e | Compute I (if flag 2 set) or _{i}P for i=X_{i} |

103 | 1 | |

104 | − | i-1 |

105 | 2 | |

106 | × | 2(i-1) |

107 | STO E | Set E to point to V,_{i}R pair_{i} |

108♦ | LBL 8 | Compute I based on index in E (also leaves _{i}R in Y)_{i} |

109 | F? 1 | Do we need to recompute V?_{node} |

110 | GSB 5 | Recompute denominator of solution |

111 | RCL E | |

112 | x↔I | |

113 | STO A | |

114 | RCL (i) | (V)_{i} |

115 | ISZ I | |

116 | RCL (i) | (R _{i}V)_{i} |

117 | x=0? | Is this a current instead of a voltage,resistance pair? |

118 | GTO 6 | |

119 | x↔y | (V _{i}R)_{i} |

120 | RCL D | (V _{node}V _{i}R)_{i} |

121 | 0 | (0 V _{node}V _{i}R)_{i} |

122 | + | (V _{node}V _{i}R _{i}R)_{i} |

123 | − | (V–_{i}V _{node}R _{i}R _{i}R)_{i} |

124 | x↔y | |

125 | ÷ | (I _{i}R _{i}R _{i}R)_{i} |

126 | GTO 7 | |

127♦ | LBL 6 | This input is a fixed current |

128 | x↔y | (I 0)_{i} |

129♦ | LBL 7 | Restore saved value of I |

130 | RCL A | |

131 | ST I | |

132 | R↓ | (I _{i}R)_{i} |

133 | F? 2 | Do we only want I? (This test clears the flag)_{i} |

134 | RTN | |

135 | x^{2} |
(I^2 _{i}R)_{i} |

136 | × | P_{i} |

137 | RTN | |

138♦ | LBL c | Clear all data |

139 | CLREG | |

140 | P↔S | |

141 | CLREG | |

142 | GTO 1 | Display number of inputs so far (zero) |

143♦ | LBL b | Prepare to swap nodes |

144 | 9 | |

145 | RC I | |

146 | − | |

147 | x<0? | Throw an error if more than 5 inputs |

148 | √x | |

149 | RC I | Save number of inputs in V of active node_{1} |

150 | STO 0 | |

151 | RCL 1 | Get R to the stack before swapping_{1} |

152 | P↔S | Swap networks |

153 | STO 1 | Set R equal to _{1}R from the other node_{1} |

154 | 2 | |

155 | RCL 0 | Retrieve number of inputs of other node |

156 | x=0? | New node? |

157 | + | Skip R,_{1}V since they are copied from other node_{1} |

158 | ST I | |

159 | RCL D | Set V equal to _{1}V from the other node_{node} |

160 | STO 0 | |

161 | SF 1 | Force recomputation |

162 | GTO 1 | Display number of inputs to this node |

An interesting feature of this program is that it takes advantages of the HP-67’s primary and secondary register sets when solving problems with two nodes. Most of the work of switching from one node to the other is done by a `P↔S`

instruction (additional instructions are used to keep R1 consistent between the two nodes, and to save the number of inputs of the inactive node in its V1 register).

## Registers and Flags

Register | Use |
---|---|

0, 2, 4, …, 18 | V or _{i}I_{i} |

1, 3, 5, …, 19 | R or 0_{i} |

A | Holds N during computation of V_{node} |

B | Accumulator for numerator of solution |

C | Accumulator for denominator of solution |

D | Computed value of V_{node} |

E | Loop index |

Flag | Meaning |
---|---|

0 | Selects between computing I or _{i}P_{i} |

1 | Indicates that V needs to be recomputed_{node} |

2 | Temporarily selects between computing I or _{i}P_{i} |

## Revision History

2008-Nov-18 — Initial release.

## Possible Improvements

The process for solving a network with two nodes is a bit tedious, because the user must manually switch back and forth between the two nodes (by pressing `f b`

) and compute the node voltage (by pressing `C`

). It is up to the user to decide when the solution has stabilized. This sort of thing is easy for a programmable calculator to do, but the program uses all the calculator’s memory registers already, so none are left to perform the necessary book-keeping to automatically control such an iteration. Hopefully, I’ll think of some clever “trick” to get around this problem. If any one reading this has a suggestion, please leave a comment below.

## Related Articles

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- Low-Sensitivity Sallen-Key Filter Design with the HP-41C Programmable Calculator
- Low-Sensitivity Sallen-Key Filter Design with the HP-67 Programmable Calculator
- Op-Amp Gain and Offset Design with the HP-67 Programmable Calculator
- Op-Amp Oscillator Design with the HP-67 Programmable Calculator
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Pedro Daniel LEIVA

July 22, 2013

Dear Dr. Stefan Vorkoetter:

Im an Agronomist researcher working for the Argentine government.

Since 1976 I become an HP calculator user, from HP 25C (1976), HP-67 (later) and HP 35s (at present time). Also, some thing else in common, my relation with aviation is by pesticide application

I enjoy very much your high quality programming techniques, and would like to contact you by email

Congratulations for your excellent job

Sincerely, Daniel