Introduction

A generally encountered school-level Physics sensible is the willpower of the interior resistance of a battery – sometimes an AA or D cell. Usually that is based mostly round a easy mannequin of such a cell as a supply emf in collection with a small resistor. The cell is linked to a resistive load and (within the easiest case the place load resistance is understood) solely open circuit voltage and cargo volts must be measured.

Then: $$r_i=frac{V_I}{I} =frac{Delta V}{I} $$ the place $$Delta V =  V_{OC}-V_{L}$$ and $$I=frac{V_L}{R_L} $$.

It must be famous that the above illustration of a battery as a easy fixed voltage supply in collection with a small resistor is a handy simplification of fairly complicated battery chemistry. In follow, one finds that neither the open circuit voltage (emf) nor the interior resistance stays fixed as is assumed on this mannequin. Each fluctuate relying upon the “state of cost” (SOC) or “depth of discharge” (DOD) of the battery. So experiments to find out inner resistance must be confined to masses for which there’s minimal or negligible discharge throughout measurement. There are extra complicated electrical fashions of battery chemistry and these will higher approximate the battery’s time-dependent electrical output throughout discharge.

On this article, we start by describing difficulties skilled making an attempt the direct measurement described above. We then talk about the “traditional” faculty experiment for the willpower of inner resistance and develop some approximation formulation. These could also be used to find out circuit parameters from regression evaluation of present vs conductance or load voltage vs conductance in addition to the “conventional”  load voltage vs amps.

Lastly, we revert to the unique easy technique of direct measurement albeit by exploiting the “battery check” setting discovered on (some) multimeters. This setting does not more than present a resistive load for the battery below check and the “battery check” voltage studying is subsequently a load voltage. As meter leads carry present from the battery terminals via the load, we have now to take into consideration lead resistance when utilizing this technique.

Experimental Temporary

Within the specific “experimental transient” given to some college students attending classes right here, inner resistance was to be decided utilizing a “home made” resistor of roughly ##;4 Omega## for the load.  Measurements of open circuit voltage, terminal (load) voltage, and present had been to be made with a single multimeter.  One quickly found that an ##approx 4 Omega## resistor linked to a typical AA cell would draw a substantive present and that measurement of terminal voltage was problematic on account of that voltage dropping even because the measurement was taken. Present measurements suffered equally.  Then again, if load resistance had been elevated to attempt to ‘stabilize’ the readings, one would discover there was scarcely any terminal voltage drop. So the requirement is to attempt to discover a resistance or resistance vary whereby the present drawn over the time span of measurement wouldn’t considerably have an effect on open circuit (and therefore terminal) voltage while on the similar time offering a measurable terminal voltage drop.

Revised Experimental Temporary

The traditional faculty experiment to find out the interior resistance of a cell doesn’t require the measurement of open circuit voltage since this parameter is as a substitute decided from linear regression. It’s the y-intercept of a straight line having a gradient equal to the cell’s inner resistance based on the equation: $$V_L = textbf{ℇ}  – Ir_i$$ Right here ##I## and ##V_L## are the x and y variables respectively with ##textbf{ℇ}## – the cell emf – and ##r_i## – inner resistance – being obtained as regression coefficients of the y-intercept and gradient respectively. In accordance with the unique experimental transient (use a “home made” resistor), the present was assorted through the use of an uncovered pencil lead as a  variable resistor. A second pencil lead was used as a present limiting resistor (##R ge 8.8 Omega##). It was not anticipated that college students would have a couple of multimeter at their disposal.

Strictly talking, it is just essential to take present and voltage readings however right here we embody resistance readings as effectively. We have to talk about why there’s a discrepancy between the measured resistance values (first column of the desk beneath) and Ohm’s legislation calculated values  (final column of the desk beneath).  One other salient level about this dataset is that the open circuit voltage has remained roughly fixed all through the set of readings. This is essential since a continuing cell emf is assumed within the equation ##V_L = textbf{ℇ}  – Ir_i.##

Opposite to the unique experiment transient (indicating using just one meter), it’s essential to make use of two meters – one for the amps and one for the volts. The reason being that readings of voltage and present must be taken concurrently to make sure that the pairs match up appropriately. If (for instance) a multimeter was used first to measure volts after which (individually) amps, the readings would NOT match up as a result of when the multimeter measures present, it routinely introduces non-negligible resistance into the circuit.  This experimental element was noticed by the writer. Nonetheless, the explanation was not appreciated till being suggested about it in a PF dialogue thread on the identical matter (battery inner resistance).

Desk of Measurements and Calculated Values

Regression Evaluation:  Measured  vs Ohm’s Legislation calculated Resistance

A plot of calculated resistance vs measured resistance yields (as anticipated) a gradient ##approx 1## however there’s a particular “offset” or y-intercept worth (##approx 1.9 Omega##) which this writer (anyway!) couldn’t discover any rationalization for.

https://www.desmos.com/calculator/mw5bq32df9

A question was placed on PF and the next rationalization was obtained from PF person @TomG:

You talked about that you just used the identical meter to measure the present. The Present vary of a digital ammeter has some low, non-zero, inner resistance, which might add to your efficient load resistance. Has this been taken into consideration?

https://www.physicsforums.com/threads/battery-internal-resistance.1016677/post-6650450

So the offset indicated within the regression evaluation above arises from the truth that when the present was measured, a further resistance (that of the ammeter) was launched into the circuit. Once we plot load voltage vs present, this doesn’t matter because the measured present is for the whole circuit resistance and corresponds to an identical load voltage measurement. Nonetheless, we should always definitely be asking questions as to why Ohm’s legislation calculation of resistance yields values which are almost 2 ohms totally different from the measured values.

Even with an ##8.8;Omega## present limiting resistor, there have been nonetheless issues with fluctuating voltage measurements when utilizing a single AA cell. For that reason, the above dataset was measured with 2 comparable AA cells in parallel on the understanding that the interior resistance obtained could be half that for a single AA cell. Nonetheless, this setup isn’t ideally suited since it’s in no way assured that even comparable AA cells can have the identical inner resistance.  The parallel connection additionally introduces issues with ‘fitment resistance’ – a subject to be additional mentioned on this article. Be that as it might, it’s nonetheless instructive to look at graphs and regression evaluation of the information obtained.

Regression Evaluation: Load Voltage vs Present

https://www.desmos.com/calculator/3cqsrmwbvv

The plot above is typical of what one would possibly count on from the “traditional faculty experiment” described within the video hyperlink above. For the parallel pair the regression fitted ‘trendline’ exhibits an inner resistance of ##approx 0.235 ;Omega## and a (parallel) cell emf of  ##approx 1518.3; mV##. The latter could be very near the measured open circuit voltage(s) within the dataset above.  The discrepancy (about 3 milli-volts) is most certainly attributable to “fitment resistance”. When making an attempt to find out a resistance worth as little as that obtained above, one must be very conscious of varied small resistance values that might have an effect on the worth obtained. For instance, one set of check leads that I used has a resistance of about ##0.113 Omega## and one has to rigorously take into account when and when not lead resistance will impression upon the willpower of an AA cell’s inner resistance.

Regression Evaluation: Present vs Conductance

Think about the equation: $$ I = frac {textbf{ℇ}}{R+r_i}=frac {textbf{ℇ}}{Rleft(1+frac{r_i}{R}proper)}approx frac{textbf{ℇ}}{R}left(1-frac{r_i}{R}proper).$$ The approximation is obtained from a one-term binomial enlargement of ##left(1+frac{r_i}{R}proper)^{-1}## and shall be more and more correct the smaller ##r_i## is towards R – ie for ##r_i<<R##. When it comes to conductance G (reciprocal of resistance), we might re-write the approximation as $$Iapproxtextbf{ℇ}G-textbf{ℇ}r_i G^2$$ and a quadratic regression evaluation of the plot I vs G ought to yield the identical parameters (##r_i## and ##textbf{ℇ}##) because the linear mannequin. To acquire conductance readings, we use the present dataset values by calculating the quotient ##frac{I}{V_L}##. We beforehand indicated why there could be an issue with the extra direct technique of calculating conductance because the reciprocal of resistance. All the identical, this specific regression evaluation is extra suited to figuring out inner resistance from direct measurements of present and conductance. Maybe through the use of a non-intrusive Corridor impact present sensor.

https://www.desmos.com/calculator/qhojl7ly1k

The graph is strikingly linear and we should always count on that since from the equation the time period ##textbf{ℇ}G## (representing a perfect battery) is by far the main part with the quadratic “correction” time period being very small. Nonetheless provided that the quadratic co-efficient is the product ##textbf{ℇ}r_i## and the co-efficient of the linear time period is simply ##textbf{ℇ}##,  we are able to divide the primary by the second to yield a worth of  ##0.22;Omega## for ##r_i## in good settlement with the sooner worth obtained by linear regression of load voltage vs present. Whether or not quadratic regression has any specific statistical benefit as in comparison with linear regression is a query we should depart to the statistics ‘gurus’.  When it comes to the Excel ‘trendline’ match above, we have now set a zero intercept since theoretically there isn’t any fixed time period within the equation. Nonetheless, a pressured zero intercept is probably not ideally suited from a statistical perspective.

In precept, we should always have the ability to use only one meter for this technique since technically solely a present studying is required. Nonetheless, this may imply that we’re utilizing exactly identified values of resistance/conductance in addition to a equally exact worth of ammeter resistance.  This isn’t that simple to understand in follow until one has an instrument comparable to a milli-ohm meter (a second meter anyway!). And ideally a non-intrusive Corridor impact present sensor!

Regression Evaluation: Load Voltage vs Conductance

Think about once more the equation: $$ I = frac {textbf{ℇ}}{R+r_i}=frac {textbf{ℇ}}{Rleft(1+frac{r_i}{R}proper)}.$$ Given ##r_i<<R##, a mathematical  ‘trick’ we are able to use to good impact right here is to write down $$1+frac{r_i}{R} =1 + r_iG approx e^{r_iG}$$ in order that: $$ I approx {ℇ}{G}e^{-r_iG}implies V_L= {ℇ}e^{-r_iG}. $$ The benefit of this re-arrangement is that exponential regression on load volts vs conductance will instantly yield a worth for ##r_i##.  And for plotting functions, one might calculate conductance instantly because the reciprocal of measured resistance albeit once more needing to know the resistance values as precisely as potential.

In precept, we don’t want a present measurement and therefore needn’t be involved with the related complication of ammeter resistance. Nonetheless, on this specific case, our desk of readings above was already obtained utilizing the “two-meter” technique so we’re once more figuring out conductance as ##frac{I}{V_load}##.  Ideally, this specific regression evaluation must be used to find out inner resistance from direct measurements of load voltage and conductance if we’re capable of precisely measure conductance or can supply high-accuracy resistors.

https://www.desmos.com/calculator/kngmrtkriz

The graph appears linear moderately than exponential for the straightforward motive that over the actual area of conductance values, the exponential is certainly effectively approximated by a 1-term linear enlargement. Nonetheless the exponential regression match instantly yields ##r_i=0.231 Omega## and ##textbf{ℇ}=1518.14;mV##.  The fixed ought to point out the open circuit voltage however right here there’s a discrepancy because the measured open circuit voltage is about 1521 millivolts. The most certainly explanation for this discrepancy is “fitment resistance” whereby there’s a quick stretch of conductive materials from the battery terminal to the purpose at which the voltmeter (measuring load volts) is connected. When the battery attracts present to produce a load, this introduces a small voltage drop which might not be there if the voltmeter had been linked on to the battery’s optimistic and unfavorable terminals.

An Alternate Methodology utilizing Battery Take a look at Mode

On one of many two meters used for this experiment, there’s a “battery check” setting for both 1.5-volt or 9-volt batteries. It seems that this setting locations a load of 30.5 Ohms onto the battery (the 30.5-ohm worth was obtained from an Ohm’s legislation regression evaluation) and the voltage measurement (within the battery check setting) is instantly throughout this resistance. That is very helpful since now one is instantly capable of calculate circuit present by dividing the measured voltage by 30.5.  The open circuit voltage of the battery will also be measured and the voltage drop (below load) can therefore be calculated.  Dividing this voltage drop by circuit present yields inner resistance. A word on the load resistor worth of 30.5 ohms is that it was in all probability rigorously chosen as being consultant of a “mid-range” AA battery load. The corresponding present draw is about 50 milliamps and an AA cell datasheet will present a  “mid-range” discharge curve as measured at this present.

Sadly nonetheless, as identified by PF member @sophiecentaur, there’s a complication with this technique primarily as a result of it’s NOT a four-wire measurement (we might make it in order described within the Appendix beneath). Load present flows from the battery (below check) via the meter leads en path to the load resistance (contained in the meter) so the worth of  ‘inner resistance’ obtained would be the sum of lead resistance and battery inner resistance.

This complication could also be handled by taking measurements utilizing 2 separate units of leads (ideally equivalent pairs) and a 3rd measurement wherein each pairs of leads are linked in collection. The circuit diagrams beneath present these 3 measurements. Be aware that open circuit voltage is nominally proven as 1.5 volts whereas the precise worth relies upon upon the AA cell one is testing/measuring. Be aware additionally that within the diagrams, every pair of leads is represented by two resistors however ##R_2## and ##R_1## seek advice from the resistance of the respective lead pairs. ##R_{int}## is the interior resistance of the AA cell. The ##30.5;Omega## resistor is inner to the multimeter and connects when one change from studying DC voltage (open circuit) to DC load voltage below the “battery check” setting.

$$r_i = (r_i + R_1) + (r_i + R_2) – (r_i+R_1+R_2) $$ $$R_1 = (r_i+R_1) – r_i$$ $$R_2= (r_i+R_2) – r_i$$

Serendipitous by-products of this set of measurements are resistance values for the 2 units of lead getting used. It must be famous that since a 1 millivolt drop corresponds (roughly) to a resistance worth of twenty-two ohms, the latter is the decision restrict of this measuring system and the resistance values are all ##pm 11;mOmega##. As soon as lead resistance is understood, a single measurement will suffice for measuring the interior resistance of some other AA cell.

Desk of Outcomes: 4 Varta AA 2100 mAh batteries

Low Resistance Measurement

Within the above setup, there may be after all nothing to stop us from changing the second set of leads with any low resistance worth. So we have now one more serendipitous by-product of this measurement approach. Particularly that it may be used to measure any low resistance values to an accuracy of ##pm 11 mOmega##. That is an order of magnitude higher than the resistance scale on the meter itself which is just good to ##0.1 Omega##.

An instance of sensible use of this system arose when the writer tried to measure the resistance of a set of 12-volt battery leap leads. It was found that the resistance was far too excessive (about 800 milli-ohms) and subsequent investigation revealed very poor crimp connections to the terminating alligator clips. The leads had been taken to a TV restore store the place the connections had been soldered down correctly and the leap leads now measure in at simply 45 milliohms.

A number of totally different lengths of galvanized metal wire had been measured and the proportionality between resistance and size was verified. In fact, a milliohm meter will do that process equally effectively however take into account value issue vs a ‘run of the mill’ multimeter such because the one this writer has used for the above measurements! One brings inside simple sight,  the necessary resistance vary between 0 and 1 ohms whereby lies a treasure trove of attention-grabbing scientific and/or engineering data!

Thévenin Equal Circuit

Circuit illustration as a resistance (impedance within the case of a sinusoidal supply voltage) in collection with a continuing voltage supply is called a Thévenin equal circuit and finds in depth use in lots of scientific and/or engineering functions wherein complicated bodily/electrical programs are simplified accordingly.  The battery circuit mentioned on this article is one such instance.  One other very helpful instance is that of a transformer wherein, wanting into the 2 terminal secondary aspect, we see a simplified Thévenin equal consisting of an AC voltage supply (main voltage instances turns ratio) in collection with an impedance R + jwL the place R is a sum of secondary winding resistance and  ‘stepped up’ main winding resistance.  The identical applies to the inductance L. The ‘step up’ co-efficient is the sq. of the turns ratio okay.

Comparable circuits are used to signify every part of 3-phase  synchronous motors and 3-phase induction motors

Abstract and Conclusion

On this article, we have now explored varied types of regression evaluation which can be used to find out the interior resistance of a battery (AA cell specifically). Given the three variables current in Ohm’s Legislation calculations, we are able to plot any two towards one another and procure inner resistance by applicable regression evaluation. Now we have talked about varied sensible issues (and options) related to accumulating such information units.

Now we have additionally described a wholly totally different approach for figuring out inner resistance by exploiting the “Battery Take a look at” setting which some digital multimeters make obtainable. This method additionally allows the measurement of low resistance values usually. Low resistance measurement makes potential a variety of various faculty science practicals in addition to discovering functions in measuring the resistance of (for instance) battery jumper cables! Or for measurements of motor and transformer windings and for checking the standard {of electrical} connections.

The modeling circuit for a  battery consists of a continuing voltage supply in collection with a resistor. This is called a Thévenin equal circuit and it’s usually potential to interrupt down complicated circuitry into its Thévenin equal. When the supply voltage is sinusoidal,  the Thévenin equal will present an AC supply with mounted frequency (sometimes 50Hz)  in collection with an impedance represented by a posh quantity.

Acknowledgments

This text stems from a protracted thread on battery inner resistance which the writer initiated after experiencing some technical difficulties with an related faculty sensible  A number of PF mentors: contributed to this thread and I wish to thank every of them for the various useful ideas and sensible recommendation on how finest to conduct experiments/measurements of this nature. I’ve taken the freedom of together with hyperlinks to specific posts from PF customers @DaveE and @Tom.G while the event of  “An Alternate Methodology” was assisted in no small measure by feedback from @SophieCentaur.

The identical PF customers contributed additional ideas once I posted this text for assessment. As well as, PF person @Anorlunda steered the inclusion of a paragraph or two on the “Thévenin equal circuit” noting that {the electrical} mannequin of a battery is an instance of such. PF person @Rive steered an early dialogue of this mannequin’s limitations and PF person  @.Scott made the purpose that inner resistance is finest measured utilizing a consultant load. @Steve4Physics strengthened the purpose that measurements must be taken shortly to keep away from modifications in battery emf in addition to battery heating which might have an effect on the resistance measurement.

Maybe too usually, we take this without any consideration however with out @Greg  there could be no PF and no platform for insights such because the above.

Lastly, I wish to sincerely thank Clinton and Douglas at Armcoil Afrika who had been extraordinarily tolerant placing up with some type of loopy nerd desirous to do low resistance measurement! They allowed me to measure a few transformer windings (the readings in contrast favorably with these obtained on Armcoil’s personal equipment) in addition to utilizing their very own gear to confirm resistance readings I had obtained on varied lengths of galvanized metal wire. Some commonplace resistors had been additionally measured. For instance a ##4.7 Omega## resistor was measured at ##4.669 Omega##.

Appendix A: Measurement of Inner Resistance and Low Resistance

Think about the above circuit which represents an AA cell being measured for open circuit voltage (##S_1## open) and below load (##S_1## closed).  ##S_1 ## open corresponds to the multimeter in bizarre DC voltage measurement setting while ##S_1## closed corresponds to the meter being within the “Battery Take a look at” setting. The present flowing within the circuit with ##S_1## closed is well calculated by dividing load volts by the mounted resistance worth of 30.5 ohms. The distinction between open circuit voltage and cargo voltage ##Delta V## ought to correspond to the voltage drop throughout the battery’s inner resistance and dividing that by the present yields inner resistance. On this case, the 2 voltmeters proven are redundant since they’re each (in impact) measuring instantly throughout the battery terminals.

The issue with the above circuit diagram is that it exhibits the battery connecting to the voltmeter through a pair of leads of zero resistance. Within the subsequent diagram, the circuit is redrawn to indicate (extra realistically) the battery linked to the voltmeter through a pair of leads of mounted resistance.

The voltage distinction between cell emf and cargo voltage ##V_2## ie: ##Delta V_2## now represents the voltage drop throughout battery inner resistance in collection with lead resistance. So in essence the battery’s inner resistance has been ‘prolonged’ to incorporate lead resistance.  In impact, we might return to our first circuit and substitute the battery’s inner resistance with its authentic worth plus the mixed lead resistance.

If we had been incautious sufficient to utilize a battery holder (moderately than putting meter probes instantly onto the battery terminals), we might introduce an additional ‘extension’ of inner resistance on account of the battery holder’s “fitment resistance”.  For that reason, the probes of Voltmeter ##V_1## should certainly be in direct contact with the battery terminals in order that they don’t measure something apart from the voltage drop throughout the battery’s inner resistance.

The above association is equal to a normal 4-wire measurement of inner resistance since Voltmeter ##V_2## may very well be changed by an ammeter in order that present is instantly measured. It’s not a priority that the ammeter would add resistance to the circuit so long as we guarantee voltmeter ##V_1## measures instantly throughout the battery terminals. Then ##r_i=frac{Delta V_1}{I}## as earlier than.  An in depth description of the 4-wire measurement of inner resistance is offered by PF person  @DaveE in one of many posts from a thread on this similar matter (battery inner resistance).  The reader could be well-advised to review this description taking cautious word of the experimental element. We regard this because the “gold commonplace”  for measuring the interior resistance of AA cells.

The benefit of the association within the circuit diagram(s) above is that it allows extra low resistance measurements different than simply battery inner resistance.  For instance within the second circuit diagram above, we are able to simply receive the meter’s lead resistance because the voltage drop throughout the (mixed) leads is just ##V_1-V_2##. Present (as earlier than) is obtained as ##frac{V_2}{30.5}## and a easy Ohm’s legislation calculation then offers the (mixed) lead resistance. As soon as we have now established a worth for lead resistance, there may be nothing to stop us from inserting extra resistance between ##V_1## and ##V_2## and measuring the modified voltage drop. The worth of complete circuit resistance could also be simply calculated by Ohm’s legislation and the deduction of lead resistance plus battery inner resistance offers a measure of the extra inserted resistance.

A remaining level price noting is that an ‘upmarket’ business meter such because the Fluke 87 can measure low resistance to the identical or higher order of accuracy.  A selected instruction given in an “Functions Be aware” on low resistance measurement,  is to attach leads collectively and internally file lead resistance in order that its worth might subsequently be “zeroed out”.  The explanation for doing that is precisely as described above.

Appendix B: Measuring “Battery Take a look at” Resistance

That is in all probability the only ever software of Ohm’s Legislation! When set to “battery check”, the multimeter reads a voltage throughout a hard and fast resistive load. All that must be completed is to put an ammeter and variable resistor  (an uncovered pencil lead was used) in collection with the multimeter’s inner “battery check” resistor.  When this circuit is powered from an AA battery, present varies with (exterior) resistance so a set of present/voltage readings could also be taken and plotted on a regression curve. Since ##V=IR##, the gradient of the plot offers the worth of the “battery check” resistance worth.

https://www.desmos.com/calculator/7zojmulw95

References