Networks of linear time-invariant (LTI) resistors are a classical topic with many applications, used to model a variety of flow phenomena. In this short communication, we provide conceptually simple proofs of the following intuitively appealing bounds, which also imply the classical Rayleigh monotonicity law (RML): (i) For an LTI resistor network, driven by some node voltages, when a set of resistors is removed the total power dissipation is reduced by at least an amount that was consumed in the removed resistors. (ii) The complementary or dual result for such a network, but when current driven, is that shorting some resistors decreases the total power dissipation by at least as much as used to occur in these resistors before shorting. In fact, ideal diodes can be allowed as network branches and the same results hold. The short derivations assume that Dirichlet's and Thomson's minimum principles are known. With the expanding applications of a linear network theory, we hope that these variants of the monotonicity theorem will prove pedagogically useful.

## I. DERIVATION OF THE BOUNDS

Let us begin by proving the two statements given in the abstract. We start with the voltage driven case, to which Dirichlet's minimum principle (DMP) is applied.

Consider a network of LTI resistors with *N* nodes, labelled by integers $1,2,\u2026,N$, and conductances $(Ci,j)i,j=1N$ between the nodes. Denote the potential at node *i* by *v _{i}*, and assume that the network is

*voltage driven*, i.e., the potentials $(vi)i\u2208S$ at some collection of source nodes $S\u2282{1,2,\u2026,N}$ are determined.

The well-known Dirichlet's minimum principle (DMP) states that the node voltages $(vi)i=1N$ of a voltage driven LTI resistor network are those that conditional on the source potentials $(vi)i\u2208S$ minimize the total energy dissipation *D* by all the resistors

The factor $12$ above is needed because the double summation will count every resistor twice. The minimizing voltages $(vi)i=1N$ are also the unique ones^{1} for which the resistor currents

satisfy Kirchhoff's current law (KCL) in the non-source nodes ${1,2,\u2026,N}\u2216S$, while the total currents in and out of the network at the source nodes are equally large.

With this minimum principle in mind, let us prove the first statement given in the abstract. Consider first removing only one resistor $Ra,b$ with end nodes *a* and *b*. By Dirichlet's principle, in the initial state, the rest of the network has the least power dissipation among all the possibilities where the node voltages at the nodes *S*, *a*, and *b* are held constant. If the resistor $Ra,b$ is removed while the node voltages at *S*, *a*, and *b* are held fixed, the rest of the network sees no change, but the dissipation is diminished by that in $Ra,b$. If the node voltages at *a* and *b* are now released, keeping still the driving voltages in *S* fixed, the remaining network finds a better optimum, so its dissipation decreases further—the overall decrease is hence at least what occurred in resistor $Ra,b$.

The proof for multiple removed resistors is exactly similar: one first removes the resistors, keeping the voltages at nodes connecting to the removed resistors constrained. Then, removing these constraints allows a better minimum, which solves the new remainder network.

Let us now consider the second statement in the abstract, addressing *current driven* networks, i.e., instead of potentials $(vi)i\u2208S$ we now fix the total currents

flowing into the network from the source nodes $i\u2208S$. In order to be consistent with Kirchhoff's current law, the source currents altogether must cancel out, $\u2211i\u2208SIi(tot)=0$.

The principle that is complementary or dual to DMP is Thomson's minimum principle (TMP), stating that in a current driven network of linear resistors, the branch currents $Ii,j$ are those minimizing the energy dissipation

among all currents (i.e., collections $(Ii,j)i,j=1N$ with $Ij,i=\u2212Ii,j$) that satisfy KCL in the non-source nodes and Eq. (2) in the source nodes.

The second statement in the abstract now follows easily: take the currents $(Ii,j)i,j=1N$ solving the original network. Fixing these currents but shorting some resistors, i.e., taking their resistances to zero, the dissipation reduces by the amount that used to occur in the now shorted resistors. Releasing the currents, they will by TMP find a better minimum, solving the shorted network.

We conclude this section by remarking that instead of removing (shorting) some resistors in a voltage-driven (current-driven) network, one can consider only decreasing the conductances (resistances) of the resistors. Straightforward generalizations for the statements and proofs then apply.

## II. CONNECTION TO RAYLEIGH'S MONOTONICITY LAW

Consider the setup of Sec. I, but assume in addition that the source nodes *S* consist of only two nodes, say *x* and *y*. Recall that there exists an effective resistance $Reff>0$ between these two nodes, such that if a voltage $U=vx\u2212vy$ is applied between the nodes *x* and *y*, then a current *I* flows into the resistor network at *x* and out at *y*, and furthermore *U*, *I*, and the network's total dissipation *D* are related by

Likewise, if the network is instead driven by a current *I* from *x* to *y*, then the voltage $U=vx\u2212vy$ and dissipation *D* satisfy the above equations.

Rayleigh's monotonicity law (RML) now states that if any resistors of the resistor network are removed, then the effective resistance $Reff$ does not decrease; and it does not increase if any resistors are shorted. The expressions for *D* above give these as immediate consequences of our estimates on the decrease of dissipation, by considering voltage driven case for removal of a resistor and current driven case for shorting. In addition, the dissipation approach yields an intuitive bound which also applies for networks with more than two source nodes.

Nice textbooks introducing Dirichlet's minimum principle, effective resistance, and Rayleigh's monotonicity law are, e.g., Refs. 2 and 3. RML has a variety of useful consequences. For instance, in connection with interval arithmetic type computations with inaccurate “interval resistors,”^{4} or in proving the series-parallel inequality for parallel addition of matrices, which generalizes Lehman's inequality,^{5} or in generating estimates more generally with shorts and cuts.^{2}

## III. A CLASSROOM EXAMPLE

The practical use of these abstract principles is now demonstrated with an example that cannot be addressed using RML without our quantitative bounds; and also showing that the DMP and TMP are more powerful than the special bounds derived with these minimum principles.

**Example.** A resistor network was probed with a 10-V dc source and gave an 0.5-A current. Further, one resistor in it was measured to have a 5-V voltage drop across it, and the current through this resistor was 0.1 A. Give bounds for the effective resistance of the network, if

- (a)
the measured resistor is cut off,

- (b)
the measured resistor is shorted, and

- (c)
the measured resistor is replaced with a 100 Ω resistor.

Keeping the voltage fixed at 10 V, the initial total dissipation was 5 W, while the single resistor consumed 0.5 W. After removal the total dissipation is at most 4.5 W. At 10 V this corresponds to effective resistance at least 100/4.5 Ω, or about 22.2 Ω.

For shorting the resistor, we consider the network driven by a constant 0.5 A current. Now after shorting the total dissipation is, again, at most 4.5 W. Based on 0.5 A current this corresponds to an at most 18 Ω effective resistance.

Finally, replacing the original 50 Ω resistor with a 100 Ω alternative could be represented as “removal of resistor,” by representing the original one as two 100 Ω resistors in parallel and cutting off one of these. However, it is instructive to revert to the fundamental principles, DMP and TMP. Denote the initial total dissipation by

*D*and the final by $D\u2032$.By DMP for the voltage-driven case, the final dissipation (in watts) is less than that with the initial node voltages, $D\u2032\u2264D\u22120.5+25/100=D\u22120.25=4.75$. We first subtracted the dissipation in the removed resistor, then added that in its 100 Ω replacement under a 5 V voltage. So 10 V gives at most 4.75 W, and the effective resistance $R\u2032\u2265100/4.75=21.0526$ Ω.

By TMP for the current-driven case, in quite similar fashion $D\u2032\u2264D\u22120.5+0.01\xd7100=D+0.5=5.5$ W. Since this is with an 0.5 A current, $R\u2032\u226422$ Ω. Please note that on right side of the inequality we use the initial current distribution without change—this satisfies the KCL as required by TMP, while the solution also satisfying KVL gives the minimal dissipation $D\u2032$.

In summary, for case c) we get fairly tight bounds and $21\u2264R\u2032\u226422$ in ohms, while the initial network was effectively 20 Ω.

**Notes.** The cases (a) and (b) above show that a single adjustable resistor could control the effective resistance of the overall network from 18 to 22.2 Ω, reaching at least both of these values and by continuity everything in between. Case (c) demonstrates the dual nature of DMP and TMP in providing lower and upper bounds to the effective resistance, so that we get an interval estimate of it when the resistance of a single resistor is increased or decreased by some amount. Generating these estimates required no knowledge about the overall network configuration, only that it should consist of LTI resistors.

## IV. ON GENERALIZATIONS

There are variants of resistor network problems including other network elements than LTI resistors, such as diodes or non-linear resistors. Let us briefly discuss extensions to such cases.

The variational formulations for LTI resistor networks including ideal diodes are explicitly given in Ref. 6, pp 3–8. From the dual formulation on that page (applied to networks driven by only voltages and not currents), one observes that a diode-resistor network minimizes the Dirichlet-type expression

where the sum runs over the resistors of the network. (Indeed, there is no power loss in an ideal diode.) The similarity of the expressions (1) and (3) readily implies that our bound remains valid despite ideal diode branches in the network. Similarly, the primal formulation on that page shows that TMP is valid when diodes are allowed in a current driven network.

We have earlier described DMP and TMP as “dual,” as is justified by appeal to this reference. That DMP and TMP remain valid with diode branches appears to be little known, but diodes would allow for example modeling one-way streets in a traffic-flow problem.

As regards non-linear resistors, in our proof of the dissipation inequality we applied the Dirichlet principle only to the remaining part of the network. Hence, that needs to be composed only of linear resistors and ideal diodes. However, it does not matter whether the removed part of the network contained non-linear resistors. The same applies to the dual results on shorting.

More general variational principles for non-linear networks are studied in Ref. 7, which has been followed by plenty of more recent work. A convenient and short modern exposition is available in Chapter 4 of Ref. 8. In that wider context, the functionals minimized no longer represent dissipation, although otherwise the approach taken here may be applicable. In fact, unless the resistors have a certain monotonicity property, the variational principles may yield a saddle point instead of a minimum as a stationary-point solution.

The approach we have taken here allows deriving bounds for effects of configuration changes, when a suitable variational principle is known that applies to a fixed configuration.

## V. FURTHER READING AND HISTORIC NOTES

A convenient short exposition to variational principles in relation to Kirchhoff's equations, for readers of this journal, is the paper by Ercan.^{9} Van Baak^{10} has advocated the use of minimization to solve electric networks, with edge currents as the decision variables: while both DMP and TMP are limited, Maxwell's minimum heat principle allows simultaneous voltage and current sources. Perez^{11} has examined minimizing the rate of entropy production in dc circuits. Kagan^{12} applied a matrix approach, similar in part to that in Maxwell's book,^{13} showing among other things that the equivalent resistance has a bilinear dependency on any single resistor.

The Lord Rayleigh to whom the monotonicity theorem is attributed went as John William Strutt, until he was conferred the title on his father's death; at any time in history there was only one Lord Rayleigh, and this one was the third. The two-volume book “The theory of sound”^{14} written by him discusses variational theorems for linear systems in generalized coordinates, particularly in sections 74 and 75 of volume 1. Then section 305 in volume 2 gives the classical RML, to which Strutt/Rayleigh refers for analogy on estimating fluid flow around the neck of a Helmholtz resonator.

Maxwell in his Treatise^{13} refers to the monotonicity theorem, attributing it to Strutt in section 306, for fairly sophisticated estimates of conductivity of a 3D body by shorting along estimated equipotential surfaces. On the other hand, he considers that “This principle may be regarded as self-evident.”

Thomson's minimum principle was originally stated by William Thomson,^{2} also known as Lord Kelvin.

These physicists of the time were generalists, working on a multitude of topics. An early attempt to extract and collect the theory together into one rigorous presentation only on electricity and magnetism is the book by Jeans.^{15} The Rayleigh monotonicity, specifically for decreasing some resistances, is proven in section 359, using Thomson's principle.

With such long history, some electric network theorems have undoubtedly been not only discovered, but also re-rediscovered. We dare hope that our elucidation of the monotonicity theorems has been entertaining, whether with any novelty or not.

## VI. CONCLUSION

Based on Dirichlet's and Thomson's principles of minimum dissipation, respectively, for voltage or current driven linear resistor networks, we derived an appealing variant of Rayleigh's monotonicity law, with a bound for the change in dissipation to strengthen that classic theorem.

Verbally our variant is stated as follows. When a set of resistors is removed from (shorted in) a voltage (current) driven LTI resistor network, the total dissipation diminishes by at least the amount that used to occur in those resistors.

We believe this variant, which we have not found in prior literature, is both pedagogically appealing and useful in various applications of the network theory.

## ACKNOWLEDGMENTS

The authors are indebted to two anonymous expert reviewers for their substantial contributions to this manuscript. A.K. was supported by the Vilho, Yrjö and Kalle Väisälä Foundation.