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What's the equivalent resistance in this circuit (between points A and B)? enter image description here

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You can solve this in the same way you can solve any resistor problem: write down Kirchoff's laws. The tricks for equivalent resistance for series and parallel circuits are useful shortcuts, but they don't provide a general algorithm for solving these problems. Kirchoff's laws do. –  kleingordon Mar 12 '12 at 5:54
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Ordinarily I close these kinds of questions, but (a) you've been around long enough that I shouldn't need to be trigger-happy with the close button ;-) and (b) you're basically asking about a Wheatstone bridge, which is pretty much the canonical example of a circuit element that can't be reduced by the series and parallel rules. I think asking how to find the resistance of that configuration is generally applicable enough that it's fine. Though I still do feel the question would be better with a bit more explanation. (Maybe it's just me) –  David Z Mar 12 '12 at 6:17
    
David, you should add whatever text you think it needs to improve the question. To me, it's just a cute application of mathematics. The purpose is to point out kleingordon's comment. –  Carl Brannen Mar 12 '12 at 7:24

4 Answers 4

Here is how I would do it, following the method outlined by kleingordon in a comment. This method is less cool but more general than Carl Brannen's answer, because it will work even in the case where there are crossing wires and you can't rearrange it into a single sheet of resistive material.

Let the electric potential at $A$ be $V_A$ and that at $B$ be $V_B$. Also, let the potential on the wire that connects $R_1$ to $R_2$ and $R_3$ be $V_C$ and let the potential on the wire connecting $R_4$ to $R_3$ and $R_5$ be $V_D$. We know that the current across each resistor must equal the potential difference divided by the resistance, so we have $$I_1 = R_1(V_A - V_C)$$ $$I_2 = R_2(V_C - V_B)$$ $$I_3 = R_3(V_C - V_D)$$ $$I_4 = R_4(V_A - V_D)$$ $$I_5 = R_5(V_D - V_B).$$

We also know that the current must be conserved at every junction, which gives us $$ I_1 + I_2 = I_4 + I_5 $$ $$ I_1 = I_2 + I_3 $$ $$ I_4 + I_3 = I_5, $$ but the last of these three equations is redundant because it can be derived from the other two, so there are seven equations in total, in nine unknowns (five currents and four potentials).

We want to calculate the resistance, which is given by $(V_A-V_B)/(I_1+I_2).$ Since everything's linear we can assume without loss of generality that $V_B=0$ and $V_A=1$. This gives us seven equations in seven unknowns, which we can solve to find the answer.

I haven't worked it through because it's a bit laborious (I'd probably use a computer algebra system rather than doing it by hand) but it should give the same answer as Carl Brannen's method.

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As suggested by Manishearth, one can perform a $Y$-$\Delta$ transform from $Y$-resistances $R_1$, $R_2$ and $R_3$, to $\Delta$-conductances $G_1$, $G_2$ and $G_3$ (using a $123$ symmetric labeling convention), cf. Fig.1 below.

 A x----x------x-----[3]-----x------x----x B
        |      |             |      |
       [4]    [2]           [1]    [5]
        |      |             |      |
        |------x-------------x------|    

$\uparrow$ Fig.1. A $\Delta$-equivalent circuit to OP's original circuit.

In terms of formulas, the $Y$-$\Delta$ transformation is given as $$ G_i ~:=~ R_i \frac{\sum_{j=1}^3 R_j}{\prod_{k=1}^3 R_k}~=~ R_i \frac{R_1+R_2+R_3}{R_1 R_2 R_3},\qquad\qquad i=1,2,3. $$

The $\Delta$-equivalent circuit in Fig.1 can be viewed as composed of only series and parallel resistors. The equivalent conductance between $A$ and $B$ therefore becomes

$$ \frac{1}{R}~=~G_3+\frac{1}{\frac{1}{G_2+\frac{1}{R_4}} +\frac{1}{G_1+\frac{1}{R_5}}}. $$

(Finally let us mention that it is also possible to apply the $Y$-$\Delta$ transform to other triples of the five resistors than $123$.)

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I'll give the answer to this question using an unusual method that showed up in the American Mathematical Monthly's problem section perhaps in the late 1970s. This is not necessarily the easy way to solve the problem, but it works out nicely from an algebraic point of view.

The way most people solve most resistance problems is to use series and parallel resistor rules. These are mathematically elegant in that they involve only resistance. But this circuit cannot be reduced to series and parallel rules (is this true if you write an infinite series in R3, perhaps?), so probably the most straightforward method is to apply a voltage of V to the circuit and use algebra to work out the total current. This is inelegant (but physical) in that it introduces ideas other than resistance itself.

The "delta" method mentioned by Manishearth, (but at this time not actually worked out to the final answer) is how an EE would solve the problem. It has the advantage of sticking with resistance, but it involves somewhat unintuitive changes to the topology of the circuit.

The method I'm giving here uses only resistances and illustrates a general solution to this sort of problem. If one generalizes the $R_k$ to complex numbers $Z_k$, it can be used for general impedances (as can the delta method), but it is more general than the delta method. It also may help with student's understanding of sheet resistance so I think it's worth my time to type it in:


First, we replace the resistors with thin flat material that happens to have a "sheet resistance" of 1 ohm per square. With such a material, if we cut out a rectangle of dimensions 1 x R, we will obtain a resistance of R ohms between two conductors attached to the 1 length sides:
R ohm resistor made from Rx1 rectangle

Now the thing about sheet resistance is that you can scale the resistor to whatever size you like; so long as you keep the ratio of the side lengths as "R", the resulting resistor will have resistance R. The sheet can be made up of little sheets that are pasted together. To do the pasting correctly, we need to use insulating glue for the horizontal connections and conducting glue for the vertical connections. This is because current only flows from left to right. So the insulating glue doesn't help or hinder the current flow, and the vertical connections don't matter because all the conducting glue has the same voltage anyway. I saw this method of computing resistors in a solution to problem E2459 at the American Mathematical Monthly, February 1975.

So replace the given circuit with one where each resistor is replaced by a rectangular region with dimensions appropriate for its resistance. In doing this, we have to make an assumption about which way current flows through resistor R3. I'll assume it flows from top to bottom. And in order to set a scale for the whole thing, let's make the vertical dimension of R3 to be length 1. This gives us the following drawing:
Circuit rewritten as rectangular sheet resistance

Now the overall circuit has a resistance given by the ratio of its length to its width:
$$R = L/W = (R_1 x_1 + R_2x_2)/(x_1+x_4)$$ There are four unknowns, $\{x_1,x_2,x_4,x_5\}$. Comparing horizontal dimensions gives two independent equations:
$$R_1x_1 = R_4x_4 + R_3,$$ $$R_5x_5 = R_3 + R_2x_2.$$ And comparing vertical dimensions gives:
$$x_1 = 1 + x_2,$$ $$x_5 = 1 + x_4.$$
This eliminates $x_1$ and $x_5$ to give two independent equations in two unknowns:
$$R_1 + R_1x_2 = R_4x_4 + R_3,$$ $$R_5 + R_5x_4 = R_3 + R_2x_2.$$ Or:
$$ R_1x_2 - R_4x_4 = R_3-R_1,$$ $$-R_2x_2+ R_5x_4 = R_3-R_5 .$$ These solve to give:
$$ x_2 = \frac{R_3R_5-R_1R_5 + R_3R_4-R_4R_5}{R_1R_5-R_2R_4},$$
$$ x_4 = \frac{R_1R_3-R_1R_5 + R_2R_3-R_1R_2}{R_1R_5-R_2R_4},$$
and so
$$ x_1 = \frac{R_3R_5-R_2R_4 + R_3R_4-R_4R_5}{R_1R_5-R_2R_4},$$
$$ x_5 = \frac{R_1R_3-R_2R_4 + R_2R_3-R_1R_2}{R_1R_5-R_2R_4},$$
We need $W=x_1+x_4:$
$$W = \frac{R_3(R_1+R_2+R_4+R_5)-(R_1+R_4)(R_2+R_5)}{R_1R_5-R_2R_4}$$ and $L = R_1x_1+R_2x_2:$
$$L = \frac{R_3(R_4+R_5)(R_1+R_2)-R_1R_2(R_4+R_5)-R_4R_5(R_1+R_2)}{R_1R_5-R_2R_4}$$ so the total resistance is:
$$R = L/W = \frac{R_1R_4(R_2+R_5)+R_2R_5(R_1+R_4)-R_3(R_4+R_5)(R_1+R_2)}{(R_2+R_5)(R_1+R_4)-R_3((R_1+R_2)+(R_4+R_5))}.$$ In the above, I've grouped terms in order to make it clear that this gives the correct answer in the limit of $R_3$ goes to $0$ or $\infty$.

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Awesome trick! If you want, I can work it out using the $Y-\Delta$ method. It may take me a while though, I'm slow with TeX. –  Manishearth Mar 15 '12 at 9:10
    
The answer may be correct(have not verified it yet), but I don't think the transformation if fully justified. It seems to be making assumptions of the current path. Also, the resistance of a sheet will only be the ratio of its lengths if the terminals are parallel. Here, at the R4-R3 interface(and some other places), this is not the case. –  Manishearth Mar 15 '12 at 9:26
    
@Manishearth; I'd appreciate seeing the Delta method completed. (And maybe someone should type up the current method.) I'll add a correction for the connections, i.e. the vertical connections are conductors while the horizontal connections are insulators. And it's true that the choice of diagram depends on which way current flows through R3. You can determine the correct direction by comparing the ratios R1/R2 to R4/R5. But you end up with the same equations (as it is unchanged on swapping R1 for R4 and R2 for R5). –  Carl Brannen Mar 15 '12 at 19:57
    
Now its much clearer! (Also, it seems that Qmechanic has done the job of doing a bit of Y-$\Delta$ derivation (without every simplification step) –  Manishearth Mar 16 '12 at 0:47
    
But I think $R_1x_1=R_4x_4-R_3$ and $R_5x_5=R_2x_2-R_3$. What I'm doing wrong? Please explain. –  A Googler Oct 3 '13 at 7:07

Use a star-delta transform to simplify part of the circuit. You may also use the principle of superposition.

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I'd originally seen this in a math problem that was given as "what is the minimum number of resistors needed to obtain a resistance of pi to an accuracy of 1 part in a million?" The solution there transformed the problem into one of tiling a rectangular array with a minimum number of square tiles (of arbitrary size) with the ratio of the length and width being an integer approximation of pi. Translated back into a circuit, the winning circuit was of this form (with the resistances each a small multiple of one ohm). –  Carl Brannen Mar 12 '12 at 8:04
    
@CarlBrannen Hmm... Such a problem I would try to solve by some infinite series of resistors. $\zeta(2)$ pops into my mind, as it's relatively simple to construct with integral resistors, but unfortunately you will finally get a resistance of $6/\pi^2$. Tiling six of these in parallel gets you $1/\pi^2$. I doubt that resistors can square-root stuff. –  Manishearth Mar 12 '12 at 8:20
    
Or just take a material of constant resistivity+cross section, draw a (semi)circle with radius=length of $1\Omega$ resistor. Lay your material on this circle. =D –  Manishearth Mar 12 '12 at 8:22
    
Another way to do it would be to use the expansion of $\arctan x$, but it has negatives in it. –  Manishearth Mar 12 '12 at 8:23
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@CarlBrannen: This is puzzling, since you can perform a continued fraction expansion using series and parallel resistors, and this is an extremely economical way to produce pi. I would need only as many 1 ohm resistors as the sum of the continued fraction denominators up to the required accuracy. Pi has a large denominator early on, so this might be the sticking point. –  Ron Maimon Mar 13 '12 at 7:28

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