Solving Multiloop Circuit Problems

Guide 21-5a. Solving Multiloop Circuit Problems

The method of solving problems below applies to multiloop circuits with a single source of emf and several resistors.

In order to simplify notation, we adopt the convention in which the symbol V represents either a voltage gain (as for a battery) or a voltage drop (as for a resistor). V relates to prior terminology as follows:

V_b = ΔV_b for a battery
V_r = -ΔV_r for a resistor

In this usage, V will always represent a positive number.

Problem: Find the potential difference across and the current through each resistor of the circuit shown below. Given values are listed to the right of Circuit Diagram 1.

V_b	13 V
R₁	1.0 Ω
R₂	3.0 Ω
R₃	2.0 Ω
R₄	3.0 Ω

Circuit Diagram 1

Begin (as we've already done) by assigning each circuit component a unique symbol and listing the given values. We've also marked particular points of the circuit. This is for easy referencing in the discussion below.

Finding the equivalent resistance

The next step is to find the equivalent resistance of the circuit. You can then use that result to find the total current. In order to find the equivalent resistance, we're going to redraw the circuit in a series of steps to simplify it. Refer to Circuit Diagram 2 for the first step. Part of the process is determining which resistors are in parallel and which are in series. In this case, R₃ and R₄ are in parallel, because points b and c are connected by a wire, assumed to have 0 resistance, and are therefore at the same potential. Likewise, points e and d are at the same potential, although it's a different potential than points b and c. We redraw the circuit as shown below with R₃₄ representing the combined resistance of R₃ and R₄. Note that in Circuit diagram 2, points b and c have been collapsed to the single point b₂, and points e and d have been collapsed to e₂.

	By the formula for resistors in series. R₃₄ = [1/R₃ + 1/R₄]^-1
Circuit Diagram 2

Now we can see that R₁, R₂, and R₃₄ are in series as they are part of a single path, and the same current must pass through all of them. We can then represent the equivalent circuit in terms of a single resistance as shown in Circuit Diagram 3.

	By the formula for resistors in series, R_eq = R₁ + R₂ + R₃₄
Circuit Diagram 3

Substituting in the expression for R₃₄, we have R_eq = R₁ + R₂ + [1/R₃ + 1/R₄]^-1.

Substituting given values yields the following:

R₃₄ = 6/5 Ω

R_eq = 26/5 Ω

Note that we express the resistances as reduced fractions in order to eliminate round-off error in calculations to come. (This is not always feasible, depending on how simple the fractions are. Sometimes, it's easier to express results as decimals to one digit more than the number of significant digits.)

Applying the loop rule to find the total current

In Circuit Diagram 3, shown again to the right for reference, the circuit is reduced to a single loop. Let's define I_b as the representing the current in the loop. Traversing the loop counterclockwise, we have for the loop rule the following.

V_b - V_eq = 0

Now we substitute V_eq = I_bR_eq and solve for I_b to obtain:

I_b = V_b/R_eq

Substituting values yields I_b = (13.0 V)/(26/5 Ω) = 5/2 A.

Finding the remaining currents and potential voltages

We can now work backwards to find the currents in and the potential differences across the individual resistors. Refer to Circuit Diagram 2 to the right. Since there's a single loop, the same current passes through each of the resistors, R₁, R₂, and R₃₄. (This is an application of the loop rule with a single loop.) Thus, we can say:

I_b = I₁ = I₂ = I₃₄ = 5/2 A.

Knowing the currents, we can solve for the voltages using V = IR.

V₁ = I₁R₁ = (5/2 A)(1.0 Ω) = 5/2 V

V₂ = I₂R₂ = (5/2 A)(3.0 Ω) = 15/2 V

V₃₄ = I₃₄R₃₄ = (5/2 A)(6/5 Ω) = 3.0 V

As a check, the loop rule must be satifsied:

V_b - V₁ - V₂- V₃₄= 0

(13 - 5/2 - 15/2 - 6/2) V = 0

(13 - 26/2) V = 0

The loop rule checks out. We can move on to deal with R₃ and R₄ individually. Refer to Circuit Diagram 4 below. Note that we've added arrows and labels indicating the currents in the resistors.

Circuit Diagram 4

First we apply the junction rule. We could use either point e or point b as the junction. For point e, the junction rule gives I₂ = I₃ + I₄. The total current entering the junction is equal to the total current leaving the junction. If we used point b, we'd have I₃ + I₄ = I₁. Note that these two results are identical since I₂ = I₁. We have two unknowns in I₂ = I₃ + I₄. I₂ is known but I₃ and I₄ are not. Thus, we need more information in order to determine the two unknown currents. We will get that information from application of the loop rule and V = IR.

For the loop, we'll use the loop that contains only the resistors in question, namely R₃ and R₄. We traverse the loop counterclockwise through the points e-d-c-b. The loop rule applied to this loop is then: V₄ - V₃ = 0. Thus, V₃ = V₄. Of course, we already knew this must be the case, since we had identified these two resistors as being in parallel.

Now using V = IR, we have the following:

V₃ = I₃R₃

V₄ = I₄R₄

Uisng the fact that V₃ = V₄, we can say that I₃R₃ = I₄R₄.

Next we use the loop rule and solve for I₄. (We could have solved for I₃ instead. The goal is to eliminate one of the unknown currents.) This gives I₄ = I₂ - I₃. Substitute this into I₃R₃ = I₄R₄ and solve for I₃.

I₃R₃ = (I₂ - I₃)R₄

       = I₂R₄ - I₃R₄

I₃(R₃ +R₄) = I₂R₄

I₃ = I₂R₄/(R₃ +R₄)

    = (5/2 A)(3.0 Ω)/(2.0 Ω + 3.0 Ω)

    = 3/2 A

We substitute the result for I₃ back into the junction rule to obtain I₄.

I₄ = I₂ - I₃

= (5/2 A) - (3/2 A)

= 1.0 A

Knowing the currents, we solve for the voltages V₃ and V₄.

V₃ = I₃R₃ = (3/2 A)(2.0 Ω) = 3.0 V

V₄ = I₄R₄ = (1.0 A)(3.0 Ω) = 3.0 V

As a check, we see that V₃ = V₄ as expected. Another check is to revisit a previous result obtained much earlier, V₃₄ = 3.0 V. For yet one more check, I₃ + I₄ = 2.5 A = I₂= I₁.

Summary

All of the numerical results are listed in the table below in decimal form to display significant figures.

V_b = 13 V
R_eq = 5.2 Ω
I_b = 2.5 A
	R ( Ω)	I (A)	V (V)
1	1.0	2.5	2.5
2	3.0	2.5	7.5
3	2.0	1.5	3.0
4	3.0	1.0	3.0

Summary

As you solve problems like the one above, follow the method described. While the order in which you calculate some of the individual potential differences and currents will vary depending on the specifics of the resistor arrangement, the plan summarized below applies in general.

Use distinguishing subscripts on resistors, voltages, and currents. Use corresponding subscripts on the resistance, potential difference, and current for each component. (For example, R₁, V₁, and I₁ would be used for the same component. This may seem obvious, but we've seen people who don't follow that rule and get themselves hopelessly confused.)
Redraw the circuit in a sequence of diagrams in order to combine resistances in series and parallel and determine the equivalent resistance.
Knowing the equivalent resistance, determine the total current from the battery.
Work backwards through your circuit diagrams to determine currents and potential differences. Make use of the loop rule and junction rule both to calculate values and to check results.

In some problems, you may have different unknowns than those above. For example, you may be given an individual current or potential difference and have to find the potential difference across the battery. Nevertheless, the general method above still applies. You just have to keep in mind what your given and unknowns are and do a bit of algebra.

Here's a complete analysis for a more complex circuit.