In Chapter 1, we presented Kirchhoff's laws (which govern the interaction between circuit elements) and Ohm's law (which governs the voltage-current relationships for resistors). These analytical tools provide us with the ability to analyze any circuit containing only resistors and ideal power supplies. However, we also saw in Chapter 1 that a circuit analysis, which relies strictly on a brute-force application of these tools can become complex rapidly - we essentially must use as our unknowns the voltage differences across all resistors and the currents through all resistors. This generally results in a large number of unknowns and a correspondingly large number of equations, which must be written and solved in order to analyze any but the simplest circuit.

In the next few chapters, we will still apply Kirchhoff's laws and Ohm's law in our circuit analysis, but we will focus on improving the efficiency of our analyses. Typically, this improvement in efficiency is achieved by reducing the number of unknowns in the circuit, which reduces the number of equations, which must be written to describe the circuit's operation.

In this chapter, we introduce analysis methods based on circuit reduction. Circuit reduction consists of combining resistances in a circuit to a smaller number of resistors, which are (in some sense) equivalent to the original resistive network. Reducing the number of resistors, of course, reduces the number of unknowns in a circuit.

We begin our discussion of circuit reduction techniques by presenting two specific, but very useful, concepts: Series and parallel resistors. These concepts will lead us to voltage and current divider formulas. We then consider reduction of more general circuits, which typically corresponds to identifying multiple sets of series and parallel resistances in a complex resistive network. This chapter then concludes with two important examples of the application of circuit reduction techniques: the analysis of non-ideal power sources and non-ideal measurement devices; without an understanding of these devices, it is impossible to build practical circuits or understand the consequences of a voltage or current measurement.

• Identify series and parallel combinations of circuit elements
• Determine the equivalent resistance of series resistor combinations
• Determine the equivalent resistance of parallel resistor combinations
• State voltage and current divider relationships from memory
• Determine the equivalent resistance of electrical circuits consisting of series and parallel combinations of resistors
• Sketch equivalent circuits for non-ideal voltage and current meters
• Analyze circuits containing non-ideal voltage or current sources
• Determine the effect of non-ideal meters on the parameter being measured

There are a number of common circuit element combinations that are quite easily analyzed. These “special cases” are worth noting since many complicated circuits contain these circuit combinations as sub-circuits. Recognizing these sub-circuits and analyzing them appropriately can significantly simplify the analysis of a circuit.

This chapter emphasizes two important circuit element combinations: elements in series and elements in parallel. Also discussed is the use of these circuit element combinations to reduce the complexity of a circuit’s analysis.

Circuit elements are said to be connected in series if all of the elements carry the same current. An example of two circuit elements connected in series is shown in Fig. 2.1. Applying KCL at node a and taking currents out of the node as positive we see that:

$$-i_1+i_2=0$$

or

$$i_1=i_2 (Eq. 2.1)$$

Equation (2.1) is a direct outcome of the fact that the (single) node a in Fig. 2.1 interconnects only two elements - there are no other elements connected to this node through which curren can be diverted. This observation is so apparent (in many cases¹⁾) that equation (2.1) is generally written by inspection for series elements such as those shown in Fig. 2.1 without explicitly writing KCL.

When resistors are connected in series, a simplification of the circuit is possible. Consider the resistive circuit shown in Fig. 2.2(a). Since the resistors are in series, they both carry the same current. Ohm's law gives:

\begin{align*} v_1 = R_1i \\ v_2=R_2i \\ (Eq.2.2) \end{align*}

Applying KVL around the loop:

$$ -v+v_1+v_2=0 \Rightarrow v=v_1+v_2 (Eq. 2.3)$$

Substituting equations (2.2) into equation (2.3) and solving for the current i results in:

$$ i = \frac{v}{R_1+R_2} (Eq. 2.4) $$

Now cinsider the circuit of Fig. 2.2(b). Application of Ohm's law to this circuit and solution for the current i gives:

$$ i = \frac{v}{R_{eq}} (Eq. 2.5) $$

Comparing equation (2.4) with equation (2.5), we can see that the circuits of Figs. 2.2(a) and 2.2(b) are indistinguishable if we select:

$$R_{eq}=R_1+R_2 (Eq. 2.6)$$

Figures 2.2(a) and 2.2(b) are called equivalent circuits if the equivalent resistance of Fig. 2.2(b) is chosen as shown in equation (2.6). R_eq of equation (2.6) is called the equivalent resistance of the series combination of resistors R₁ and R₂.

This result can be generalized to a series combination of N resistances as follows:

A series combination of N resistors R₁, R₂, …, R_N can be replaced with a single equivalent resistance $R_{eq}=R_1+R_2+ \cdots + R_N$. The equivalent circuit can be analyzed to determine the current through the series combination of resistors.

Combining equations (2.2) with equation (2.4) results in the following expressions for V_i and v₂:

$$v_1= \frac{R_1}{R_1+R_2}v (Eq. 2.7)$$

$$v_2= \frac{R_2}{R_1+R_2}v (Eq. 2.8)$$

These results are commonly called voltage divider relationships, because they state that the total voltage drop across a series combination of resistors is divided among the individual resistors in the combination. The ratio of each individual resistor's voltage drop to the overall voltage drop is the same as the ratio of the individual resistance to the total resistance.

The above results can be generalized for a series combination of N resistance as follows:

The voltage drop across any resistor in a series combination of N resistances is proportional to the total voltage drop across the combination of resistors. The constant of proportionality is the same as the ratio of the individual resistor value to the total resistance of the series combination. For example, the voltage drop of the k^th resistance in a series combination of resistors given by:

$$ v_k={R_k}{R_1+R_2+ \cdots +R_N}v (Eq. 2.9)$$

For the circuit below, determine the voltage across the 5Ω resistor, v, the current supplied by the source, i, and the power supplied by the source.

The voltage across the 5Ω resistor can be determined from our voltage divider relationship:

$$ v = [\frac{5Ω}{5Ω + 15Ω + 10Ω}] \cdot 15V= \frac{5}{30} \cdot 15V = 2.5V $$

The current supplied by the source can be determined by dividing the total voltage by the equivalent resistance:

$$i= \frac{15V}{R_{eq}}= \frac{15V}{5Ω+15Ω+10Ω}= \frac{15V}{30Ω}=0.5A$$

The power supplied by the source is the product of the source voltage and the source current:

$$P=iv=(0.5A)(15V)=7.5W$$

We can double-check the consistency between the voltage v and the current i with Ohm's law. Applying Ohm's law to the 5Ω resistor, with a 0.5A current, results in $v=(5Ω)(0.5A)=2.5V$, which agrees with the result obtained using the voltage divider relationship.

• If only two elements connect at a single node, the two elements are in series. A more general definition, however, is that circuit elements in series all share the same current - this definition allows us to determine series combinations that contain more than two elements. Identification of series circuit elements allows us to simplify our analysis, since there is a reduction in the number of unknowns: there is only a single unknown current for all series elements.
• A series combination of resistors can be replaced by a single equivalent resistance, if desired. The equivalent resistance is simply the sum of the individual resistances in the series combination. Therefore, a series combination of N resistors $R_1,R_2, \cdots, R_N$ can be replaced with a single equivalent resistance $R_{eq}=R_1+R_2+ \cdots + R_N$.
• If the total voltage difference across a set of series is known, the voltage differences across any individual resistor can be determined by the concept of voltage division. The term voltage division comes from the fact that the voltage drop across a series combination of resistors is divide among the individual resistors. The ratio between the voltage difference across a particular resistor and the total voltage difference is the same as the ratio between the resistance of that resistor and the total resistance of the combination. If v_k is the voltage across the k_th resistor, and R_TOT is the total resistance of the series combination, the mathematical statement of this concept is:

$$ \frac{v_k}{v_{TOT}} = \frac{R_k}{R_{TOT}} $$

1. Determine the voltage V₁ in the circuit below.

Circuit elements are said to be connected in parallel if all of the elements share the same pair of nodes. An example of two circuit elements connected in parallel is shown in Fig. 2.3. Applying KVL around the loop of Fig. 2.3 results in:

$$ v_1=v_2 (Eq. 2.10)$$

This result is so common that equation (2.10) is generally written by inspection fro parallel elements such as those shown in Fig. 2.3 without explicitly writing KVL.

We can simplify circuits, which consist of resistors connected in parallel. Consider the resistive circuit shown in Fig. 2.4(a). The ressitors are connected in parallel, so both resistors have a voltage difference of v. Ohm's law applied to each resistor results in:

$$ i_1= \frac{v}{R_1} \\ i_2= \frac{v}{R_2} \\ (Eq. 2.11) $$

Applying KCL at node a:

$$i=i_1 + i_2 (Eq. 2.12)$$

Substituting equations (2.11) into equation (2.12):

$$ i = \left[ \frac{1}{R_1} + \frac{1}{R_2} \right] v (Eq. 2.13)$$

or

$$v= \frac{1}{\frac{1}{R_1}+\frac{1}{R_2}}\cdot i (Eq. 2.14)$$

If we set $ R_{eq} = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2}}$, we can draw Fig. 2.4(b) as being equivalent to Fig. 2.4(b).

We can generalize this result for N parallel resistances:

A parallel combination of N resistors $R_1, R_2, \cdots, R_N$ can be replaced with a single equivalent resistance:

$$ R_{eq} = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2}+ \cdots \frac{1}{R_N}} (Eq. 2.15)$$

For the special case of two parallel resistances, $R_1$ and $R_2$, the equivalent resistance is commonly written as:

$$R_{eq}= \frac{R_1R_2}{R_1+R_2} (Eq. 2.16)$$

This alternative way to calculate $R_{eq}$ can be also used to calculate $R_{eq}$ for larger numbers of parallel resistors since any number of resistors could be combined two at a time.

Substituting equation (2.14) into equations (2.11) results in:

$$i_1 = \frac{1}{R_1}\cdot \frac{i}{ \frac{1}{R_1} + \frac{1}{R_2}} (Eq. 2.17)$$

Simplifying:

$$i_1=\frac{R_2}{R_1+R_2} (Eq. 2.18)$$

Likewise, for the current $i_2$:

$$i_2 = \frac{R_1}{R_1+R_2} (Eq. 2.19)$$

Equations (2.18) and (2.19) are the current divider relationships for two parallel resistances, so called because the current into the parallel resistance combination is divided between the two resistors. The ratio of one resistor's current to the overall current in the same as the ratio of the other resistance to the total resistance.

The above results can be generalized for a series combination of N resistances. By Ohm's law, $v = R_{eq} i$. Substituting our previous result for the equivalent resistance for a parallel combination of N resistors results in:

$$v= \frac{1}{\frac{1}{R_1}+\frac{1}{R_2} + \cdots \frac{1}{R_N}} \cdot i (Eq. 2.20)$$

Since the voltage difference across all resistors is the same, the current through the k^th resistor is, by Ohm's law:

$$i_k=\frac{v}{R_k} (Eq. 2.21)$$

Where $R_k$ is the resistance of the k^th resistor. Combining equations (2.20) and (2.21) gives:

$$i_k= \frac{\frac{1}{R_k}}{\frac{1}{R_1}+\frac{1}{R_2}+ \cdots \frac{1}{R_N}} \cdot i (Eq. 2.22)$$

It is often more convenient to provide the generalized result of equation (2.20) in terms of the conductance of the individual resistors. Recall that the conductance is the reciprocal of the resistance, $G=\frac{1}{R}$. Thus, equation (2.22) can be re-expressed as follows:

The Current through any resistor in a parallel combination of N resistances is proportional to the total current into the combination of resistors. The constant of proportionality is the same as the ratio of the conductance of the individual resistor value to the total conductance of the parallel combination. For example, the current through the k^th resistance in a parallel combination of resistors is given by:

$$i_k=\frac{G_k}{G_1+G_2+ \cdots + G_N}i (Eq. 2.23)$$

Where i is the total current through the parallel combination of resistors.

One final comment about notation: two parallel bars are commonly used as shorthand notation to indicate that two circuit elements are in parallel. For example, the notation $R_1 \parallel R_2$ indicates that the resistors $R_2$ and $R_2$ are in parallel. The notation $R_1 \parallel R_2$ is often used as shorthand notation for the equivalent resistance of the parallel resistance combination, in lieu of equation (2.16).

Double-checking results for parallel resistances:

• The equivalent resistance for a parallel combination of N resistors will always be less than the smallest resistance in the combination. In fact, the equivalent resistance will always obey the following inequalities:

$$\frac{R_{min}}{N} \leq R_{eq} \leq R_{min}$$

• Where $R_{min}$ is the smallest resistance value in the parallel combination.
• In a parallel combination of resistances, the resistor with the smallest resistance will have the largest current and the resistor with the largest resistance will have the smallest current.

• If several elements interconnect the same two nodes, the two elements are in parallel. A more general definition, however, is that circuit elements in parallel all share the same voltage difference. As with series circuit elements, identification of parallel circuit elements allows us to simplify our analysis, since there is a reduction in the number of unknowns: there is only a single unknown voltage difference for all of the parallel elements.
• A parallel combination of resistors can be replaced by a single equivalent resistance, if desired. The conductance of the parallel combination is simply the sum of the individual conductance of the parallel resistors. Therefore, a parallel combination of N resistors $R_1$, $R_2$,