CHAPTER 2

Basic Laws

Here we explore two fundamental laws that govern electric circuits (Ohm’s law and Kirchhoff’s laws) and discuss some techniques commonly applied in circuit design and analysis.

2.1. Ohm’s Law Ohm’s law shows a relationship between voltage and current of a resis- tive element such as conducting wire or light bulb.

2.1.1. Ohm’s Law: The voltage v across a resistor is directly propor- tional to the current i flowing through the resistor. v = iR, where R = resistance of the resistor, denoting its ability to resist the flow of electric current. The resistance is measured in ohms (Ω). • To apply Ohm’s law, the direction of current i and the polarity of voltage v must conform with the passive sign convention. This im- plies that current flows from a higher potential to a lower potential

15 16 2. BASIC LAWS

in order for v = iR. If current flows from a lower potential to a higher potential, v = −iR.

l i

+ v R – Material with Cross-sectional resistivity r area A

2.1.2. The resistance R of a cylindrical conductor of cross-sectional area A, length L, and conductivity σ is given by L R = . σA Alternatively, L R = ρ A where ρ is known as the resistivity of the material in ohm-meters. Good conductors, such as copper and aluminum, have low resistivities, while insulators, such as mica and paper, have high resistivities. 2.1.3. Remarks: (a) R = v/i (b) Conductance : 1 i G = = R v 1 2 The unit of G is the mho (f) or siemens (S)

1Yes, this is NOT a typo! It was derived from spelling ohm backwards. 2In English, the term siemens is used both for the singular and plural. 2.1. OHM’S LAW 17

(c) The two extreme possible values of R. (i) When R = 0, we have a short circuit and v = iR = 0 showing that v = 0 for any i.

+ i

v = 0 R = 0

–

(ii) When R = ∞, we have an open circuit and v i = lim = 0 R→∞ R indicating that i = 0 for any v.

+ i = 0

v R = ∞

–

2.1.4. A resistor is either fixed or variable. Most resistors are of the fixed type, meaning their resistance remains constant. 18 2. BASIC LAWS

A common variable resistor is known as a potentiometer or pot for short

2.1.5. Not all resistors obey Ohms law. A resistor that obeys Ohms law is known as a linear resistor. • A nonlinear resistor does not obey Ohms law.

• Examples of devices with nonlinear resistance are the lightbulb and the diode. • Although all practical resistors may exhibit nonlinear behavior un- der certain conditions, we will assume in this class that all elements actually designated as resistors are linear. 2.1. OHM’S LAW 19

2.1.6. Using Ohm’s law, the power p dissipated by a resistor R is v2 p = vi = i2R = . R Example 2.1.7. In the circuit below, calculate the current i, and the power p.

i

+ 30 V DC 5 kΩ v –

Definition 2.1.8. The power rating is the maximum allowable power dissipation in the resistor. Exceeding this power rating leads to overheating and can cause the resistor to burn up. Example 2.1.9. Determine the minimum resistor size that can be con- 1 nected to a 1.5V battery without exceeding the resistor’s 4-W power rating. 54 CHAPTER TWO resistive networks

illustrative circuit. We will then apply the same systematic method to solve more complicated examples, including the one shown in Figure 2.1.

202.1 TERMINOLOGY 2. BASIC LAWS Lumped circuit elements are the fundamental building blocks of electronic cir- cuits. Virtually2.2. all of Node, our analyses Branches will be conducted and Loops on circuits containing two-terminal elements; multi-terminal elements will be modeled using combi- Definitionnations of two-terminal2.2.1. Since elements. the elements We have already of an seen electric several two-terminalcircuit can be in- terconnectedelements in such several as resistors, ways, voltage we need sources, to and understand current sources. some Electronic basic concept access to an element is made through its terminals. of networkAn topology. electronic circuit is constructed by connecting together a collection of • Networkseparate elements = interconnection at their terminals, as of shown elements in Figure or 2.2. devices The junction points • Circuitat which the=a terminals network of two with or more closed elements paths are connected are referred to as the nodes of a circuit. Similarly, the connections between the nodes are referred Definitionto as the edges2.2.2 or. branchesBranchof: a A circuit. branch Note represents that each element a single in Figure element 2.2 such forms a single branch. Thus an element and a branch are the same for circuits as a voltagecomprising source only two-terminalor a resistor. elements. A branch Finally, circuit representsloops are defined any two-terminal to be element.closed paths through a circuit along its branches. Several nodes, branches, and loops are identified in Figure 2.2. In the circuit Definitionin Figure 2.2,2.2.3 there. Node are 10 branches: A node (and is thus, the 10 “point” elements) of and connection 6 nodes. between two or moreAs branches. another example, a is a node in the circuit depicted in Figure 2.1 at • Itwhich is usually three branches indicated meet. Similarly, by a dotb is in a node a circuit. at which two branches meet. ab and bc are examples of branches in the circuit. The circuit has five branches • Ifand a four short nodes. circuit (a connecting wire) connects two nodes, the two nodesSince constitute we assume that a single the interconnections node. between the elements in a circuit are perfect (i.e., the wires are ideal), then it is not necessary for a set of elements Definitionto be joined2.2.4 together. Loop at a single: A point loop in isspace any for closed their interconnection path in a to circuit. be A closed pathconsidered is formed a single by node. starting An example at a of node, this is passing shown in through Figure 2.3. a While set of nodes and returningthe four to elements the starting in the figure node are connected without together, passing their through connection any does node more not occur at a single point in space. Rather, it is a distributed connection. than once.

Nodes

Loop Elements

FIGURE 2.2 An arbitrary circuit. Branch

Definition 2.2.5. Series: Two or more elements are in series if they are cascaded or connected sequentially and consequently carry the same current. Definition 2.2.6. Parallel: Two or more elements are in parallel if they are connected to the same two nodes and consequently have the same voltage across them. 2.2. NODE, BRANCHES AND LOOPS 21

2.2.7. Elements may be connected in a way that they are neither in series nor in parallel. Example 2.2.8. How many branches and nodes does the circuit in the following figure have? Identify the elements that are in series and in par- allel.

5 Ω

1 Ω 2 Ω DC 10 V 4 Ω

2.2.9. A loop is said to be independent if it contains a branch which 2.2 Kirchhoff’s Laws CHAPTER TWO 55 is not in any other loop. Independent loops or paths result in independent sets of equations. A network with b branches, n nodes, and ` independent loops will satisfy the fundamental theorem of network topology:

Elements B b = ` + n − 1. B FIGURE 2.3 Distributed C Definition 2.2.10. The primary signals withinA a circuitC are its currents interconnections of four circuit and voltages,A which we denote by the symbols i andD v, respectively. We elements that nonetheless occur define a branch currentD asDistributed the current node along a branch of the circuit, and at a single node. Ideal wires a branch voltage as the potential difference measured across a branch.

- FIGURE 2.4 Voltage and current i v definitions illustrated on a branch in Branch a circuit. current Branch + voltage

Nonetheless, because the interconnections are perfect, the connection can be considered to be a single node, as indicated in the figure. The primary signals within a circuit are its currents and voltages, which we denote by the symbols i and v, respectively. We define a branch current as the current along a branch of the circuit (see Figure 2.4), and a branch voltage as the potential difference measured across a branch. Since elements and branches are the same for circuits formed of two-terminal elements, the branch voltages and currents are the same as the corresponding terminal variables for the elements forming the branches. Recall, as defined in Chapter 1, the terminal variables for an element are the voltage across and the current through the element. As an example, i4 is a branch current that flows through branch bc in the circuit in Figure 2.1. Similarly, v4 is the branch voltage for the branch bc.

2.2 KIRCHHOFF’S LAWS Kirchhoff’s current law and Kirchhoff’s voltage law describe how lumped- parameter circuit elements couple at their terminals when they are assembled into a circuit. KCL and KVL are themselves lumped-parameter simplifications of Maxwell’s Equations. This section defines KCL and KVL and justifies that they are reasonable using intuitive arguments.1 These laws are employed in circuit analysis throughout this book.

1. The interested reader can refer to Section A.2 in Appendix A for a derivation of Kirchhoff’s laws from Maxwell’s Equations under the lumped matter discipline. 22 2. BASIC LAWS

2.3. Kirchhoff’s Laws Ohm’s law coupled with Kirchhoff’s two laws gives a sufficient, powerful set of tools for analyzing a large variety of electric circuits. 2.3.1. Kirchhoff’s current law (KCL): the algebraic sum of currents departing a node (or a closed boundary) is zero. Mathematically, X in = 0 n KCL is based on the law of conservation of charge. An alternative form of KCL is Sum of currents (or charges) drawn as entering a node = Sum of the currents (charges) drawn as leaving the node.

i5 i1

i4

i2 i3

Note that KCL also applies to a closed boundary. This may be regarded as a generalized case, because a node may be regarded as a closed surface shrunk to a point. In two dimensions, a closed boundary is the same as a closed path. The total current entering the closed surface is equal to the total current leaving the surface.

Closed boundary 2.3. KIRCHHOFF’S LAWS 23

Example 2.3.2. A simple application of KCL is combining current sources in parallel.

IT a

I1 I2 I3

b (a)

IT a

IT = I1 – I2 + I3

b (b)

A Kirchhoff’s voltage law (KVL): the algebraic sum of all voltages around a closed path (or loop) is zero. Mathematically, M X vm = 0 m=1 KVL is based on the law of conservation of energy. An alternative form of KVL is Sum of voltage drops = Sum of voltage rises.

v v + 2 – + 3 –

v1 v4

– v5 + 24 2. BASIC LAWS

Example 2.3.3. When voltage sources are connected in series, KVL can be applied to obtain the total voltage.

a +

V1

Vab V2 a +

V3 Vab VS = V1 + V2 – V3

– – b b

Example 2.3.4. Find v1 and v2 in the following circuit. 4 Ω

+ v1 –

10 V 8 V

v + 2 – 2 Ω

Example 2.3.5. halliday_c27_705-734v2.qxd 23-11-2009 14:35 Page 725

HALLIDAY REVISED

PART 3 QUESTIONS 725

Pr at which energy is dissipated as thermal energy in the battery is RC Circuits When an emf Ᏹ is applied to a resistance R and ca- 2 pacitance C in series, as in Fig. 27-15 with the switch at a, the charge Pr i r. (27-16) on the capacitor increases according to The rate Pemf at which the chemical energy in the battery changes is q C Ᏹ(1 et/RC) (charging a capacitor), (27-33) Ᏹ Pemf i . (27-17) Ᏹ in which C q0 is the equilibrium (final) charge and RC t is the capacitive time constant of the circuit. During the charging, the Series Resistances When resistances are in series, they have current is the same current. The equivalent resistance that can replace a se- ries combination of resistances is dq Ᏹ i ΂ ΃et/RC (charging a capacitor). (27-34) n dt R Req ͚ Rj (n resistances in series). (27-7) j1 When a capacitor discharges through a resistance R, the charge on the capacitor decays according to Parallel Resistances When resistances are in parallel, t/RC they have the same potential difference. The equivalent resistance q q0e (discharging a capacitor). (27-39) that can replace a parallel combination of resistances is given by During the discharging, the current is 1 n 1 dq q0 ͚ (n resistances in parallel). (27-24) i ΂ ΃e t/RC (discharging a capacitor). (27-40) Req j1 Rj dt RC

and then again rank the segments. (c) What is the direction of the 1 (a) In Fig. 27-18a, with R1 R2, is the potential difference electric field along the x axis? across R2 more than, less than, or equal to that across R1? (b) Is the current through resistor R2 more than, less than, or equal to that 5 For each circuit in Fig. 27-20, are the resistors connected in through resistor R1? series, in parallel, or neither? R 3 +–

R1 R2 + – +– + R2 R1 – – + R3 (a) (b) (a)(b)(c) R1 Fig. 27-20 Question 5. – R 2.3. KIRCHHOFF’S LAWS 25 3 + 6 Res-monster maze. In Fig. 27-21, all the resistors have a R1 R2 resistance of 4.0 and all the (ideal) batteries have an emf of 4.0 + R ExampleV. What is2.3.6 the current (HRW through p. 725) .resistorIn the R figure? (If below,you can all find the the resistors – 3 (c) (d) haveproper a resistance loop through of 4.0 this Ω and maze, all you the can (ideal) answer batteries the question are 4.0 with V. Whata is the currentfew seconds through of mental resistor calculation.) R? R2 Fig. 27-18 Questions 1 and 2.

2 (a) In Fig. 27-18a, are resistors R1 and R3 in series? (b) Are resistors R1 and R2 in parallel? (c) Rank the equivalent resistances of the four circuits shown in Fig. 27-18, greatest first. 3 You are to connect resistors R1 and R2, with R1 R2, to a bat- tery, first individually, then in series, and then in parallel. Rank R those arrangements according to the amount of current through the battery, greatest first. +– 4 In Fig. 27-19, a circuit consists of a battery and two uniform resistors, R1 R2 and the section lying along an x axis x is divided into five segments of abcde equal lengths. (a) Assume that R1 Fig. 27-21 Question 6. R2 and rank the segments according Fig. 27-19 Question 4. to the magnitude of the average 7 A resistor R1 is wired to a battery, then resistor R2 is added in electric field in them, greatest first. (b) Now assume that R1 R2 series. Are (a) the potential difference across R1 and (b) the cur- 26 2. BASIC LAWS

2.4. Series Resistors and Voltage Division

2.4.1. When two resistors R1 and R2 ohms are connected in series, they can be replaced by an equivalent resistor Req where

Req = R1 + R2 . In particular, the two resistors in series shown in the following circuit

i a R1 R2

+ v1 – + v2 –

v

b can be replaced by an equivalent resistor Req where Req = R1 +R2 as shown below.

i a Req + v –

v

b

The two circuits above are equivalent in the sense that they exhibit the same voltage-current relationships at the terminals a-b.

Voltage Divider: If R1 and R2 are connected in series with a voltage source v volts, the voltage drops across R1 and R2 are

R1 R2 v1 = v and v2 = v R1 + R2 R1 + R2 2.5. PARALLEL RESISTORS AND CURRENT DIVISION 27

Note: The source voltage v is divided among the resistors in direct propor- tion to their resistances.

2.4.2. In general, for N resistors whose values are R1,R2,...,RN ohms connected in series, they can be replaced by an equivalent resistor Req where N X Req = R1 + R2 + ··· + RN = Rj j=1 If a circuit has N resistors in series with a voltage source v, the jth resistor Rj has a voltage drop of

Rj vj = v R1 + R2 + ··· + RN 2.5. Parallel Resistors and Current Division

When two resistors R1 and R2 ohms are connected in parallel, they can be replaced by an equivalent resistor Req where 1 1 1 = + Req R1 R2 or R1R2 Req = R1kR2 = R1 + R2

i Node a

i1 i2

v R1 R2

Node b

Current Divider: If R1 and R2 are connected in parallel with a current source i, the current passing through R1 and R2 are R2 R1 i1 = i and i2 = i R1 + R2 R1 + R2 Note: The source current i is divided among the resistors in inverse pro- portion to their resistances. 28 2. BASIC LAWS

Example 2.5.1.

Example 2.5.2. 6k3 = Example 2.5.3. (a)k(na) = Example 2.5.4. (ma)k(na) = In general, for N resistors connected in parallel, the equivalent resistor Req = R1kR2k · · · kRN is 1 1 1 1 = + + ··· + Req R1 R2 RN

Example 2.5.5. Find Req for the following circuit.

4 Ω 1 Ω

2 Ω Req 5 Ω

6 Ω 3 Ω 8 Ω

Example 2.5.6. Find the equivalent resistance of the following circuit.

2.5. PARALLEL RESISTORS AND CURRENT DIVISION 29

i 4 Ω a i0

+ 12 V 6 Ω v0 3 Ω –

b

Example 2.5.7. Find io, vo, po (power dissipated in the 3Ω resistor).

Example 2.5.8. Three light bulbs are connected to a 9V battery as shown below. Calculate: (a) the total current supplied by the battery, (b) the current through each bulb, (c) the resistance of each bulb.

I I1

I2 + V2 R2 – + 9 V V1 R1 15 W + – 9 V 20 W V3 R3 10 W – 30 2. BASIC LAWS

2.6. Practical Voltage and Current Sources An ideal voltage source is assumed to supply a constant voltage. This implies that it can supply very large current even when the load resistance is very small. However, a practical voltage source can supply only a finite amount of current. To reflect this limitation, we model a practical voltage source as an ideal voltage source connected in series with an internal resistance rs, as follows:

Similarly, a practical current source can be modeled as an ideal current source connected in parallel with an internal resistance rs. 2.7. Measuring Devices Ohmmeter: measures the resistance of the element. Important rule: Measure the resistance only when the element is discon- nected from circuits. Ammeter: connected in series with an element to measure current flowing through that element. Since an ideal ammeter should not restrict the flow of current, (i.e., cause a voltage drop), an ideal ammeter has zero internal resistance. Voltmeter:connected in parallel with an element to measure voltage across that element. Since an ideal voltmeter should not draw current away from the element, an ideal voltmeter has infinite internal resistance.