Direct-Current Circuits Chapter 26 Direct-Current Circuits

Learning Goals for Chapter 26 Looking forward at … how to analyze circuits with multiple resistors in series or parallel. rules that you can apply to any circuit with more than one loop. how to use an ammeter, voltmeter, ohmmeter, or potentiometer in a circuit. how to analyze circuits that include both a resistor and a capacitor. how electric power is distributed in the home.

What is an electric circuit, and how can the idea of electromotive force be used to trace the movement of charge in a circuit?

Electromotive force and circuits Just as a water fountain requires a pump, an electric circuit requires a source of electromotive force to sustain a steady current.

Electromotive force and circuits The influence that makes current flow from lower to higher potential is called electromotive force (abbreviated emf and pronounced “ee-em-eff”), and a circuit device that provides emf is called a source of emf. Note that “electromotive force” is a poor term because emf is not a force but an energy-per-unit-charge quantity, like potential. The SI unit of emf is the same as that for potential, the volt (1 V = 1 J/C). A typical flashlight battery has an emf of 1.5 V; this means that the battery does 1.5 J of work on every coulomb of charge that passes through it. We’ll use the symbol (a script capital E) for emf.

Internal resistance Real sources of emf actually contain some internal resistance r. The terminal voltage of the 12-V battery shown at the right is less than 12 V when it is connected to the light bulb.

Table 25.4 — Symbols for circuit diagrams

Potential changes The figure shows how the potential varies as we go around a complete circuit. The potential rises when the current goes through a battery, and drops when it goes through a resistor. Going all the way around the loop brings the potential back to where it started.

As current flows in a branch of most circuits the amount of current does not change, but the energy of the charges as they flow does. Resistances cause the flowing charges to lose energy and voltage sources like batteries can cause charges to gain energy or lose energy. A loss in energy is called a ‘voltage drop’ and a gain in energy a ‘voltage gain’. + and – signs are often shown. They indicate charges come into the resistor with more energy than they come out with. It must therefore be -8 V here. 10 W Say it is 12 V here 2 A From Ohm’s Law, there is a voltage drop of (2A)(10 W) = 20 V across this resistor. So the potential drops by 20 V. 20 V 2 A Say it is 12 V here 10 W What is the potential here? battery

Metallic conduction Electrons in a conductor are free to move through the crystal, colliding at intervals with the stationary positive ions. The motion of the electrons is analogous to the motion of a ball rolling down an inclined plane and bouncing off pegs in its path.

dc versus ac Our principal concern in this chapter is with direct-current (dc) circuits, in which the direction of the current does not change with time. Flashlights and automobile wiring systems are examples of direct-current circuits. Household electrical power is supplied in the form of alternating current (ac), in which the current oscillates back and forth. The same principles for analyzing networks apply to both kinds of circuits, and we conclude this chapter with a look at household wiring systems.

Resistors in series Resistors are in series if they are connected one after the other so the current is the same in all of them. The equivalent resistance of a series combination is the sum of the individual resistances:

R’s in Series Series: Every loop with resistor 1 also has resistor 2. (they all have the same current)

Example: A Series Resistor Circuit

Real Batteries (1) An ideal battery provides a potential difference that is a constant, independent of current flow or duration of use. But real batteries “sag” under load and become “weak” or “dead” as their chemical energy is used up. How can we include such effects? A reasonable approximation is to include an internal resistance rint. The internal resistance may increase as the battery ages and supplies energy. The rule is that the larger and more expensive the battery, the lower is rint. A regulated electronic power supply provides a very good approximation to a zero- resistance constant-potential ideal battery. Note: A real battery would read a voltage E if no current was flowing.

Real Batteries (2) DVbat Question: How can you measure rint? Answer: One (rather brutal) way is to vary an external load resistance R until the potential drop across R is ½E. Then R=rint because each drops ½E.

Resistors in parallel If the resistors are in parallel, the current through each resistor need not be the same, but the potential difference between the terminals of each resistor must be the same, and equal to Vab. The reciprocal of the equivalent resistance of a parallel combination equals the sum of the reciprocals of the individual resistances:

R’s in Parallel Parallel: Can make a loop that contains only those two resistors If only 2 R’s in parallel: (they all have the same voltage)

Example: A Parallel Resistor Circuit Three resistors are connected across a 9 V battery. Find the current through the battery. Find the potential differences across and currents through each resistor. I I1 I2 I3 DVbat

Voltage Divider and Current Divider Voltage Divider gives how the source voltage is split between the two resistors: R1 R2 Vs V1 V2 Current divider gives how the current splits between two resistors: R1 Vs R2 I I1 I2

Series versus parallel combinations When connected to the same source, two incandescent light bulbs in series (shown at top) draw less power and glow less brightly than when they are in parallel (shown at bottom).

Series and parallel combinations: Example 1 Resistors can be connected in combinations of series and parallel, as shown. In this case, try reducing the circuit to series and parallel combinations. For the example shown, we first replace the parallel combination of R2 and R3 with its equivalent resistance; this then forms a series combination with R1.

Series and parallel combinations: Example 2 Resistors can be connected in combinations of series and parallel, as shown. In this case, try reducing the circuit to series and parallel combinations. For the example shown, we first replace the series combination of R2 and R3 with its equivalent resistance; this then forms a parallel combination with R1.

Example: Analyzing a Complex Circuit (1) Four resistors are connected to a 2 V battery as shown. Find the current through the battery. Find the potential differences across and currents through each resistor.

Example: Analyzing a Complex Circuit (2)

Kirchhoff’s rules Many practical resistor networks cannot be reduced to simple series-parallel combinations. To analyze these networks, we’ll use the techniques developed by Kirchhoff.

Kirchhoff’s Laws Also called current law (KCL): S Iin = S Iout Also called the voltage law (KVL): S DVaround loop = 0 Gustav Kirchhoff discovered these laws while a student at Albertus University in Konigsberg in 1845.

Sign conventions for the loop rule Use these sign conventions when you apply Kirchhoff’s loop rule. In each part of the figure, “Travel” is the direction that we imagine going around the loop, which is not necessarily the direction of the current.

A single-loop circuit The circuit shown contains two batteries, each with an emf and an internal resistance, and two resistors. Using Kirchhoff’s rules, you can find the current in the circuit, the potential difference Vab, and the power output of the emf of each battery.

Multiloop Circuits Example: Find the unknown currents in the circuit below: You can't say any of the resistors are in series or parallel. You must use Kirchhoff's Laws. 3.0 v 10 W 6.0 v 20 W 4.0 v 25 W Step 1: Pick and label the unknown currents. The directions and names don't matter. Step 2: Put + and – signs. Current goes in + out – for resistors and + on long bar of battery and – on short bar. Step 3: Write KCL equation at a node where 3 or more wires come together. Step 4: Pick loop directions (arbitrary). Step 5: Apply KVL for each loop, taking – to + as positive voltage and + to – as negative voltage (or vice versa).

Step 1: Pick and label the unknown currents Step 1: Pick and label the unknown currents. The directions and names don't matter. Node A 3.0 v 10 W 6.0 v 20 W 4.0 v 25 W i1 i2 i3 Step 2: Put + and – signs. Current goes in + out – for resistors and + on long bar of battery and – on short bar. Step3: Write KCL equation at a node where 3 or more wires come together. Loop 1 Loop 2 Step 4: Pick loop directions (arbitrary). KCL Node A: i1 + i3 = i2 Step 5: Apply KVL for each loop following the loop direction from step 4, taking – to + as positive voltage and + to – as negative voltage (or vice versa). In Matlab: a = [1 -1 1;-20 -25 0;0 25 10] b=[0;-10;7] inv(a)*b ans = 0.1842 A 0.2526 A 0.0684 A KVL Loop 1: + 6.0 v – i1(20 W) + 4.0 v – i2(25 W) = 0 KVL loop 2: - 3.0 v + i3(10 W) + i2(25 W) - 4.0 v = 0 You now have 3 equations with 3 unknowns. Solve using algebra, your calculator or Matlab: = i1 = i2 = i3 If a current comes out negative it means that it is really going in the opposite direction (the numbers would be correct though).

D’Arsonval galvanometer A galvanometer measures the current that passes through it. Many electrical instruments, such as ammeters and voltmeters, use a galvanometer in their design.

Ammeters and voltmeters An ammeter measures the current passing through it. A voltmeter measures the potential difference between two points. Both instruments contain a galvanometer.

Ammeters x x I = ? Answer: You must break the circuit and insert an ammeter into the line of current flow. Question: How do you measure the current in a circuit? Ideal Ammeter: To have a minimum effect on the circuit being measured, the inserted ammeter must have zero resistance, so that there is zero potential difference across the ammeter. Electronic ammeters can give good approximations to this condition, but electro-mechanical ammeters may not. Note: “Clip on” ammeters that measure AC current without breaking the circuit are commercially available. They use magnetic induction (ch 29, 30).

Voltmeters Question: How do you measure the potential difference between two points in a circuit? Answer: You can connect one lead of a voltmeter to each point. Ideal Voltmeter: To have a minimum effect of the circuit being measured, the connected voltmeter must have infinite resistance, so that no current is diverted through the voltmeter. Electronic volt-meters can give good approxima-tions to this condition, but electro-mechanical voltmeters may not.

Ammeters and voltmeters in combination An ammeter and a voltmeter may be used together to measure resistance and power. Two ways to do this are shown below. Either way, we have to correct the reading of one instrument or the other unless the corrections are small enough to be negligible.

Voltmeters vs. Ammeters An ideal voltmeter has infinite internal resistance. It must be connected between circuit elements to measure the potential difference between two points in the circuit. V An ideal ammeter has zero internal resistance. It must be inserted by breaking a circuit connection to measure the current flowing through that connection in the circuit. A I X

Ohmmeters An ohmmeter consists of a meter, a resistor, and a source (often a flashlight battery) connected in series. The resistor Rs has a variable resistance, as is indicated by the arrow through the resistor symbol. To use the ohmmeter, first connect x directly to y and adjust Rs until the meter reads zero. Then connect x and y across the resistor R and read the scale.

Digital multimeters A digital multimeter can measure voltage, current, or resistance over a wide range.

The potentiometer The potentiometer is an instrument that can be used to measure the emf of a source without drawing any current from the source. Essentially, it balances an unknown potential difference against an adjustable, measurable potential difference. The term potentiometer is also used for any variable resistor, usually having a circular resistance element and a sliding contact controlled by a rotating shaft and knob. The circuit symbol for a potentiometer is shown below.