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

Energy and power in electric circuits The box represents a circuit element with potential difference Vab = Va − Vb between its terminals and current I passing through it in the direction from a toward b. If the potential at a is lower than at b, then there is a net transfer of energy out of the circuit element. The time rate of energy transfer is power, denoted by P, so we write:

Power The upper rectangle represents a source with emf and internal resistance r, connected by ideal wires to an external circuit represented by the lower box. Point a is at higher potential than point b, so Va > Vb and Vab is positive. P = VabI

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.

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.

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 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.

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.

R-C circuits: Charging a capacitor: Slide 1 of 4 Shown is a simple R-C circuit for charging a capacitor. We idealize the battery to have a constant emf and zero internal resistance, and we ignore the resistance of all the connecting conductors. We begin with the capacitor initially uncharged.

R-C circuits: Charging a capacitor: Slide 2 of 4 At some initial time t = 0 we close the switch, completing the circuit and permitting current around the loop to begin charging the capacitor. As t increases, the charge on the capacitor increases, while the current decreases.

R-C circuits: Charging a capacitor: Slide 3 of 4 The charge on the capacitor in a charging R-C circuit increases exponentially, with a time constant τ = RC.

R-C circuits: Charging a capacitor: Slide 4 of 4 The current through the resistor in a charging R-C circuit decreases exponentially, with a time constant τ = RC.

R-C circuits: Discharging a capacitor: Slide 1 of 4 Shown is a simple R-C circuit for discharging a capacitor. Before the switch is closed, the capacitor charge is Q0, and the current is zero.

R-C circuits: Discharging a capacitor: Slide 2 of 4 At some initial time t = 0 we close the switch, allowing the capacitor to discharge through the resistor. As t increases, the magnitude of the current decreases, while the charge on the capacitor also decreases.

R-C circuits: Discharging a capacitor: Slide 3 of 4 The charge on the capacitor in a discharging R-C circuit decreases exponentially, with a time constant τ = RC.

R-C circuits: Discharging a capacitor: Slide 4 of 4 The magnitude of the current through the resistor in a discharging R-C circuit decreases exponentially, with a time constant τ = RC.

Power distribution systems The figure below shows the basic idea of house wiring. The “hot line” has an alternating sinusoidal voltage with a root-mean-square value of 120 V. The “neutral line” is connected to “ground,” which is usually an electrode driven into the earth.

Circuit overloads A fuse (Figure a) contains a link of lead–tin alloy with a very low melting temperature; the link melts and breaks the circuit when its rated current is exceeded. A circuit breaker (Figure b) is an electromechanical device that performs the same function, using an electromagnet or a bimetallic strip to “trip” the breaker and interrupt the circuit when the current exceeds a specified value. Circuit breakers have the advantage that they can be reset after they are tripped, while a blown fuse must be replaced.

Why it is safer to use a three-prong plug

Potential across an inductor Potential across an inductor depends on rate of change of current through it. Self-induced EMF does not oppose current, but does opposes a change in current. If i doesn’t change in time, EMF of inductor = 0! Ɛ = – L 𝒅𝒊 𝒅𝒕

Potential across an inductor The potential across a resistor drops in the direction of current flow Vab = Va – Vb > 0 Three cases of the potential across an inductor: No change Increasing current => Va > Vb Decreasing current => Va < Vb

Potential across an inductor The potential across an inductor depends on the rate of change of the current through it. No change Increasing current => Va > Vb Decreasing current => Va < Vb No change in current: EMF of inductor = 0

Potential across an inductor The potential across an inductor depends on the rate of change of the current through it. No change Increasing current => Va > Vb Decreasing current => Va < Vb Increasing current: EMF of inductor against current Inductor acts like a temporary voltage source OPPOSITE to change.

Potential across an inductor Inductor acts like a temporary voltage source OPPOSITE to change. This implies the inductor looks like a battery pointing the other way! Note Va > Vb! Direction of current flow from a battery oriented this way

Potential across an inductor Wait a minute! Wasn’t it EMF = – L di/dt ? Yes! Sign depends upon di/dt, always opposing) Vab = Va – Vb = L di/dt If di/dt = 0, Vab = 0 If di/dt > 0, Vab > 0 Va – Vb > 0 implies Va > Vb! Vab = L di/dt If current increases, induced EMF pushes back on circuit as if inductor was a battery oriented backwards!

Potential across an inductor What if current was decreasing? Same result! The inductor acts like a temporary voltage source pointing OPPOSITE to the change. Now inductor pushes current in original direction Note Va < Vb!

The R-L circuit An R-L circuit contains a resistor and inductor and possibly an emf source. Start with both switches open Close Switch S1: Current flows Inductor resists flow Actual current less than maximum E/R E – i(t)R – L(di/dt) = 0 di/dt = E /L – (R/L) i(t) First-order differential equation

The R-L circuit Close Switch S1: E – i(t)R – L(di/dt) = 0 di/dt = E /L – (R/L) i(t) Boundary Conditions At t=0, di/dt = E /L i() = E /R Solve this 1st order diff eq: i(t) = (E /R) (1 – e –(R/L)t) Looks an awful lot like the RC circuit current!

Current growth in an R-L circuit i(t) = (E /R) (1 – e –(R/L)t) The time constant for an R-L circuit is  = L/R. Compare with the time constant for an R-C circuit:  = RC [ ]= L/R = Henrys/Ohm = (Tesla-m2/Amp)/Ohm = (Newtons/Amp-m) (m2/Amp)/Ohm = (Newton-meter) / (Amp2-Ohm) = Joule/Watt = Joule/(Joule/sec) = seconds! 

The R-L circuit E = i(t) R + L (di/dt) Power in circuit = E I E i = i2R + Li (di/dt) Some power radiated in resistor Some power stored in inductor

The R-L circuit example R = 175 W; i = 36 mA; current limited to 4.9 mA in first 58 ms. What is required EMF? What is required inductor? What is the time constant?

The R-L circuit Example R = 175 W; i = 36 mA; current limited to 4.9 mA in first 58 ms. What is required EMF? What is required inductor? What is the time constant? EMF = IR = (0.36 mA)x(175W ) = 6.3 V i(t) = E /R (1-e –(R/L)t) i(58ms) = 4.9 mA 4.9mA = 6.3V(1-e –(175/L)0.000058) L = 69 mH  = L/R = 390 ms

Current decay in an R-L circuit Now close the second switch! Current decrease is opposed by inductor EMF is generated to keep current flowing in the same direction Current doesn’t drop to zero immediately

Current decay in an R-L circuit Now close the second switch! –i(t)R - L(di/dt) = 0 Note di/dt is NEGATIVE! i(t) = – L/R (di/dt) i(t) = i(0) e –(R/L)t i(0) = maximum current before the second switch is closed (this is a boundary condition)

Current decay in an R-L circuit i(t) = i(0)e –(R/L)t

Current decay in an R-L circuit Test yourself! Signs of Vab and Vbc when S1 is closed? Vab > 0; Vbc > 0 Vab > 0, Vbc < 0 Vab < 0, Vbc > 0 Vab < 0, Vbc < 0

Current decay in an R-L circuit Test yourself! Signs of Vab and Vbc when S1 is closed? Vab > 0; Vbc > 0 WHY? Current increases suddenly, so inductor resists change

Current decay in an R-L circuit Test yourself! Signs of Vab and Vbc when S1 is closed? Vab > 0; Vbc > 0 WHY? Current still flows around the circuit counterclockwise through resistor EMF generated in L is from c to b So Vb> Vc!

Current decay in an R-L circuit Test yourself! Signs of Vab and Vbc when S2 is closed, S1 open? Vab > 0; Vbc > 0 Vab > 0, Vbc < 0 Vab < 0, Vbc > 0 Vab < 0, Vbc < 0

Current decay in an R-L circuit Test yourself! Signs of Vab and Vbc when S2 is closed, S1 open? Vab > 0, Vbc < 0 WHY? Current still flows counterclockwise di/dt <0; EMF generated in L is from b to c! So Vb < Vc!

The L-C circuit An L-C circuit contains an inductor and a capacitor and is an oscillating circuit. Initially capacitor fully charged; close switch Charge flows FROM capacitor, but inductor resists that increased flow. Current builds in time. At maximum current, charge flow now decreases through inductor Inductor now resists decreased flow, and keeps pushing charge in the original direction i

The L-C circuit An L-C circuit contains an inductor and a capacitor and is an oscillating circuit. Initially capacitor fully charged; close switch Charge flows FROM capacitor, but inductor resists that increased flow. Current builds in time. Capacitor slowly discharges At maximum current, no charge is left on capacitor; current now decreases through inductor Inductor now resists decreased flow, and keeps pushing charge in the original direction i

The L-C circuit An L-C circuit contains an inductor and a capacitor and is an oscillating circuit.

The L-C circuit An L-C circuit contains an inductor and a capacitor and is an oscillating circuit. Now capacitor fully drained; Inductor keeps pushing charge in the original direction Capacitor charge builds up on other side to a maximum value While that side charges, “back EMF” from capacitor tries to slow charge build-up Inductor keeps pushing to resist that change. i

The L-C circuit An L-C circuit contains an inductor and a capacitor and is an oscillating circuit.

The L-C circuit An L-C circuit contains an inductor and a capacitor and is an oscillating circuit. Now capacitor charged on opposite side; Current reverses direction! System repeats in the opposite direction i

The L-C circuit An L-C circuit contains an inductor and a capacitor and is an oscillating circuit.

Electrical oscillations in an L-C circuit Analyze the current and charge as a function of time. Do a Kirchoff Loop around the circuit in the direction shown. Remember i can be + or – Recall C = q/V For this loop: – L (di/dt) – qC = 0

Electrical oscillations in an L-C circuit – L (di/dt) – qC = 0 i(t) = dq/dt L (d2q/dt2) + C q = 0 This second-order differential equation is a case of simple harmonic motion (SHM), as we saw for pendulums and springs The solution of the equation is q(t)= Qmax cos(w t + f) where w = 1/(LC)½

Electrical oscillations in an L-C circuit q(t)= Qmax cos(w t + f) So i(t) = – w Qmax sin(w t + f) (based on this ASSUMED direction!) w = 1/(LC)½ = angular frequency 𝜙 = phase angle

The L-C circuit An L-C circuit contains an inductor and a capacitor and is an oscillating circuit.

Electrical and mechanical oscillations SHM and L-C circuit oscillations are analogous, and various quantities are directly analogous: for instance, the amplitude A of motion of a spring is analogous to the charge Q in an L-C circuit.

The L-R-C series circuit An L-R-C circuit exhibits damped harmonic motion if the resistance is not too large.