Circuit BasicsDirect Current (DC) Circuits

1.5 V+–

wire

open switch

closed switch

2-way switch

ideal battery

capacitor

resistor

47 F

4.7 k

These circuit elements and many others can be combined to produce a limitless variety of useful devices

•Two devices are in series if they are connected at one end, and nothing else is connected there

•Two devices are in parallel if they are connected at both ends

Resistors in Parallel and in Series

•When resistors are in series, the same current must go through both of them•The total voltage difference is

•The two resistors act like one with resistance

1 1V IR 2 2V IR

R1 R2

1 2V V V 1 2I R R

1 2R R R

•When resistors are in parallel, the same potential is across both of them•The total current through them is

•The two resistors act like one with resistance

R1 R2

1 1

2 2

V I R

V I R

1 2I I I 1 2

V V

R R

V

R

1 2

1 1 1

R R R

Parallel and Series - Formulas

Capacitor Resistor Inductor*

Series

Parallel

Fundamental Formula

1 2R R R

1 2

1 1 1

R R R 1 2C C C

1 2

1 1 1

C C C

1 2L L L

1 2

1 1 1

L L L

QV

C V IR L

dIL

dtE

* To be defined in a later chapter

The Voltage Divider

+–

•Many circuits can be thought of as a voltage divider•Intentionally or unintentionally

R1

R2

If Mr. Curious has a resistance of 10 k and the light bulb has a resistance of 240 , how bright is Mr.

Curious?

1 2R R R 1 2

IR R

E1

1 11 2

RV IR

R R

E

22 2

1 2

RV IR

R R

E

What’s the voltage drop across each of the resistors?

+–120 V curious

10000117 V

10240V E=

The larger resistor gets most of the voltage

Not very bright

Ideal vs. Non-Ideal Batteries•Up until now, we’ve treated a battery as if it produced a fixed voltage, no matter what we demand of it•Real batteries also have resistance

•It limits the current and therefore the power that can be delivered

•If the internal resistance r is small compared to other resistances in the problem, we can ignore it

ideal battery

+–

realistic battery

+–r

+–30 V10

50

A 30 V battery with 10 of internal resistance is connected to a 50 resistor. What is the actual voltage across the 50 resistor?

11

1 2

RV

R R

E 25 V

Kirchoff’s Laws

Kirchoff’s First Law:The total current into any

vertex equals the current out of that

vertex

•Kirchoff’s Laws help us figure out where and how much current is flowing in a circuit•The first step is to assign a direction and a current to every part of a circuit

•Items in series must have the same current in them•Then you apply the two laws, which can be thought of as conservation of charge and conservation of voltage, which you apply to vertices and loops respectively.

+–

+ –

3

5

4

12 V

6 V

I1

I3

I2

Kirchoff’s Second Law:The total voltage change

around a loop is always zero

•These yield a series of equations, which you then solve

in outI I

Kirchoff’s First LawKirchoff’s First Law:

The total current into any vertex equals the current out of that

vertex

•A vertex is any place where three or more wires come together•For example, at point A, this gives the equation:

•At point B, this gives the equation:

+–

+ –

3

5

4

12 V

6 V

I1

I3

I2

1 2 3I I I

AB

3 1 2I I I

You always get oneredundant equation

+–

+ –

3

5

4

12 V

6 V

I1

I3

I2

•First, pick a direction for everyloop

•I always pick clockwise•Start anywhere, and set 0 equal to sum of potential change from each piece:•For batteries: V =

•It is an increase if you go from – to +•It is a decrease if you go from + to –

•For resistors: V = IR•It is a decrease if you go with the current•It is an increase if you go against the current

Kirchoff’s Second Law:The total voltage change

around a loop is always zero

Kirchoff’s Second Law

0 12 13I 6 25I

1 20 18 3 5I I

Kirchoff’s Second Law

How to apply it:•First, assign a current and a direction to every pathway

•Two components in series will always have the same current•At every vertex, write the equation:

in outI I Which equation do you get for point A?

A) I1 + I2 = I3 B) I2 + I3 = I1 C) I1 + I3 = I2 D) I1 + I2 + I3 = 0

•The equation from point B is

3 1 2I I I You always get oneredundant equation

+–

+ –

3

5

4

12 V

6 V

I1

I3

AB

I2

Kirchoff’s Law- Final Step

•You have derived three equations in three unknowns +–

+ –

3

5

4

12 V

6 V

I1

I3

A

I2

2 30 5 6 4I I 1 20 18 3 5I I 3 1 2I I I

•We now solve these simultaneously•We can let Maple do it for us if we want:

> solve({i3=i1+i2,0=-5*i2-6.-4*i3,0=18-3*i1+5*i2},[i1,i2,i3]);

•Negative currents mean we guessed the wrong way•Not a problem

Kirchoff’s Laws with Capacitors

•Pick one side to put the charge on•The voltage change is given by V = Q/C

•It is a decrease if Q is the side you are going in•It is an increase if Q is the side you are going out

•The current is related to the time change of Q•Add a minus sign if I isn’t on the same side as Q

•If you are in a steady state, the current through a capacitor is always zero

C

Q

+–

dQI

dt

+–+

–

In this circuit, in the steady state, where is

current flowing?

It’s really just a battery and two resistors in series!

The Simplest RC Circuit

Q0

C

RIn the circuit shown at left, the capacitor starts

with charge Q0. At time t = 0, the switch is closed. What happens to the charge Q?

I

•Current begins to flow around the loop, so the charge Q will change

0Q

RIC

dQI

dt Q

RC

•This is a differential equation, and therefore hard to solve

dQ dt

Q RC

dQ dt

Q RC ln

tQ k

RC

t RCQ e

0t RCQ Q e Check the units: FRC

V C

A V

C

C s s

Charging and Discharging Capacitors

0tQ Q e

0t RCQ Q e

RC

•The combination RC = is called the time constant•It’s the characteristic time it takes to discharge

•We can work out the current fromdQ

Idt

0tQ e

In this circuit, the capacitor is initially

uncharged, but at t = 0 the switch is closed.

What happens?

Q

C R +–

I

dQI

dt

0Q

IRC

E

dQ Q

dt RC R

E

1 t RCQ C e E

Ammeters and Voltmeters

•An ammeter is a device that measures the current (amps) anywhere in a circuit

•To use it, you must route the current through it•A perfect ammeter should have zero resistance

•A voltmeter is a device that measures the potential difference (volts) between any two points in a circuit

•To use it, you can simply connect to any two points •A perfect voltmeter has infinite resistance

V

A

Household Wiring•All household appliances consume electrical power

•Think of them as resistors with fixed resistance R•Devices are designed to operate at 120 V*

•Often, they give the wattage at this voltage•Can easily get the effective resistance from

•To make sure power is given to each device, they are all placed in parallel

+–

A

2V

R

P

•If you put too many things on at once, a lot of current is drawn•The wires, which have some resistance, will start to get hot•To avoid setting the house on fire, add a fuse (or a circuit breaker)

Fuse

box

Inside House

*Actually, this is alternating current, later

chapter

Why three wires?•If a device is functioning properly, you need only two wires

•“Live” and “Neutral” wires

Toaster

•If the live wire accidentally touches the casing, the person can be electrocuted•The wrong solution – connect the neutral to the casing•Now imagine the neutral wire breaks

•The person again can be electrocuted•The right solution: Add a third “ground” wire connected directly to ground

•Normally no current will flow in this wire•If the hot wire touches the casing, it will trigger the fuse/circuit breaker and protect the person