Chapter 18Direct Current Circuits

Chapter 18 Objectives

• Compare emf v potential difference

• Construct circuit diagrams

• Open v Closed circuits

• Potential difference in a circuit

• Resistors in series

• Resistors in parallel

• Equivalent resistance

• Current across a resistor

• Voltage across a resistor

Sources of emf

• Any source that maintains a constant current in a closed circuit is called an emf source.

– The letters emf used to stand for electromotive force.• But the source does not provide the force, the difference in charge position does.

So we no longer use the long version of the name.

• The difference between potential difference and emf is that the potential difference is the pure movement of charges in space without any ability to move back on their own.

– Thus, emf is the constant resupplying of charges to create a potential difference.

• The symbol for emf is

– • The SI units for emf is

– V• volts

• Examples of emf sources are– batteries– generators

Internal Resistance and emf

• Even though a battery is rated at say 12 V, not all 12 V gets into the circuit.• This is due to the internal resistance of the battery.

– Every emf source has an internal resistance based on the materials that make up the source and its internal charge path.

• Thus the terminal voltage, V, is always less than the advertised emf of a source.

V = - Ir• r is the internal resistance of the emf source

• From the expression, the terminal voltage will be equal to when the current is zero.

– That would mean there is no load on the source.

• Once an external resistance is placed on the source, the circuit is complete and will allow current to flow both through the internal resistance and the external resistance.

– The external resistance is also referred to as the load resistance, R.

• So = IR + Ir

– And is what we will use to calculate or voltage in a circuit.• Note: If r << R, then we will disregard r in all calculations!

Schematic Diagrams

• A schematic diagram depicts the construction of an electrical circuit.

• There is a standard set of symbols used to eliminate any confusion.

• Each symbol represents a component of an electrical circuit such as a resistor, a light bulb, a battery, a switch, etc.

Symbols for Schematic Diagrams

Component Symbol Explanation

Wire or

The wire connects the components of

the circuit.

Resistor or Load

A resistor is any load that disrupts

the flow of electricity.

Battery

The larger line represents + charge and the smaller is

- .

Derived Symbols

Component Symbol Explanation

Switch The circles represent two connection points to the

circuit.

Light A light is treated like a resistor.

Plug Looks like the end of a plug, and like a battery.

Capacitor Two parallel lines show the capacitor plates.

Semiconductor Can control the flow of electricity

Electrical Circuits

• An electrical circuit is a set of electrical components connected so that they provide one or more complete paths for the movement of charges.

• A closed circuit is one with at least one continuous loop from one terminal of the power source to the other.

• An open circuit is one that does not contain a continuous loop.

Short Circuit

• A short circuit is when there is a direct path from one terminal to the other terminal of the power source.

• That direct path contains no resistors, and therefore provides the least resistance of flow for the charge carriers.

• A short circuit increases the current flow, which can cause damage to the power source as well as create a great deal of heat in the wiring.

Schematic Examples

Closed Circuit

Open Circuit

Short Circuit

Resistors in Series

• A series of resistors describes a circuit or portion of a circuit that provides a single conducting path of all resistors being in line with each other.

• When resistors are in series, the amount of charge over a certain time period entering and exiting the first resistor is equal to the amount of charge entering and exiting the second resistor and so on.

• Thus the total current remains constant when resistors are in series.

• Series circuits require all elements to conduct. As soon as there is a gap, the entire circuit goes out.

Equivalent Resistance of Resistors in Series

• Since resistors in series line up one after another, the equivalent resistance of resistors in series is the sum of the individual resistances.– The potential drop over the entire circuit must be equal to the potential difference of

the voltage source• Therefore each resistor will use up as many volts as its resistance requires to abide by

Ohm’s Law.

– Also, since charge must be conserved and it only has one path to follow, the charge remains constant.

• And if charge is constant, then current is constant!

• The equivalent resistance for series circuit will always be greater than any individual resistance.

Req = R1 + R2 + R3 + …

Total Current in aSeries Circuit

• To find the total current in a series circuit, simplify the circuit to a single equivalent resistance.

• Then use ΔV = IR to calculate the current.

I = ΔVΔVReq

Potential Difference Across Each Resistor

• Because the current in each resistor is equal to the total current in the circuit, you can use ΔV = IR to calculate the potential difference across each resistor.

ΔVΔV11 = IR = IR11 ΔVΔV22 = IR = IR22

Resistors in Parallel

• A parallel circuit describes two or more components in a circuit that connected across common points providing separate conducting paths for the charges.– The parallel nature comes from looking at schematic diagrams

and showing the alternate paths for charges to flow as being parallel to each other, whether they physically are or not.

• Due to the differing paths, the charge through each path will sum to the total charge of the circuit.– Therefore, the current in each path will sum to the total current of

the entire circuit.

• The potential difference across the resistors in parallel will remain constant.

• If there is a gap in a parallel circuit, the rest of the circuit will still conduct.

Equivalent Resistance of Resistors in Parallel

• Observe what happens to ΔV = IR when we solve for the current.– We solve for current because the sum of

each branch of current gives the total.

Itotal =ΔV1 ΔV2

R1 R2+

• Notice how the inverses of the resistances are added together.•The equivalent resistance of a parallel circuit is always less than the smallest resistance of the group.

Req R1 R2 R3+ …++

1 111 =

Current in Individual Resistors of a Parallel Circuit

• The current in each resistor is found by

• The potential difference remains constant.

ΔV

Rn

I =

Series and Parallel Schematics

Series

Parallel

Assembly of Circuits

• A series circuit connects the resistors one after another.

• So the positive lead should connect to one side of the resistor.

• The negative lead goes from the other side of the resistor to one side of the next resistor.

• The circuit should look like one continuous loop of alternating positive and negative leads.

• A parallel circuit connects the resistors as parts of separate branches of the circuit.

• All the positive leads branch off from the same junction point.

• The negatives then join at the same junction point of their own.

• Simply put, positive connects to positive for each resistor.

Kirchhoff’s Rules

• Gustav Kirchhoff (1824-1887) came up with an order of operations for electrical circuits

1. The sum of the currents entering any junction must equal the sum of the currents leaving that junction.

1. Called the junction rule.

2. The sum of the potential differences across all the elements around any closed-circuit loop must be zero.

2. Called the loop rule.

Applying Kirchhoff’s Rules

1. Draw the circuit diagram in a way that is simple to understand.

1. Try to draw each resistor sequence so it is either vertical or horizontal.

1. You must designate a current flow for each loop.1. If you designate it incorrectly, it will show up as a negative

value, but the magnitude will still be correct.1. If this is the case, leave it negative for all calculations later

on.

2. Apply the junction rule for every new junction individually.3. Apply Kirchhoff’s loop rule for each loop necessary in the

circuit.3. Be sure to identify whether it is a voltage drop of voltage

gain as you pass through a voltage source.

4. Solve the equations simultaneously for the unknown quantities.

RC Circuits

• Once a capacitor is inserted into the circuit, the current is no longer constant.

• The current will gradually decrease as the capacitor charges.– Once the capacitor is fully charged, the current will

be zero.• q = Q(1 - e(-t / RC))

• But this takes time to charge the capacitor– = RC

• Where is the time constant for a capacitor to charge to 63.2% of its maximum equilibrium charge.

– Plug t = = RC and notice what happens!

Circuit Breaker v Fuse

• A circuit breaker is typically used in large circuit applications such has a house or building.

– The typical tripping level for a circuit breaker is 15 A.

• A circuit breaker acts like a switch to open the circuit when an overload is sensed.

– The switch consists of a bimetallic strip that heats up as current travels through the circuit.

– When the heat is too much, the strip pushes on the switch itself to open the circuit.

– The circuit cannot be closed until the strip has cooled down.

• A fuse is typically used in small circuits that need a quick break in the current flow.

– A fuse can be used for a large range of current levels.

• A fuse is a small metallic strip that is part of the current path.

– When the current level becomes too high, the fuse will burn out from the excess heat.

– The metallic strip will melt or break away from the heat, opening the circuit.

– The fuse must be replaced with a new one if it has blown.