Chapter 20: Circuits

1
Chapter 20 Circuits

• Current and EMF
• Ohms Law and Resistance
• Electrical Power
• Alternating Current
• Series and Parallel Connection
• Kirchoffs Rules and Circuit Analysis
• Internal Resistance
• Capacitors in Circuits

2
Current and EMF

• A few definitions
• Circuit A continuous path made of conducting
  materials
• EMF an alternative term for potential
  difference or voltage, especially when applied
  to something that acts as a source of electrical
  power in a circuit (such as a battery)
• Current the rate of motion of charge in a
  circuit
• Symbol I (or sometimes i).
• SI units C/s ampere (A)

3
Current and EMF

• Conventional current assumed to consist of the
  motion of positive charges.
• Conventional current flows from higher to lower
  potential.

4
Current and EMF

• Direct current (DC) flows in one direction around
  the circuit
• Alternating current (AC) sloshes back and
  forth, due to a time-varying EMF that changes its
  sign periodically

5
Chapter 20 Circuits

• Current and EMF
• Ohms Law and Resistance
• Electrical Power
• Alternating Current
• Series and Parallel Connection
• Kirchoffs Rules and Circuit Analysis
• Internal Resistance
• Capacitors in Circuits

6
Ohms Law and Resistance

• The current that flows through an object is
  directly proportional to the voltage applied
  across the object
• A constant of proportionality makes this an
  equation

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Ohms Law and Resistance

• The constant of proportionality, R, is called the
  resistance of the object.
• SI unit of resistance ohm (W)

8
Ohms Law and Resistance

• Resistance depends on the geometry of the object,
  and a property, resistivity, of the material from
  which it is made
• Resistivity symbol r
• SI units of resistivity ohm m (W m)

9
Ohms Law and Resistance

• In most materials, resistivity increases with
  temperature, according to the materials
  temperature coefficient of resistivity, a
• (resistivity is r0, and R R0, at temperature
  T0)
• SI units of a (C)-1

10
Chapter 20 Circuits

• Current and EMF
• Ohms Law and Resistance
• Electrical Power
• Alternating Current
• Series and Parallel Connection
• Kirchoffs Rules and Circuit Analysis
• Internal Resistance
• Capacitors in Circuits

11
Electrical Power

• Power is the time rate of doing work
• Voltage is the work done per unit charge
• Current is the time rate at which charge goes by
• Combining

12
Electrical Power

• Ohms Law substitutions allow us to write several
  equivalent expressions for power
• Regardless of how specified, power always has SI
  units of watts (W)

13
Chapter 20 Circuits

• Current and EMF
• Ohms Law and Resistance
• Electrical Power
• Alternating Current
• Series and Parallel Connection
• Kirchoffs Rules and Circuit Analysis
• Internal Resistance
• Capacitors in Circuits

14
Alternating Current

• EMF can be produced by rotating a coil of wire in
  a magnetic field.
• This results in a time-varying EMF

time, s
peak voltage
frequency (Hz)
15
Alternating Current
16
Alternating Current

• The time-varying voltage produces a time-varying
  current, according to Ohms Law

peak current
time, s
frequency (Hz)
17
Alternating Current
18
Alternating Current

• Calculate the power

19
Alternating Current
20
Alternating Current

• Calculate the power
• Average power

21
Chapter 20 Circuits

• Current and EMF
• Ohms Law and Resistance
• Electrical Power
• Alternating Current
• Series and Parallel Connection
• Kirchoffs Rules and Circuit Analysis
• Internal Resistance
• Capacitors in Circuits

22
Series Connection

• A circuit, or a set of circuit elements, are said
  to be connected in series if there is only one
  electrical path through them.

23
Series Connection

• A circuit, or a set of circuit elements, are said
  to be connected in series if there is only one
  electrical path through them.
• The same current flows through all
  series-connected elements. (Equation of
  continuity)

24
Series Connection

• A circuit, or a set of circuit elements, are said
  to be connected in series if there is only one
  electrical path through them.
• The same current flows through all
  series-connected elements. (Equation of
  continuity)
• A set of series-connected resistors is equivalent
  to a single resistor having the sum of the
  resistance values in the set.

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Series Connection
26
Series Connection

• Potential drops add in series.

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Parallel Connection

• A circuit, or a set of circuit elements, are said
  to be connected in parallel if the circuit
  current is divided among them.
• The same potential difference exists across all
  parallel-connected elements.

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Parallel Connection
29
Parallel Connection

• What is the equivalent resistance?
• The equation of continuity requires that I I1
  I2 I3

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Parallel Connection

• Applying Ohms Law

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Series - Parallel Networks

• Resistive loads may be so connected that both
  series and parallel connections are present.

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Series - Parallel Networks

• Simplify this network by small steps

33
Series - Parallel Networks

• Continue the simplification

34
Series - Parallel Networks

• Finally

35
Chapter 20 Circuits

• Current and EMF
• Ohms Law and Resistance
• Electrical Power
• Alternating Current
• Series and Parallel Connection
• Kirchoffs Rules and Circuit Analysis
• Internal Resistance
• Capacitors in Circuits

36
Kirchoffs Rules and Circuit Analysis

• The Loop Rule
• Around any closed loop in a circuit, the sum of
  the potential drops and the potential rises are
  equal and opposite or
• Around any closed loop in a circuit, the sum of
  the potential changes must equal zero.
• (Energy conservation)

37
Kirchoffs Rules and Circuit Analysis

• The Junction Rule
• At any point in a circuit, the total of the
  currents flowing into that point must be equal to
  the total of the currents flowing out of that
  point.
• (Charge conservation equation of continuity)

38
Chapter 20 Circuits

• Current and EMF
• Ohms Law and Resistance
• Electrical Power
• Alternating Current
• Series and Parallel Connection
• Kirchoffs Rules and Circuit Analysis
• Internal Resistance
• Capacitors in Circuits

39
Internal Resistance

• An ideal battery has a constant potential
  difference between its terminals, no matter what
  current flows through it.
• This is not true of a real battery. The voltage
  of a real battery decreases as more current is
  drawn from it.

40
Internal Resistance

• A real battery can be modeled as ideal one,
  connected in series with a small resistor
  (representing the internal resistance of the
  battery). The voltage drop with increased
  current is due to Ohms Law in the internal
  resistance.

41
Chapter 20 Circuits

• Current and EMF
• Ohms Law and Resistance
• Electrical Power
• Alternating Current
• Series and Parallel Connection
• Kirchoffs Rules and Circuit Analysis
• Internal Resistance
• Capacitors in Circuits

42
Capacitors in Circuits

• Like resistors, capacitors in circuits can be
  connected in series, in parallel, or in
  more-complex networks containing both series and
  parallel connections.

43
Capacitors in Parallel

• Parallel-connected capacitors all have the same
  potential difference across their terminals.

44
Capacitors in Series

• Capacitors in series all have the same charge,
  but different potential differences.

V1
V2
V3
45
RC Circuits

• A capacitor connected in series with a resistor
  is part of an RC circuit.
• Resistance limits charging current
• Capacitance determines ultimate charge

46
RC Circuits

• At the instant when the circuit is first
  completed, there is no potential difference
  across the capacitor.
• At that time, the current charging the capacitor
  is determined by Ohms Law at the resistor.

47
RC Circuits

• In the final steady state, the capacitor is fully
  charged.
• The full potential difference appears across the
  capacitor.
• There is no charging current.
• There is no potential difference across the
  resistor.

48
RC Circuits

• Between the initial state and the final state,
  the charge approaches its final value according
  to
• The product RC is the time constant of the
  circuit.

49
RC Circuits
R 100 KW C 10 mF (RC 1 s) V 12 V
50
RC Circuits

• During discharge, the time dependence of the
  capacitor charge is

51
RC Circuits
R 100 KW C 10 mF (RC 1 s) V 12 V