Download Introduction to circuit analysis Classification of Materials Electric

Introduction to circuit analysis
• Electrical effects are due to
OUTLINE
• Electrical quantities
–
–
–
–
Electric Charge
– separation of charge Æ electric force (voltage)
– charges in motion Æ electric flow (current)
Charge
Current
Voltage
Power
• Macroscopically, most matter is electrically
neutral most of the time.
• The ideal basic circuit element
• Sign conventions
• Current versus voltage (I-V) graph
– Exceptions: clouds in a thunderstorm, people on
carpets in dry weather, plates of a charged capacitor,
etc.
• Microscopically, matter is full of electric charges
– Electric charge exists in discrete quantities, integral
multiples of the electronic charge -1.6 x 10-19
Coulomb
Reading: 1.2, 1.3,1.6
EE40 Fall 2005
Lecture 2, Slide 1
Prof. Neureuther
EE40 Fall 2005
Classification of Materials
Lecture 2, Slide 4
Prof. Neureuther
Electric Current
• Solids in which all electrons are tightly bound to
atoms are insulators.
– Examples:
• Solids in which the outermost atomic electrons
are free to move around are metals.
– Metals typically have ~1 “free electron” per atom
– Examples:
Definition: rate of positive charge flow
Symbol: i
Units: Coulombs per second ≡ Amperes (A)
Note: Current has polarity.
i = dq/dt
where
q = charge (Coulombs)
t = time (in seconds)
• Electrons in semiconductors are not tightly
bound and can be easily “promoted” to a free
state.
– Examples:
André-Marie Ampère's
1775-1836
EE40 Fall 2005
Lecture 2, Slide 2
Prof. Neureuther
EE40 Fall 2005
Lecture 2, Slide 5
Prof. Neureuther
Electric Current Examples
1.
105 positively charged particles (each with charge
1.6×10-19 C) flow to the right (+x direction) every
nanosecond
I=
2.
105 electrons flow to the right (+x direction) every
microsecond
I=
EE40 Fall 2005
Lecture 2, Slide 3
Prof. Neureuther
Q
105 ×1.6 ×10−19
=+
= 1.6 ×10−5 A
t
10−9
Q
105 × 1.6 × 10−19
=−
= −1.6 ×10−5 A
10−9
t
EE40 Fall 2005
Lecture 2, Slide 6
Prof. Neureuther
1
Current Density
Electric Power
Definition: rate of positive charge flow per unit area
Symbol: J
Units: A / cm2
• Definition: transfer of energy per unit time
• Symbol: p
• Units: Joules per second ≡ Watts (W)
Semiconductor with 1018 “free
electrons” per cm3
Example 1:
Wire attached
to end
m
2c
1 cm
C2
C1
10 cm
X
Suppose we force a current of 1 A to flow from C1 to C2:
• Electron flow is in -x direction:
Lecture 2, Slide 7
• Concept:
As a positive charge q moves through a James Watt
1736 - 1819
drop in voltage v, it loses energy
ƒ energy change = qv
ƒ rate is proportional to # charges/sec
1C / sec
electrons
= −6.25 ×1018
− 1.6 × 10 −19 C / electron
sec
EE40 Fall 2005
p = dw/dt = (dw/dq)(dq/dt) = vi
Prof. Neureuther
EE40 Fall 2005
Current Density Example (cont’d)
• Example 2:
Typical dimensions of integrated circuit
components are in the range of 1 µm. What is
the current density in a wire with 1 µm² area
carrying 5 mA?
Lecture 2, Slide 10
Prof. Neureuther
The Ideal Basic Circuit Element
i
+
v
_
• Polarity reference for voltage can be
indicated by plus and minus signs
• Reference direction for the current
is indicated by an arrow
Attributes:
• Two terminals (points of connection)
• Mathematically described in terms of current
and/or voltage
• Cannot be subdivided into other elements
EE40 Fall 2005
Lecture 2, Slide 8
Prof. Neureuther
Electric Potential (Voltage)
-
b
Subscript convention:
vab means the potential at a
minus the potential at b.
vab ≡ va - vb
Lecture 2, Slide 9
v
+
Alessandro Volta
(1745–1827)
Note: Potential is always referenced to some point.
EE40 Fall 2005
Prof. Neureuther
• A problem like “Find the current” or “Find the
voltage” is always accompanied by a definition
of the direction:
i
where w = energy (in Joules), q = charge (in Coulombs)
a
Lecture 2, Slide 11
A Note about Reference Directions
• Definition: energy per unit charge
• Symbol: v
• Units: Joules/Coulomb ≡ Volts (V)
v = dw/dq
EE40 Fall 2005
Prof. Neureuther
• In this case, if the current turns out to be 1 mA
flowing to the left, we would say i = -1 mA.
• In order to perform circuit analysis to determine
the voltages and currents in an electric circuit,
you need to specify reference directions.
• There is no need to guess the reference
direction so that the answers come out positive.
EE40 Fall 2005
Lecture 2, Slide 12
Prof. Neureuther
2
Sign Convention Example
Power Calculation Example
Suppose you have an unlabelled battery and you measure
its voltage with a digital voltmeter (DVM). It will tell you the
magnitude and sign of the voltage.
Find the power absorbed by each element:
Conservation of energy
Î total power delivered
equals
total power absorbed
With this circuit, you are
measuring vab.
a
The DVM indicates −1.401, so
va is lower than vb by 1.401 V.
−1.401
DVM
+
b
Aside: For electronics these are unrealistically
large currents – milliamperes or smaller is more
typical
p (W)
vi (W)
918
- 810
- 12
- 400
- 224
1116
Which is the positive battery
terminal?
Note that we have used the “ground” symbol ( ) for the reference
node on the DVM. Often it is labeled “C” for “common.”
EE40 Fall 2005
Lecture 2, Slide 13
Prof. Neureuther
EE40 Fall 2005
Another Example
Find vab, vca, vcb
a
+
2V
−
• 5 ideal basic circuit elements:
+
−1 V
–
–
–
–
–
vcd
−
−
b
+
vbd
−
Prof. Neureuther
Circuit Elements
c
+
Lecture 2, Slide 16
d
voltage source
current source
resistor
inductor
capacitor
active elements, capable of
generating electric energy
passive elements, incapable of
generating electric energy
• Many practical systems can be modeled with
just sources and resistors
Note that the labeling convention has nothing to do with
whether or not v is positive or negative.
EE40 Fall 2005
Lecture 2, Slide 14
Prof. Neureuther
• The basic analytical techniques for solving
circuits with inductors and capacitors are
similar to those for resistive circuits
EE40 Fall 2005
Lecture 2, Slide 17
Prof. Neureuther
Power
Electrical Sources
If an element is absorbing power (i.e. if p > 0), positive
charge is flowing from higher potential to lower potential.
• An electrical source is a device that is capable
of converting non-electric energy to electric
energy and vice versa.
p = vi if the “passive sign convention” is used:
i
i
_
+
v
or
v
_
+
Examples:
– battery: chemical
electric
– dynamo (generator/motor): mechanical
electric
(Ex. gasoline-powered generator, Bonneville dam)
ÆElectrical sources can either deliver or absorb power
How can a circuit element absorb power?
By converting electrical energy into heat (resistors in toasters),
light (light bulbs), or acoustic energy (speakers); by storing
energy (charging a battery).
EE40 Fall 2005
Lecture 2, Slide 15
Prof. Neureuther
EE40 Fall 2005
Lecture 2, Slide 18
Prof. Neureuther
3
Ideal Voltage Source
Electrical Conductance
– Voltage is known, but current is determined by the
circuit to which the source is connected.
• The voltage can be either independent or
dependent on a voltage or current elsewhere in
the circuit, and can be constant or time-varying.
• Conductance is the reciprocal of resistance.
Symbol: G
Ω
• Circuit element that maintains a prescribed
voltage across its terminals, regardless of the
current flowing in those terminals.
Units: siemens (S) or mhos ( )
Example:
Consider an 8 Ω resistor. What is its conductance?
Device symbols:
vs +_
independent
EE40 Fall 2005
vs=µ vx +_
vs=ρ ix +_
voltage-controlled
current-controlled
Lecture 2, Slide 19
Prof. Neureuther
Ideal Current Source
Werner von Siemens
1816-1892
EE40 Fall 2005
– Current is known, but voltage is determined by the
circuit to which the source is connected.
• The current can be either independent or
dependent on a voltage or current elsewhere in
the circuit, and can be constant or time-varying.
Device symbols:
independent
EE40 Fall 2005
is=α vx
is=β ix
voltage-controlled
current-controlled
Lecture 2, Slide 20
Prof. Neureuther
Electrical Resistance
R
Units: Volts per Ampere ≡ ohms (Ω)
p = vi = v ( v/R ) = v2/R
Note that p > 0 always, for a resistor Æ a resistor
dissipates electric energy
Example:
a) Calculate the voltage vg and current ia.
b) Determine the power dissipated in the 80Ω resistor.
EE40 Fall 2005
Lecture 2, Slide 23
Prof. Neureuther
• Short circuit
– R = 0 Æ no voltage difference exists
– all points on the wire are at the same
potential.
– Current can flow, as determined by the circuit
• Open circuit
• The current flowing in the resistor
is proportional to the voltage
across the resistor:
v = i R (Ohm’s Law)
p = vi = ( iR )i = i2R
Short Circuit and Open Circuit
• Resistance: the ratio of voltage drop and
current. The circuit element used to model this
behavior is the resistor.
Circuit symbol:
Prof. Neureuther
Example: Power Absorbed by a Resistor
• Circuit element that maintains a prescribed
current through its terminals, regardless of the
voltage across those terminals.
is
Lecture 2, Slide 22
Georg Simon Ohm
1789-1854
– R = ∞ Æ no current flows
– Voltage difference can exist, as determined
by the circuit
where v = voltage (V), i = current (A), and R = resistance (Ω)
EE40 Fall 2005
Lecture 2, Slide 21
Prof. Neureuther
EE40 Fall 2005
Lecture 2, Slide 24
Prof. Neureuther
4
I-V Characteristic of Ideal Voltage Source
i
a i
+
Vab +_ vs
_
i=0
b
• Current = rate of charge flow i = dq/dt
• Voltage = energy per unit charge created by
charge separation
• Power = energy per unit time (i in to +v => pos)
• Ideal Basic Circuit Elements
v
– two-terminal component that cannot be sub-divided
– described mathematically in terms of its terminal
voltage and current
Vs>0
1. Plot the I-V characteristic for vs > 0. For what
values of i does the source absorb power? For
what values of i does the source release power?
2. Repeat
vs < 0.power; i>0 absorb power
Vs>0 Æ(1)
i<0for
release
3. What is the I-V characteristic for an ideal wire?
EE40 Fall 2005
Summary
Lecture 2, Slide 25
Prof. Neureuther
– An ideal voltage source maintains a prescribed voltage
regardless of the current in the device.
– An ideal current source maintains a prescribed current
regardless of the voltage across the device.
– A resistor constrains its voltage and current to be
proportional to each other:
v = iR (Ohm’s law)
EE40 Fall 2005
Lecture 2, Slide 28
Prof. Neureuther
I-V Characteristic of Ideal Current Source
i
i
+
v
_
is
v
1. Plot the I-V characteristic for is > 0. For what values
of v does the source absorb power? For what
values of v does the source release power?
V>0 absorb power; V<0 release power
EE40 Fall 2005
Lecture 2, Slide 26
Prof. Neureuther
I-V Characteristic of Ideal Resistor
a
+
v
_
i
i
R
v
b
1.
Plot the I-V characteristic for R = 1 kΩ. What is the
slope?
a
+
Vab
_
b
EE40 Fall 2005
a
R
Vab
R
b
Lecture 2, Slide 27
Prof. Neureuther
5