Download Ch 1

Basic Concepts
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Preliminary information
Circuit analysis
Definitions: current, voltage, power and energy
Passive sign convention
Circuit elements
Ideal sources
Preliminary Information
• Engineering notation will be used:
Represent a quantity with a number between 1 and 999
and a power divisible by 3.
Ex: 0.048 W would be 48 mW
124,000
=
?
6.4 x 10-8 C = ?
• Problem solving advice:
State goal of problem.
Draw a circuit diagram.
Be methodical and neat; show every step.
Verify your solution. Does it make sense?
Circuit Analysis Introduction
• We’ll use linear models to analyze circuits when possible. The more
complicated non-linear models will be used when more accuracy is
required. Linear: read 1 page in 5 minutes means read 10 pages in
50 min. Non-linear: Bench-press 135 lb ten times does not mean
you can lift 1350 lb once!
• There are 3 types of linear analysis:
Resistive: not time dependent, simple. Flashlight, toaster.
Time domain: add in capacitors and inductors. Solve for
expressions for all time.
Frequency domain: can transform tough differential
equations into easy algebraic equations.
Used for circuits that respond to different
frequencies. Radios, cell phones (EE 233).
Circuit Analysis Introduction
• An electric circuit is a collection of connected electrical
elements.
• Electric charge is a property of certain particles that make up
matter. It has units of coulombs C.
• 1 C equals the charge of 6.24 x 1018 protons. 1 C is a very large
amount of charge.
• Charge of 1 electron = -1.602 x 10-19 C = -e
• Charge of 1 proton = 1.602 x 10-19 C = +e
• The law of conservation of charge: charge cannot be created or
destroyed, only moved.
A Simple Resistive Circuit
i
switch
switch
1.5 V
Batteries,
1.5 V each
Light bulb
10
1.5 V
• This circuit has 4 circuit elements.
• We’ll use symbols to represent circuit elements (it’s simpler).
Electric Current
• Electric current is the rate of change of charge. Units are
amperes (or amps), A.
• 1 A = 1 C/s
• Two types we’ll consider:
--Direct Current (DC): the current remains constant
--Alternating Current (AC): the current varies sinusoidally with
time.
i=
dq
dt
where i = the current in amperes
q = the charge in coulombs
t = the time in seconds
Calculating Charge From Current
• You can use the definition of current to find the charge that has
been transferred over some time interval:
q(t )
t
to
i dt
q(to )
• Ex: The current in a circuit is given by i(t) = 12 t A. How much
charge has passed a point in the circuit from t = 0 s to t = 2 s?
Assume q(0) = 0.
Notes On Current
• Current is actually movement of
electrons (negatively charged particles)
• Direction of current is specified by an
arrow
• By convention, current is defined as the
flow of positive charge
• Note: positive charge isn’t really flowing.
Electrons are flowing the opposite
direction of the current.
• Note:
3A
i
e
1.5 V
10
1.5 V
-3 A
=
-
More On Current
• Due to collisions with atoms in a wire, the average drift velocity
of electrons as they flow down the wire is very low (about 1
mm/s).
• But the electric field that provides the push to those electrons is
set up very quickly (at the speed of light in that wire).
• So when a switch is thrown, current will flow almost
instantaneously.
• One convention often used is to use a capital I or a lower case i
for DC currents, but only a lower case i for AC currents (or i(t)).
Assumptions
• We’ll assume propagation is instantaneous. This means the
circuits we’ll study are small enough that the time for electrical
signals to travel through the entire circuit is essentially zero.
This assumption is called the lumped-parameter system.
• Net charge stored by each circuit elements is zero. No circuit
can store electrons.
• There is no electromagnetic coupling between the components
of the system. That is, a changing current in one part of the
circuit will not induce a current in another part of the circuit
(according to Faraday’s Law.)
Voltage
• Voltage is the energy absorbed or expended per unit of charge as
that charge moves through a circuit element.
• Also called potential difference. It’s a measure of the difference in
electric potential between two points in a circuit. Electric potential is
defined as the potential energy per unit of charge.
• In differential form voltage is
v
dw
dq
where w is the energy (work) in joules and q is the charge in
coulombs.
• Voltage exists when there is a separation of charge, even if no
current flows.
• You can think of voltage as the force that pushes the charges
through the circuit. No voltage=no current.
• Voltage has units of joules/coulomb or volts V.
Power
• Power is the rate that energy is supplied to or absorbed by a
object.
• Power has units of watts. 1 W = 1 J/s
• By convention,
--circuit elements that absorb power have a positive value of
the power.
--circuit elements that supply power have a negative value of
the power.
•
p
dw
dt
dw
dq
dq
dt
vi
Power and Energy
• From the Conservation of Energy, every watt absorbed by some
element must be supplied from some other element(s). That is,
Net power supplied = Net power absorbed
• You can calculate the energy supplied or absorbed by
rearranging the equation for power:
t
w
p dt
to
• Ex: problem 1.16
t
( vi)dt
to
The Passive Sign Convention
• We will consider circuit components as two terminal devices
(see fig.) They can be described by their current and/or voltage.
• The selection of current direction and polarity for voltage is
arbitrary. But the equations you use must agree with those
references. The most common convention used is the passive
sign convention (psc):
Any time the direction of the current is chosen so that it is in
the direction of a voltage drop across an element (as in the
fig.), use a positive sign in any formulas relating v and i.
Otherwise use a negative sign.
i
+
1
v
-
2
More on the PSC
• For the circuit element below, the power absorbed is given by
p = vi = 10 W. Alternately you can say that the element supplies
a power of p = -vi = -10 W.
• In other words, an element that is absorbing positive power
means it is also supplying negative power and vice versa.
2A
+
1
5V
-
2
More PSC
• Compute the absorbed power by each element:
3A
+
1
2V
-
1
-2 V
-
2
+
2
-3 A
2A
-5 A
+
1
1
5V
4V
-
+
2
-
2
Comments on the PSC
• Failure to follow the PSC will result in incorrect equations and
wrong answers.
• You must see how the direction of i and the polarity of v is
labeled on the circuit diagram to determine the sign used in
formulas like p
vi .
PSC Example
Ex: Problem 1.26
Voltage Signs and Terminology
a
+
3V
-
b
• The following statements are equivalent:
There is
There is
There is
There is
a
a
a
a
3 V drop from a to b
3 V rise from b to a
–3 V rise from a to b
–3 V drop from b to a
• The first expression listed is the one most commonly used.
Circuit Elements
• In Ch. 2 we’ll discuss 3 ideal basic circuit elements: voltage
sources, current sources and resistors.
• Many practical circuits can be modeled with just these three
elements.
• Only need algebra!
• Electrical sources like batteries and generators usually supply
energy to the circuit, but sometimes can absorb power (you can
charge a battery).
• Ideal sources are models used to simplify analysis. They
approximate actual physical devices.
• They don’t exist (real batteries have internal resistance that can
change as the load changes for instance).
• We’ll use more accurate models later in the term.
Ideal Voltage Sources
+
-
Vs
Vs
+
-
Independent source
(battery)
Independent source
+
-
Vs= Vx
+
-
Vs=
ix
Voltage controlled
Current controlled
dependent voltage source
dependent voltage source
• Ideal voltage source: produces voltage Vs regardless of the
current absorbed or supplied by the device.
• An independent voltage source produces a voltage that doesn’t
depend on the voltage or current in other parts of the circuit.
• For a dependent voltage source, the voltage produced depends
on the current or voltage elsewhere in the circuit.
More on Voltage Sources
• The sign of Vs can be negative.
6V
+
-
=
-6 V
6V
= -6 V
+
-
Ideal Current Sources
• Ideal current source: produces a current of Is amps regardless
of the voltage across the terminals of the device.
• An independent current source produces a current that doesn’t
depend on the voltage or current in other parts of the circuit.
• For a dependent current source, the current produced depends
on the current or voltage elsewhere in the circuit.
Is
Independent source
Is= Vx
Voltage controlled
dependent source
Is= Ix
Current controlled
dependent source
More on Current Sources
• The sign of Is can be negative.
6A
=
-6 A
Home
6A
= -6 A