EE70 Review

1
EE70 Review
2
Electrical Current
3
Circuit Elements
An electrical circuit consists of circuit
elements such as voltage sources, resistances,
inductances and capacitances that are connected
in closed paths by conductors
4
Reference Directions
The voltage vab has a reference polarity that is
positive at point a and negative at point b.
5
Reference Directions
6
Reference Directions
downhill resistor
uphill battery
Energy is transferred when charge flows through
an element having a voltage across it.
7
Power and Energy
Watts
Joules
8
Reference Direction
Current is flowing in the passive configuration

• If current flows in the passive configuration
  the power is given by p vi
• If the current flows opposite to the passive
  configuration, the power is given by p -vi

9
Dependent Sources
10
Resistors and Ohms Law
a
b
The units of resistance are Volts/Amp which are
called ohms. The symbol for ohms is omega ?
11
Resistance Related to Physical Parameters
? is the resistivity of the material used to
fabricate the resistor. The units of resitivity
are ohm-meters (?-m)
12
Power dissipation in a resistor
13
Kircohoffs Current Law (KCL)

• The net current entering a node is zero
• Alternatively, the sum of the currents entering a
  node equals the sum of the currents leaving a node

14
Kircohoffs Current Law (KCL)
15
Series Connection
16
Kircohoffs Voltage Law (KVL)

• The algebraic sum of the voltages equals zero
  for any closed path (loop) in an electrical
  circuit.

17
Kircohoffs Voltage Law (KVL)
18
Parallel Connection
KVL through A and B -vavb 0 ? va vb KVL
through A and C -va - vc 0 ? va -vc
19
Equivalent Series Resistance
20
Equivalent Parallel Resistance
21
Circuit Analysis using Series/Parallel Equivalents

1. Begin by locating a combination of resistances
   that are in series or parallel. Often the place
   to start is farthest from the source.
2. Redraw the circuit with the equivalent resistance
   for the combination found in step 1.

22
Voltage Division
Of the total voltage, the fraction that appears
across a given resistance in a series circuit is
the ratio of the given resistance to the total
series resistance.
23
Current Division
For two resistances in parallel, the fraction of
the total current flowing in a resistance is the
ratio of the other resistance to the sum of the
two resistances.
24
Node Voltage Analysis
25
Node Voltage Analysis
26
Mesh Current Analysis
27
Mesh Current Analysis
28
Thévenin Equivalent Circuits
29
Thévenin Equivalent Circuits
30
Thévenin Equivalent Circuits
31
Thévenin Equivalent Circuits
32
Thévenin Equivalent Circuits
33
Norton Equivalent Circuits
34
Norton Equivalent Circuits
35
Source Transformations
36
Maximum Power Transfer
37
Superposition Principle
The superposition principle states that the total
response is the sum of the responses to each of
the independent sources acting individually. In
equation form, this is
38
Superposition Principle
39
Superposition Principle
Current source open circuit
40
Superposition Principle
Voltage source short circuit
Req
41
Voltage-Amplifier Model
The input resistance Ri is the equivalent
resistance see when looking into the input
terminals of the amplifier. Ro is the output
resistance. Avoc is the open circuit voltage gain.
42
Voltage Gain
Ideally, an amplifier produces an output signal
with identical waveshape as the input signal, but
with a larger amplitude.
43
Current Gain
44
Power Gain
45
Operational Amplifier
46
Summing Point Constraint
Operational amplifiers are almost always used
with negative feedback, in which part of the
output signal is returned to the input in
opposition to the source signal.
47
Summing Point Constraint
In a negative feedback system, the ideal op-amp
output voltage attains the value needed to force
the differential input voltage and input current
to zero. We call this fact the summing-point
constraint.
48
Summing Point Constraint

• Verify that negative feedback is present.
• Assume that the differential input voltage and
  the input current of the op amp are forced to
  zero. (This is the summing-point constraint.)
• Apply standard circuit-analysis principles, such
  as Kirchhoffs laws and Ohms law, to solve for
  the quantities of interest.

49
The Basic Inverter
50
Applying the Summing Point Constraint
51
Inverting Amplifier
52
Summing Amplifier
53
Non-inverting Amplifiers
54
Voltage Follower
55
Capacitance
56
Capacitances in Parallel
57
Capacitances in Series
58
Inductance
59
Inductance
The polarity of the voltage is such as to oppose
the change in current (Lenzs law).
60
Series Inductances
61
Parallel Inductances
62
Mutual Inductance
Fields are aiding
Fields are opposing
Magnetic flux produced by one coil links the
other coil
63
Discharge of a Capacitance through a Resistance
KCL at the top node of the circuit
iC
iR
64
Discharge of a Capacitance through a Resistance
We need a function vC(t) that has the same form
as its derivative.
Substituting this in for vc(t)
65
Discharge of a Capacitance through a Resistance
Solving for s
Substituting into vc(t)
Initial Condition
Full Solution
66
Discharge of a Capacitance through a Resistance
To find the unknown constant K, we need to use
the boundary conditions at t0. At t0 the
capacitor is initially charged to a voltage Vi
and then discharges through the resistor.
67
Discharge of a Capacitance through a Resistance
68
Charging a Capacitance from a DC Source through a
Resistance
69
Charging a Capacitance from a DC Source through a
Resistance
Current into the capacitor
Current through the resistor
70
Charging a Capacitance from a DC Source through a
Resistance
Rearranging
This is a linear first-order differential
equation with contant coefficients.
71
Charging a Capacitance from a DC Source through a
Resistance
The boundary conditions are given by the fact
that the voltage across the capacitance cannot
change instantaneously
72
Charging a Capacitance from a DC Source through a
Resistance
Try the solution
Substituting into the differential equation
Gives
73
Charging a Capacitance from a DC Source through a
Resistance
For equality, the coefficient of est must be zero
Which gives K1VS
74
Charging a Capacitance from a DC Source through a
Resistance
Substituting in for K1 and s
Evaluating at t0 and remembering that vC(0)0
Substituting in for K2 gives
75
Charging a Capacitance from a DC Source through a
Resistance
76
DC Steady State
In steady state, the voltage is constant, so the
current through the capacitor is zero, so it
behaves as an open circuit.
77
DC Steady State
In steady state, the current is constant, so the
voltage across and inductor is zero, so it
behaves as a short circuit.
78
DC Steady State
The steps in determining the forced response for
RLC circuits with dc sources are 1. Replace
capacitances with open circuits. 2. Replace
inductances with short circuits. 3. Solve the
remaining circuit.
79
RL Transient Analysis
80
RL Transient Analysis
81
RC and RL Circuits with General Sources
First order differential equation with constant
coefficients
Forcing function
82
RC and RL Circuits with General Sources
The general solution consists of two parts.
83
The particular solution (also called the forced
response) is any expression that satisfies the
equation. In order to have a solution that
satisfies the initial conditions, we must add the
complementary solution to the particular solution.
84
The homogeneous equation is obtained by setting
the forcing function to zero. The
complementary solution (also called the natural
response) is obtained by solving the homogeneous
equation.
85
Integrators and Differentiators
Integrators produce output voltages that are
proportional to the running time integral of the
input voltages. In a running time integral, the
upper limit of integration is t .
86
(No Transcript)
87
Differentiator Circuit
88
SecondOrder Circuits
Differentiating with respect to time
89
SecondOrder Circuits
Dampening coefficient
Define
Undamped resonant frequency
Forcing function
90
Solution of the Second-Order Equation
91
Solution of the Complementary Equation
92
Solution of the Complementary Equation
Roots of the characteristic equation
Dampening ratio
93
1. Overdamped case (? gt 1). If ? gt 1 (or
equivalently, if a gt ?0), the roots of the
characteristic equation are real and distinct.
Then the complementary solution is
In this case, we say that the circuit is
overdamped.
94
2. Critically damped case (? 1). If ? 1 (or
equivalently, if a ?0 ), the roots are real
and equal. Then the complementary solution is
In this case, we say that the circuit is
critically damped.
95
3. Underdamped case (? lt 1). Finally, if ? lt 1
(or equivalently, if a lt ?0), the roots are
complex. (By the term complex, we mean that the
roots involve the square root of 1.) In other
words, the roots are of the form
in which j is the square root of -1 and the
natural frequency is given by
96
For complex roots, the complementary solution is
of the form
In this case, we say that the circuit is
underdamped.
97
Circuits with Parallel L and C
We can replace the circuit with its Norton
equivalent and then analyze the circuit by
writing KCL at the top node
98
Circuits with Parallel L and C
99
Circuits with Parallel L and C
100
Circuits with Parallel L and C
This is the same equation as we found for the
series LC circuit with the following changes for
?
101
Complex Impedances-Inductor
102
Complex Impedances-Inductor
103
Complex Impedances-Capacitor
104
Complex Impedances-Capacitor
105
Impedances-Resistor
106
Impedances-Resistor
107
Kirchhoffs Laws in Phasor Form
We can apply KVL directly to phasors. The sum of
the phasor voltages equals zero for any closed
path.
The sum of the phasor currents entering a node
must equal the sum of the phasor currents leaving.
108
Power in AC Circuits
For ?gt0 the load is called inductive since
Zj?L for an inductor For ?lt0 the load is
capacitive since Z-j/?C for a capacitor
109
Load Impedance in the Complex Plane
110
Power for a General Load
If the phase angle for the voltage is not zero,
we define the power angle ?
Power angle
111
AC Power Calculations
Average Power
Reactive Power
Apparent Power
112
Power Triangles
Average power
Reactive power
Average power
Apparent power
113
Thevenin Equivalent Circuits
The Thevenin equivalent for an ac circuit
consists of a phasor voltage source Vt in series
with a complex impedance Zt
114
Thevenin Equivalent Circuits
The Thévenin voltage is equal to the open-circuit
phasor voltage of the original circuit.
We can find the Thévenin impedance by zeroing the
independent sources and determining the impedance
looking into the circuit terminals.
115
Thevenin Equivalent Circuits
The Thévenin impedance equals the open-circuit
voltage divided by the short-circuit current.
116
Norton Equivalent Circuit
The Norton equivalent for an ac circuit consists
of a phasor current source In in parallel with a
complex impedance Zt
117
Maximum Average Power Transfer
The Thevenin equivalent of a two-terminal circuit
delivering power to a load impedance.
118
Maximum Average Power Transfer
If the load can take on any complex value,
maximum power transfer is attained for a load
impedance equal to the complex conjugate of the
Thévenin impedance.
If the load is required to be a pure resistance,
maximum power transfer is attained for a load
resistance equal to the magnitude of the Thévenin
impedance.
119
Transfer Functions
The transfer function H(f ) of the two-port
filter is defined to be the ratio of the phasor
output voltage to the phasor input voltage as a
function of frequency
120
First-Order Low Pass Filter
Half power frequency
121

First-Order Low Pass Filter
122
First-Order Low Pass Filter
For low frequency signals the magnitude of the
transfer function is unity and the phase is 0?.
Low frequency signals are passed while high
frequency signals are attenuated and phase
shifted.
123
Magnitude Bode Plot for First-Order Low Pass
Filter
124

• A horizontal line at zero for f lt fB /10.
• 2. A sloping line from zero phase at fB /10 to
  90 at 10fB.
• 3. A horizontal line at 90 for f gt 10fB.

125
First-Order High-Pass Filter
126
First-Order High-Pass Filter
127
First-Order High-Pass Filter
128
Bode Plots for the First-Order High-Pass Filter
129
Series Resonance
For resonance the reactance of the inductor and
the capacitor cancel
130
Series Resonance
Quality factor QS
131
Series Resonant Band-Pass Filter
132
Series Resonant Band-Pass Filter
133
Parallel Resonance
At resonance ZP is purely resistive
134
Parallel Resonance
Quality factor QP
135
Parallel Resonance
Vout for constant current, varying the frequency
136
Second Order Low-Pass Filter
137
Second Order Low-Pass Filter
138
Second Order Low-Pass Filter
139
Second Order High-Pass Filter
At low frequency the capacitor is an open
circuit At high frequency the capacitor is a
short and the inductor is open
140
Second Order Band-Pass Filter
At low frequency the capacitor is an open
circuit At high frequency the inductor is an open
circuit
141
Second Order Band-Reject Filter
At low frequency the capacitor is an open
circuit At high frequency the inductor is an open
circuit
142
First-Order Low-Pass Filter
A low-pass filter with a dc gain of -Rf/Ri
143
First-Order High-Pass Filter
A high-pass filter with a high frequency gain of
-Rf/Ri
144
Flux Linkages and Faradays Law
Magnetic flux passing through a surface area A
For a constant magnetic flux density
perpendicular to the surface
The flux linking a coil with N turns
145
Faradays Law
Faradays law of magnetic induction

• The voltage induced in a coil whenever its flux
  linkages are changing. Changes occur from
• Magnetic field changing in time
• Coil moving relative to magnetic field

146
Lenzs Law
Lenzs law states that the polarity of the
induced voltage is such that the voltage would
produce a current (through an external
resistance) that opposes the original change in
flux linkages.
147
Lenzs Law
148
Magnetic Field Intensity and Ampères Law
Ampères Law
149
Ampères Law
The line integral of the magnetic field intensity
around a closed path is equal to the sum of the
currents flowing through the surface bounded by
the path.
150
Magnetic Field Intensity and Ampères Law
151
Magnetic Circuits
In many engineering applications, we need to
compute the magnetic fields for structures that
lack sufficient symmetry for straight-forward
application of Ampères law. Then, we use an
approximate method known as magnetic-circuit analy
sis.
152
magnetomotive force (mmf) of an N-turn
current-carrying coil
Analog Voltage (emf)
reluctance of a path for magnetic flux
Analog Resistance
Analog Ohms Law
153
Magnetic Circuit for Toroidal Coil
154
A Magnetic Circuit with Reluctances in Series and
Parallel
155
A Magnetic Circuit with Reluctances in Series and
Parallel
156
Mutual Inductance
Self inductance for coil 1
Self inductance for coil 2
Mutual inductance between coils 1 and 2
157
Mutual Inductance
Total fluxes linking the coils
158
Mutual Inductance
Currents entering the dotted terminals produce
aiding fluxes
159
Circuit Equations for Mutual Inductance
160
Ideal Transformers
161
Ideal Transformers
162
Mechanical Analog
d1
d2
163
Impedance Transformations
164
Semiconductor Diode
165
Shockley Equation
166
Load-Line Analysis of Diode Circuits
Assume VSS and R are known. Find iD and vD
167
Load-Line Analysis of Diode Circuits
168
Ideal Diode Model
The ideal diode acts as a short circuit for
forward currents and as an open circuit
with reverse voltage applied.
iD gt 0 vD lt 0 ? diode is in the on state vD lt
0 ID 0 ? diode is in the off state
169
Assumed States for Analysis of Ideal-Diode
Circuits
1. Assume a state for each diode, either on
(i.e., a short circuit) or off (i.e., an open
circuit). For n diodes there are 2n possible
combinations of diode states. 2. Analyze the
circuit to determine the current through the
diodes assumed to be on and the voltage across
the diodes assumed to be off.
170
3. Check to see if the result is consistent with
the assumed state for each diode. Current must
flow in the forward direction for diodes assumed
to be on. Furthermore, the voltage across the
diodes assumed to be off must be positive at the
cathode (i.e., reverse bias). 4. If the results
are consistent with the assumed states, the
analysis is finished. Otherwise, return to step 1
and choose a different combination of diode
states.
171
Half-Wave Rectifier with Resistive Load
The diode is on during the positive half of the
cycle. The diode is off during the negative half
of the cycle.
172
Half-Wave Rectifier with Smoothing Capacitor
The charge removed from the capacitor in one
cycle
173
Full-Wave Rectifier
The capacitance required for a full-wave
rectifier is given by
174
Full-Wave Rectifier
175
Clipper Circuit
176
NPN Bipolar Junction Transistor
177
Bias Conditions for PN Junctions
The base emitter p-n junction of an npn
transistor is normally forward biased
The base collector p-n junction of an npn
transistor is normally reverse biased
178
Equations of Operation
From Kirchoffs current law
179
Equations of Operation
Define ? as the ratio of collector current to
emitter current
Values for ? range from 0.9 to 0.999 with 0.99
being typical. Since
Most of the emitter current comes from the
collector and very little (?1) from the base.
180
Equations of Operation
Define ?? as the ratio of collector current to
base current
Values for ? range from about 10 to 1,000 with a
common value being ? ? 100.
The collector current is an amplified version of
the base current.
181
The base region is very thin
Only a small fraction of the emitter current
flows into the base provided that the
collector-base junction is reverse biased and the
base-emitter junction is forward biased.
182
Common-Emitter Characteristics
vBC
vCE
183
Common-Emitter Input Characteristics
184
Common-Emitter Output Characteristics
185
Amplification by the BJT
A small change in vBE results in a large change
in iB if the base emitter is forward biased.
Provided vCE is more than a few tenths of a
volt, this change in iB results in a larger
change in iC since iC?iB.
186
Common-Emitter Amplifier
187
Load-Line Analysis of a Common Emitter Amplifier
(Input Circuit)
188
Load-Line Analysis of a Common Emitter Amplifier
(Output Circuit)
189
Inverting Amplifier
As vin(t) goes positive, the load line moves
upward and to the right, and the value of iB
increases. This causes the operating point on
the output to move upwards, decreasing vCE ? An
increase in vin(t) results in a much larger
decrease in vCE so that the common emitter
amplifier is an inverting amplifier
190
PNP Bipolar Junction Transistor
Except for reversal of current directions and
voltage polarities, the pnp BJT is almost
identical to the npn BJT.
191
PNP Bipolar Junction Transistor
192
NMOS Transistor
193
NMOS Transistor
194
Operation in the Cutoff Region
195
Operation Slightly Above Cut-Off
By applying a positive bias between the Gate (G)
and the body (B), electrons are attracted to the
gate to form a conducting n-type channel between
the source and drain. The positive charge on the
gate and the negative charge in the channel form
a capacitor where
196
Operation Slightly Above Cut-Off
For small values of vDS, iD is proportional to
vDS. The device behaves as a resistance whose
value depends on vGS.
197
Operation in the Triode Region
198
(No Transcript)
199
Load-Line Analysis of a Simple NMOS Circuit
200
Load-Line Analysis of a Simple NMOS Circuit
201
Load-Line Analysis of a Simple NMOS Circuit
202
CMOS Inverter
203
Two-Input CMOS NAND Gate
204
Two-Input CMOS NOR Gate