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Course: Microelectronics Circuit Professor: Truong Cong Dung Nghi
Typology: Slides
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between the two points and is the defined voltage. It is extremely important that the variables used to represent voltage between two points be defined in such a way that the solution will let us interpret which point is at the higher potential with respect to the other. Lightning bolt Large industrial motor current Typical household appliance current Causes ventricular fibrillation in humans Human threshold of sensation Integrated circuit (IC) memory cell current Synaptic current (brain cell) 106 104 102 100 10 –^2 10 –^4 10 –^6 10 –^8 10 –^10 10 –^12 10 –^14 Cu rrent in amperes (A)
C H A P T E R 1 B A S I C C O N C E P T S Figure 1. Voltage representations. +
V 1 = 2 V V 2 =– 5 V V 2 = 5 V A (^) C i r c u i t 1 C i r c u i t 2 C i r c u i t 3 A A B (a) (b) (c) B B 1-024hr.qxd 30-06-2010 13:16 Page 4
The range of magnitudes for voltage, equivalent to that for currents in Fig. 1.4, is shown in Fig. 1.6. Once again, note that this range spans many orders of magnitude.
Lightning bolt High-voltage transmission lines Voltage on a TV picture tube Large industrial motors ac outlet plug in U.S. households Car battery Voltage on integrated circuits Flashlight battery
Voltage across human chest produced by the heart (EKG) Voltage between two points on human scalp (EEG) Antenna of a radio receiver
108
106
104
102
100
10 –^2
10 –^4
10 –^6
10 –^8
10 –^10
Voltage in
volts (
V)
In their normal mode of operation, independent sources supply power to the remainder of the circuit. However, they may also be connected into a circuit in such a way that they absorb power. A simple example of this latter case is a battery-charging circuit such as that shown in Example 1.1. It is important that we pause here to interject a comment concerning a shortcoming of the models. In general, mathematical models approximate actual physical systems only under a cer- tain range of conditions. Rarely does a model accurately represent a physical system under every set of conditions. To illustrate this point, consider the model for the voltage source in Fig. 1.14a. We assume that the voltage source delivers v volts regardless of what is connected to its terminals. Theoretically, we could adjust the external circuit so that an infinite amount of current would flow, and therefore the voltage source would deliver an infinite amount of power. This is, of course, physically impossible. A similar argument could be made for the independ- ent current source. Hence, the reader is cautioned to keep in mind that models have limitations and thus are valid representations of physical systems only under certain conditions. For example, can the independent voltage source be utilized to model the battery in an automobile under all operating conditions? With the headlights on, turn on the radio. Do the headlights dim with the radio on? They probably won’t if the sound system in your automo- bile was installed at the factory. If you try to crank your car with the headlights on, you will notice that the lights dim. The starter in your car draws considerable current, thus causing the voltage at the battery terminals to drop and dimming the headlights. The independent volt- age source is a good model for the battery with the radio turned on; however, an improved model is needed for your battery to predict its performance under cranking conditions.
A
B
A
B
v( t )
i
i ( t )
(a) (b)
i
Determine the power absorbed or supplied by the elements in the network in Fig. 1.15.
D E P E N D E N T S O U R CES In contrast to the independent sources, which produce a particular voltage or current completely unaffected by what is happening in the remainder of the circuit, dependent sources generate a voltage or current that is determined by a voltage or current at a specified location in the circuit. These sources are very important because they are an integral part of the mathematical models used to describe the behavior of many elec- tronic circuit elements. For example, metal-oxide-semiconductor field-effect transistors (MOSFETs) and bipolar transistors, both of which are commonly found in a host of electronic equipment, are mod- eled with dependent sources, and therefore the analysis of electronic circuits involves the use of these controlled elements. In contrast to the circle used to represent independent sources, a diamond is used to represent a dependent or controlled source. Fig. 1.16 illustrates the four types of dependent sources. The input terminals on the left represent the voltage or current that controls the dependent source, and the output terminals on the right represent the output current or volt- age of the controlled source. Note that in Figs. 1.16a and d, the quantities! and " are dimen- sionless constants because we are transforming voltage to voltage and current to current. This is not the case in Figs. 1.16b and c; hence, when we employ these elements a short time later, we must describe the units of the factors r and g.
element is supplying (2)(24)=48 W of power. The current is into the positive terminals of elements 1 and 2, and therefore elements 1 and 2 are absorbing (2)(6)=12 W and (2)(18)=36 W, respectively. Note that the power supplied is equal to the power absorbed.
+– v=! v S +–
(a) (b)
v S
v S i^ = g v S
v= ri (^) S
i =" iS
iS
(c) (d)
i (^) S
Figure 1. erent types of dent sources.
s have
wer that is absorbed or supplied by the elements in Fig. E1.3.
g Assessment
ANSWER: Current source supplies 36 W, element 1 absorbs 54 W, and element 2 supplies 18 W. 12 V 6 V
I =3 A
3 A
I =3 A
18 V
2
1
EECE 251, Set 1^49 Resistance
rial, i.e., r
A l R v A l R r
v or equivalently, by the voltage–c relationship between the polarity note that we have tacitly assumed voltage–current characteristic is l The symbol " is used to repre Although in our analysis we w described by a straight-line chara readers realize that some very use resistance characteristic; that is, t
i ( t ) v( t )
(a) Figure 2. (a) Symbol for a resistor; (b) some practical devices. (1), (2), and (3) are high- power resistors. (4) and (5) are high-wattage fixed resistors. (6) is a high- precision resistor. (7)–(12) are fixed resistors with different power ratings. (Photo courtesy of Mark Nelms and Jo Ann Loden)
e is zero and the circuit is reduced to that shown in Fig. 2.3b; that is, the eplaced by a short circuit. On the other hand, if the resistance is increased r and larger, we finally reach a point where it is essentially infinite and the eplaced by an open circuit, as shown in Fig. 2.3c. Note that in the case of re R! 0, although the current could theoretically be any value. In the open- ent is zero regardless of the value of the voltage across the open terminals.
i(t) = v(t)!R
v(t) = Ri(t)
(a) (b) (c)
v( t )
e find the current to be
I=V/R=12/2k=6 mA
place of 1000. The power absorbed by the resistor is given by Eq. (2.2) or
S E C T I O N 2. 2 K I R C H H O F F ’ S L
1
R 5 i 1 ( t ) R 2 i 2 ( t ) i 3 ( t ) i 5 ( t ) v 1 ( t ) v 2 ( t ) i 7 ( t ) i 6 ( t )^ i^ 4 ( t ) i 8 ( t ) R 1 R 4 R 5
1 4 5 2
± – ± – ±– Figure 2.
Circuit used to i Kirchho
hm’s law. At this point we begin to expand our capabilities to handle orks that result from an interconnection of two or more of these sim- assume that the interconnection is performed by electrical conductors resistance—that is, perfect conductors. Because the wires have zero in the circuit is in essence lumped in each element, and we employ the r circuit to describe the network. iscussion, we will define a number of terms that will be employed s. As will be our approach throughout this text, we will use examples ts and define the appropriate terms. For example, the circuit shown
0 13:14 Page 31
SM (^) EECE 251, Set 1 70
Example
will be used to describe the terms node, loop , and branch. A node is simply a onnection of two or more circuit elements. The reader is cautioned to note that, ne node can be spread out with perfect conductors, it is still only one node. This ed in Fig. 2.5b, where the circuit has been redrawn. Node 5 consists of the entire nnector of the circuit. tart at some point in the circuit and move along perfect conductors in any direction encounter a circuit element, the total path we cover represents a single node. we can assume that a node is one end of a circuit element together with all the per- ctors that are attached to it. Examining the circuit, we note that there are numerous ugh it. A loop is simply any closed path through the circuit in which no node red more than once. For example, starting from node 1, one loop would contain the and i 1 ; another loop would contain and i 1 ; and so on. the path and i 1 is not a loop because we have encountered node 3 ally, a branch is a portion of a circuit containing only a single element and the nodes d of the element. The circuit in Fig. 2.5 contains eight branches. the previous definitions, we are now in a position to consider Kirchhoff’s laws, er German scientist Gustav Robert Kirchhoff. These two laws are quite simple but important. We will not attempt to prove them because the proofs are beyond our el of understanding. However, we will demonstrate their usefulness and attempt to eader proficient in their use. The first law is Kirchhoff’s current law (KCL), which the algebraic sum of the currents entering any node is zero. In mathematical form pears as a^ 2. N j = 1 ij(t) = 0 R 1 , v 1 , R 5 , v 2 , R 3 , R 1 , v 2 , R 4 , R 2 , v 1 , v 2 , R 4 , v 2 ( t ) v 1 ( t ) R (^) 1 R^^2 4^ R 5 R 3 (a) i 1 ( t ) R 2 i 2 ( t ) i (^) 3 ( t ) i (^) 5 ( t ) v 1 ( t ) v 2 ( t ) i (^) 7 ( t ) i 6 ( t )^ i^ 4 ( t ) i 8 ( t ) R 1 R 4 R 5 (b) 1 4 5 2 3 R 3 ± – ±– ±– ±– Figure 2. Circuit used to illustrate KCL KCL is an extremely important and useful law.
ts. We will assume that the interconnection is performed by electrical conductors Kirchhoff’s Law at have zero resistance—that is, perfect conductors. Because the wires have zero the energy in the circuit is in essence lumped in each element, and we employ the ed-parameter circuit to describe the network. us in our discussion, we will define a number of terms that will be employed t our analysis. As will be our approach throughout this text, we will use examples e the concepts and define the appropriate terms. For example, the circuit shown
^
and , in series, ple of conservation
or the number of 2.19. The sign on ntered first as we nch and go around pose we start with op as shown; then in that order. For l is met first; hence, rminal first; hence, (2.20) (2.21) rises (2.22) 1^ I 2
(a) (b) b a I (^) T = I 1 – I 2 + I 3 b IT
circuit, (b) equivalent circuit. KVL can be applied in two ways: by taking either a clockwise or a counter- clockwise trip around the loop. Either way, the algebraic sum of voltages around the loop is zero.
v 1 + − +^ −^ v 4 v (^) 2 v (^) 3 v 5
are in series, since the sa Ohm’s law to each of the
If we apply KVL to the l have
Combining Eqs. (2.24) an
or
Notice that Eq. (2.26) can
implying that the two res tor ; that is,
Thus, Fig. 2.29 can be re The two circuits in Figs. exhibit the same voltage- equivalent circuit such as the analysis of a circuit.
The equivalent resistan series is the sum of the i
For N resistors in series t
To determine the volt stitute Eq. (2.26) into Eq
v!
R eq! R
R eq
v!
v 1
44 Chapter 2 Basic Laws
Figure 2. Equivalent circuit of the Fig. 2.29 circuit.
v
R eq
v
i
a
b
Resistors in series behave as a single resistor whose resistance is equal to the sum of the resistances of the individual resistors.
Figure 2. A single-loop circuit with two resistors in series.
v + −
R (^) 1 v 1
R 2 v (^) 2
i
a
b
· · ·
,... ,
,... ,
cations, computers, and power system the tuning circuits of radio receivers a in computer systems. A capacitor is typically construct
Figure 6. A typical capacitor.
− − −
q − q
− v Figure 6. A capacitor with applied voltage v.
Alternatively, capacitance is the amount of charge stored per plate for a unit voltage difference in a capacitor.
A capacitor consists of two conduct lator (or dielectric).
In many practical applications, the pl the dielectric may be air, ceramic, pa When a voltage source is co Fig. 6.2, the source deposits a positive ative charge on the other. The cap charge. The amount of charge stored, portional to the applied voltage so
where C , the constant of proportiona of the capacitor. The unit of capacita the English physicist Michael Farada we may derive the following definitio
q! C
v
" q
v
Capacitance is the ratio of the charg the voltage difference between the tw
Note from Eq. (6.1) that 1 farad! 1
Introduction So far we have limited our study to resis we shall introduce two new and importa ments: the capacitor and the inductor. Unl energy, capacitors and inductors do not which can be retrieved at a later time. Fo inductors are called storage elements. The application of resistive circuits is duction of capacitors and inductors in thi analyze more important and practical circ cuit analysis techniques covered in Chapte cable to circuits with capacitors and indu We begin by introducing capacitors bine them in series or in parallel. Later, w As typical applications, we explore how c op amps to form integrators, differentiato
Capacitors A capacitor is a passive element designe tric field. Besides resistors, capacitors are components. Capacitors are used extensiv cations, computers, and power systems. F the tuning circuits of radio receivers and a in computer systems. A capacitor is typically constructed a
6.
6.
216 Chapter 6 Capacitors and Inductors
In contrast to a resistor, which spends or dissipates energy irreversibly, an inductor or capacitor stores or releases energy (i.e., has a memory).
Metal plates, each with area A
d
Dielectric with permittivity_!_
Figure 6. A typical capacitor.
ale29559_ch06.qxd 07/08/2008 10:59 AM Page 216
spacing, the greater permittivity, the t values and types. to microfarad ey are made of and 6.3 shows the cir- at according to the and the itor is discharging. capacitors. Poly- change with tem- ielectric materials acitors are rolled apacitors produce common types of padder) capacitor 0 i 6 0, (mF) Figure 6.
Circuit symbols for capacitors: (a) fixed capacitor, (b) variable capacitor. i C i
−∞
Example 6.
v ( t )
0 1 2 3 4
50
− 50
t
For Example 6.4.
i (mA)
0 1 2 3 4
10
− 10
t
For Example 6.4.
Practice Problem 6. i (mA)
0 2 4 6
100
t (ms)
222 Chapter 6 Capacitors and Inductors
Example 6.5 Obtain the energy stored in e conditions.
Figure 6.
6 mA 3 kΩ
5 kΩ 4 kΩ
2 kΩ
2 mF
4 mF
(a)
6 mA
59_ch06.qxd 07/08/2008 10:59 AM Page 222
ve the same voltage across
(6.11)
(6.12)
p (^) " CN (6.13)
" p^ " CN
dv dt
" iN
v
6.3 Series and Parallel Capacitors 223
i C 1
(a)
i 1 C 2 C 3 C (^) N
iN v
−
i
(b)
C eq v
−
i (^) 2 i 3
Figure 6. (a) Parallel-connected N capacitors, (b) equivalent circuit for the parallel capacitors.
connected capacitors is the
bine in the same manner as
onnected in series by com- the equivalent circuit in i flows (and consequently Applying KVL to the loop
(6.14)
,
(6.15)
dt " vN ( t 0 )
i ( t ) dt " v 2 ( t 0 )
p (^) " vN
v
C (^) 1
(a)
C (^) 2 C 3 CN
v (^) 1 v (^) 2 v (^) 3 v (^) N
i
v
(b)
i
−
Figure 6. (a) Series-connected N capacitors, (b) equivalent circuit for the series
p (^) " CN (6.13)
Figure 6. (a) Parallel-connected N capacitors, (b) equivalent circuit for the parallel capacitors.
connected capacitors is the
bine in the same manner as
onnected in series by com- he equivalent circuit in i flows (and consequently pplying KVL to the loop
(6.14)
(6.15)
p (^) " (6.16) 1 CN
" p^ " vN ( t 0 )
dt " v 1 ( t 0 ) " v 2 ( t 0 )
dt " vN ( t 0 )
( t ) dt " v 2 ( t 0 )
p (^) " vN
v
C (^) 1
(a)
C (^) 2 C 3 CN
v (^) 1 v (^) 2 v (^) 3 v (^) N
i
v
(b)
i
−
Figure 6. (a) Series-connected N capacitors, (b) equivalent circuit for the series capacitor.
Parallel Capacitors 225
of the circuit in Practice Problem 6.
C eq 70_!_ F 20_!_ F 120_!_ F
60_!_ F
50_!_ F
Example 6.
40 mF 20 mF
20 mF 30 mF
30 V + −
v 1 v (^) 2 v (^) 3
−
