I e=Ic+ I b=β I b+ I b

And I e=(β+1)I b (8-3)

However, since the ac beta is typically much greater than 1, we use the following approximation for the current analysis:

I e≅ β I b (8-4)

The input impedance is determined by the following ratio:

Zi=U i

I i=U be

I b

The voltage gain for the common-emitter configuration will now be determined for the configuration of Figure 8-7. For the defined direction of I o and polarity of U o, the formula is:

U o=−I oRL

Figure 8-7 Determining the voltage gain for the common-emitter transistor amplifier

The minus sing simply reflects the fact that the direction of I o in Figure 8-7 would establish a voltage U o with the opposite polarity. We can continue:

U o=−I oRL=−β I b RL

And U i=I b βr e

So that A v=U o

U i=

−βIb RLI b βre

And A v=−RLre

(8-5)

The resulting minus sing for the voltage gain reveals that the output and input voltages are 180° out of phase.

Words and Expressionsmodel n. modeloschematic a. esquemáticoformative a. formativohybrid a. híbridospecification n. especificación

drawback n. inconvenienteflaw n. fallatroublesome n. molestopredetermine

a. predeterminar

proficiency vt. competenciainvestigation n. investigacióndramatic n. dramáticoundertaking a. empresaswing n. columpiobypass n. derivaciónsubscript n. subíndiceimpedance n. impedancia

Activities

1. Complete the following sentences according to the passage above.

(1) What we should make clear first is that one must do a complete of a system before considering the ac response.

(2) The re model for BJT transistor has been introduced above with a short description of why it is good approximation to the actual of a BJT transistor.

(3) As you the modifications of the network to define the ac equivalent, it is important that the parameters of interest such as Zi,Zo,I i and I o be carried through properly.

(4) As we have known before that the base and collector currents are by the following equation: I c=β I b.

2. For the network of Figure 8-8 without C e (unbypassed), determine:

Figure 8-8

a. re b. Zi

c. Zo d. Av

3. Repeat the analysis of the problem above with C e in place.

Further ReadingHybrid π Model for a BJT transistor

The hybrid π model includes parameters that do not appear in other simple models primarily to provide a more accurate model for high-frequency effects. For lower frequencies approximations to the model can be made with the result that the re model introduced earlier will result. The hybrid π model appears in Figure 8-9 with all the parameters necessary for a full-frequency analysis.

All the capacitors that appear in Figure 8-9 are stray or parasitic capacitors between the various junctions of the device. Their capacitive effects only come into play at high frequencies. At low and medium frequencies their resistance is very large and they can be considered to be open circuits.

Figure 8-9 Hybrid π model for a transistor

The capacitor Cμ is usually just a few to a few tens of picofarads. The resistance rb includes three levels base contact, base bulk resistance and base spreading resistance. The first is due to the actual connection to the base. The second includes the resistance from the external terminal to the active region of the transistor, and the last is the actual resistance within the active base region. rb is typically a few to a few tens of ohms. The resistances r π, r μ and ro represent the resistancesbetween the indicated terminals of the device when the device is in the active region. The resistance r π is simply equal to β re as introduced for the common-emitter re model. The resistance r μ is a very large resistance, and provides a feedback path from output to input circuits in the equivalent model. Its value is placed in the megohm range, typically larger than βro. The output resistance ro is the output resistance normally appearing across an applied load, whose typically lies between 5kΩ and 40kΩ.

New Wordsstray n. errantepicofarad n. picofaradiobulk n. masa

Cμ

rμ

Uπ rπCπ

gmUπ

megohm n. megaohmio

Unit 9 Digital Concepts

After reading this unit and completing the exercises, you will be able to

Define analog Define digital Explain the differences between digital and analog quantities Define binary and its bit

Lead-in1. Look at the picture and discuss about it. Then write down the relevant terms and expressions in the space below.

Dual-in-line package

2. Read the following passage with the questions below to think about it.

(1) How to explain the advantages of digital over analog?

(2) How to determine the amplitude, period, frequency and duty cycle of a digital waveform?

Time to Read ItThe concept of a digital computer can be traced back to Charles Babbage, Who developed a crude mechanical computation device in the 1830s. The first functioning digital computer was built in 1944 at Harvard University, but it was electromechanical, not electronic. Modern digital electronics began in 1946 with an electronic digital computer called ENIAC, which was implemented with vacuum-tube circuits. Even though it took up an entire room, ENIAC didn’t have the computing power that your hand-held calculator does.

The term “digital” is derived from the way computers perform operations, by counting digits. For many years, applications of digital electronics were confined to computer systems. Today, digital technology is applied in a wide range of areas. Such applications as television, communication systems, radar, navigation and guidance system, military systems, digital techniques. Digital technology has progressed from vacuum-tube circuits to discrete transistors to complex integrated circuits, some of which contain millions of transistors.

Digital and Analog Quantities

Electronic circuits can be divided into two broad categories, digital and analog. Digital electronics involves quantities with discrete values, and analog electronics involves quantities with continuous values. For everyone who is ready to study electronics, analog electronics will always be the foundation, and then it is time to get closer to digital circuits.

An analog quantity has continuous values, and digital quantity has set of discrete values. Most things that can be measured quantitatively appear in nature in analog forms. For example, the air temperature changes over a continuous range of values. During a given day, the temperature does not go from, say, 70°F to 71°F instantaneously; it takes on all the infinite values in between. If you use the graph to display the temperature on a typical summer day, you will have a smooth, continuous curve similar to the curve in Figure 9-1.

Rather than using the graph to display the temperature on a continuous basis, suppose you just take a temperature reading every hour. Now you have sampled values representing the temperature at discrete points every hour over a 24-hours period, as indicated in Figure 9-2. You have effectively converted an analog quantity to a form that can now be digitalized, representing each sampled value by a digital code. It is important to realize that Figure 9-2 itself is not the digital representation of the analog quantity.

.

Figure 9-1 Graph of an analog quantity (temperature versus time)

Figure 9-2 Sampled-value representation (quantization) of the analog quantity in figure 9-1

The Digital Advantage

Digital has certain advantages over analog in electronics applications. For one thing, digital data can be processed and transmitted more efficiently and reliably than analog data. Also, digital data has a great advantage when storage is necessary. For example, music, when converted to digital form, can be stored more compactly and reproduced with greater accuracy and clarity than in analog form. For another, noise (unwanted voltage fluctuations) does not affect digital data as much as it affects analog signals.

Binary Digits, Logic Levels and Digital Waveforms

Digital electronics involves circuits and systems in which there are only two possible states. These two states are represented by two different voltage levels: HIGH and LOW. The two states can also be represented by current levels, open and closed switches, or lamps turned on and off. In digital systems, combinations of two states, called codes, are used to represent numbers, symbols, alphabetic characters, and other types of information. The two-state number is called binary, and its two digits 0 and 1.

Binary Digits

The two digits in the binary system, 1 and 0, are called bits, which is a contraction of the words “binary digits”. In digital circuits, two different voltage levels are used to represent the two bits. The binary digit 1 is represented by the higher voltage, which is referred to as HIGH and the binary digit 0 is represented by the lower voltage level, which as LOW. This is called positive logic. A less common system in which 1 is represented by LOW and 0 is represented by HIGH is called negative logic.

Groups of bits (combinations of 1s and 0s), called codes, are used to represent numbers, letters, symbols, instructions, and anything else required in a given applications.

Logic Levels

The voltages used to represent binary digits 1 and 0 are called logic levels. Ideally, one voltage level represents HIGH and the other represents LOW. In a practical digital circuit, however, HIGH may be any voltage between a specific minimum value and specified maximum level. Likewise,

LOW may be any voltage between a specified minimum and a specified maximum. There may be no overlap between the accepted HIGH levels and the accepted LOW levels.

Digital Waveforms

Digital waveforms consist of voltage levels that are changing back and forth between the HIGH and LOW levels or states. Figure 9-3 shows that a single positive-going pulse is generated when the voltage (or current) goes from its normally LOW level to its HIGH level and then back to its LOW level. The negative-going pulse in Figure 9-4 is generated when the voltage goes from its normally HIGH level to its LOW level and back to its HIGH level. A digital waveform is made up of a series of pulses.

As indicated in Figure 9-3 and Figure 9-4, the pulse has two edges: a leading edge that occurs first at the time t o and a trailing edge that occurs last at the time t i. For a positive-going pulse, the leading edge is a rising edge, and the trailing edge is falling edge. The pulses in Figure 9-3 and Figure 9-4 are ideal because the rising and falling edges are assumed to change in zero time (instantaneously).

Figure 9-3 Positive-going pulse

Figure 9-4 Negative-going pulse

Figure 9-5 Nonideal pulse characteristics

In practice, these transitions never occur instantaneously, although for most digital work you can assume ideal pulses.

Figure 9-5 shows a nonideal pulse. The time required for the pulse to go from its LOW level to its HIGH is called the rise time (t r), and the time required for the transition from the HIGH level to the

LOW level is called the fall time (t f ). In practice, it is common to measure the rise time from 10% to 90% of the pulse amplitude (height from baseline) and to measure the fall time from 90% to 10% of the pulse amplitude, as indicated in Figure 9-5. The bottom 10% and the top 10% of the pulse are not included in the rise and fall times because of the nonlinearities in the waveform in these areas. The pulse width (tw) is a measure of the duration of the pulse and is often defined as the time interval between the 50% points on the rising and falling edges, as indicated in Figure 9-5.

Most waveforms encountered in digital systems are composed of series of pulses, sometimes called pulse trains, and can be classified as either periodic or nonperiodic. A periodic pulse waveform is one that repeats itself at a fixed interval, called period (T). The frequency ( f ) is the rate at which it repeats itself and it is measured in hertz (Hz). A nonperiodic pulse waveform, of course, does not repeat itself at fixed intervals and may be composed of pulses with randomly different pulse widths and/or with randomly different time intervals between the pulses.

The frequency (f ) of a pulse (digital) waveform is the reciprocal of the period. The relationship between frequency and period is expressed as follows:

f= 1T (9-1)

An important characteristic of a periodic digital waveform is its duty cycle. The duty Cycle refers to the ratio of the pulse width (tw) to the period (T ) and can be expressed as a percentage.

DutyCycle=( twT )∗100% (9-2)

Example 9-1 A portion of a periodic digital waveform is shown in Figure 9-6. The measurements are in milliseconds. Determine the following:

(a) period (b) frequency (c) duty cycle

.

Figure 9-6 A periodic digital waveform

Solution

(a) The period is measured from the edge of one pulse to the corresponding edge of the next pulse. In this case T is measured from one leading edge to the next, as indicated. T equals 10 ms.

(b) f= 1T= 110ms

=100Hz

(c) DutyCycle=( twT )∗100%=( 1ms10ms )∗100%=10%

Words and Expressionscrude a. crudodigit n. dígitoinstrumentation n. instrumentacióndiscrete a. discretoanalog a. analógicocategory n. categoríafoundation n. fundacióncontinuous a. continuoinstantaneously ad. instantáneamentegraph n. gráficoversus prep. versusquantization n. cuantizacióndigitize vt. digitalizartransmit vt. transmitirboldface n. negritacompactly ad. compactareproduce vt. reproducirclarity n. claridadfluctuation n. fluctuaciónalphabetic a. alfabéticocontraction n. contracciónoverlap n. superposiciónpositive-going a. curso positivopulse n. pulsoNegative-going a. curso negativoTrail vt. rastroNonlinearity n. no linealidadBaseline n. baseAmplitude n. amplitudTrain n. trenClassify vt. clasificarperiodic a. periódico