• A signal is any waveform that serves as a means of communication. It represents a fluctuating electric quantity, such as voltage, current, electric or magnetic field strength, sound, image, or any message transmitted or received in telegraphy, telephony, radio, television, or radar.
• The average value of a waveform in the interval is defined as
• A periodic time function satisfies the expression
for all time and for all integers . The constant is the period and it is the smallest value of time which separates recurring values of the waveform.
• An alternating waveform is any periodic time function whose average value over a period is zero.
• The effective (or RMS) value of a periodic current waveform denoted as is the cur-rent that produces heat in a given resistor at the same average rate as a direct (constant) current and it is found from the expression
where RMS stands for Root Mean Squared, that is, the effective value or value of a current is computed as the square root of the mean (average) of the square of the current.
• If the peak (maximum) value of a current of a sinusoidal waveform is , then
• An amplifier is an electronic circuit which increases the magnitude of the input signal.
• An electronic (or electric) circuit which produces an output that is smaller than the input is called an attenuator. A resistive voltage divider is a typical attenuator.
• An amplifier can be classified as a voltage, current or power amplifier. The gain of an amplifier is the ratio of the output to the input. Thus, for a voltage amplifier
f t( ) a t b≤ ≤
Voltage Gain Output Voltage Input Voltage
---=
or
The current gain and power gain are defined similarly.
• The ratio of any two values of the same quantity (power, voltage or current) can be expressed in decibels (dB). By definition,
The values for voltage and current ratios are
• The bandwidth is where and are the cutoff frequencies. At these fre-quencies, and these two points are known as the 3-dB down or half-power points.
• The low-pass and high-pass filters have only one cutoff frequency whereas pass and band-stop filters have two. We may think that low-pass and high-pass filters have also two cutoff fre-quencies where in the case of the low-pass filter the second cutoff frequency is at while in a high-pass filter it is at .
• We also recall also that the output of circuit is dependent upon the frequency when the input is a sinusoidal voltage. In general form, the output voltage is expressed as
where is known as the magnitude response and is known as the phase response. These two responses together constitute the frequency response of a circuit.
• The magnitude and phase responses of a circuit are often shown with asymptotic lines as approximations and these are referred to as Bode plots.
• Two frequencies and are said to be separated by an octave if and separated
by a decade if .
• The transfer function of a system is defined as
where the numerator and denominator are as shown in the expression Gv = Vout⁄Vin
Gi Gp
dB = 10log Pout⁄Pin dB
dBv 20= log Vout⁄Vin dBi 20= log Iout⁄Iin
BW = ω2–ω1 ω1 ω2
Vout = 2 2⁄ = 0.707
ω = 0 ω = ∞
Vout( )ω = Vout( ) eω jϕ ω( )
Vout( )ω ejϕ ω( )
ω1 ω2 ω2 = 2ω1
ω2 = 10ω1
G s( ) Vout( )s Vin( )s
--- N s( ) D s( )
---= =
N s( ) D s( )
Chapter 1 Basic Electronic Concepts and Signals
• In the expression
where , the roots of the numerator are called the zeros of , and are found by letting . The roots of the denominator are called the poles of and are found by letting .
• The zeros and poles can be real and distinct, or repeated, or complex conjugates, or combina-tions of real and complex conjugates. However, in most engineering applicacombina-tions we are inter-ested in the nature of the poles.
• A system is said to be stable if a finite input produces a finite output. We can predict the sta-bility of a system from its impulse response .
• Stability can easily be determined from the transfer function on the complex frequency plane of the complex variable . A system is stable only when all poles lie on the left-hand half-plane. It is marginally stable when one or more poles lie on the axis, and unstable when one or more poles lie on the right-hand half-plane. However, the location of the zeros in the is immaterial.
• We can use the MATLAB function bode(sys) to draw the Bode plot of a system where sys = tf(num,den) creates a continuous-time transfer function sys with numerator num and denomi-nator den, and tf creates a transfer function. With this function, the frequency range and number of points are chosen automatically. The function bode(sys,{wmin,wmax}) draws the Bode plot for frequencies between wmin and wmax (in radians/second) and the function bode(sys,w) uses the user-supplied vector w of frequencies, in radians/second, at which the Bode response is to be evaluated. To generate logarithmically spaced frequency vectors, we use the command logspace(first_exponent,last_exponent, number_of_values).
The bode(sys,w) function displays both magnitude and phase. If we want to display the magni-tude only, we can use the bodemag(sys,w) function.
• Amplifiers are often represented by equivalent circuits also known as circuit models. The com-mon types are the voltage amplifier, the current amplifier, the transresistance amplifier, and the transconductance amplifier.
1.12 Exercises
1. Following the procedure of Example 1.1, derive and sketch the magnitude and phase responses for an high-pass filter.
2. Derive the transfer function for the network shown below.
3. A system has poles at , , , and zeros at , , and . Derive the transfer function of this system given that .
4. The circuit model shown below is known as a transresistance amplifier and the ideal characteris-tics for this amplifier are and .
With a voltage source in series with resistance connected on the input side and a load resistance connected to the output, the circuit is as shown below.
Find the overall voltage gain if . Then, use MATLAB to plot the RC
Chapter 1 Basic Electronic Concepts and Signals
magnitude of for the range . From the plot, estimate the cutoff fre-quency.
5. The circuit model shown below is known as a transconductance amplifier and the ideal charac-teristics for this amplifier are and .
With a voltage source in series with resistance connected on the input side and a load resistance connected to the output, the circuit is as shown below.
Derive an expression for the overall voltage gain
Av 103≤ ≤ω 108 3 dB
Rin→∞ Rout→∞
vin Rin G
mvin Rout vout iout
Gm iout vin
---vout=0
=
vs Rs
Rload
Gmvin Rout Rload vload
vs Rs
vin Rin
iout
Av = vload⁄vs
1.13 Solutions to End-of-Chapter Exercises Dear Reader:
The remaining pages on this chapter contain solutions to all end-of-chapter exercises.
You must, for your benefit, make an honest effort to solve these exercises without first looking at the solutions that follow. It is recommended that first you go through and solve those you feel that you know. For your solutions that you are uncertain, look over your procedures for inconsistencies and computational errors, review the chapter, and try again. Refer to the solutions as a last resort and rework those problems at a later date.
You should follow this practice with all end-of-chapter exercises in this book.
Chapter 1 Basic Electronic Concepts and Signals