Radio Frequency Electronics Preliminaries I Invented Regenerative circuit while an undergraduate (1914) Superheterodyne receiver (1918) Super-regenerative circuit (1922) Frequency modulation (FM) radio Edwin H. Armstrong Professor at Columbia University Held 42 patents Awards IRE (now IEEE) Medal of Honor French Legion of Honor, National Inventors Hall of Fame Image from Wikipedia 1
Calculators Calculators should be a real calculator. Smartphone apps the simulate calculators are not allowed. If your calculator has wireless/network functionality, turn the wireless off during exams. Be sure to set your calculator to display in engineering mode. Unless stated otherwise, provide answer to 3 significant figures. Be sure that you know how to do complex number arithmetic on your calculator. Be sure that you know how to do 2 2 and 3 3 matrix arithmetic on your calculator. Consider programming frequently-used formulas into your calculator. I can t teach you how to use your calculator. Read the manual or look for tutorials on the web. There are some decent tutorial for TI calculators on YouTube. 2
Matlab, Mathematica, SPICE, Some homework assignments will require SPICE simulations. Some homework will involve writing Matlab scripts. The use of tools such as Matlab, Mathematica, Maple, SPICE, the programmable features of your calculator, etc. is strongly encouraged. If you solve homework problems using these tools, or check your answers with these tools, you are eligible for extra credit. 3
Metric/Engineering Prefixes Since we are engineers, be sure to express numbers in formats that other engineers do. When in Rome, do as the Romans do. If your calculator displays 5.2E 4 set it so it displays the number as 520E 6 If this were the answer to a voltage calculation, you would write V o = 520 μv Note space between number and units Note V is not in italics Grayed-out prefixes are generally not used in electrical engineering, but we have decibel and use phrases such as 20 db per decade. 4
Metric/Engineering Formats There are of course exceptions. For example, the speed of light is c = 3 10 8 m/s, and the permeability of vacuum is 4π 10 7 H/m and the charge on an electron is 1.6 10 19 C, and so on. You can tell Matlab to format its answer in engineering format >> format shorteng >> u = 4*pi*1e-7 u = 1.2566e-00 To make Microsoft Excel format cells in engineering format Right click Format Custom and then type in ##0.0E+0 To increase the number of decimals just add more zero's i.e., ##0.00E+0 5
Complex Number Review Imaginary Axis Rectangular Polar Exponential b z = a + jb z = z θ z = z e jθ θ z = a 2 + +b 2 a Real Axis θ = tan 1 b a Euler s identity e jθ = cos θ + j sin θ cos θ = ejθ + e jθ + 2 sin θ = ejθ e jθ 2j Let z 1 = a + jb z 2 = c + jd z 3 = e + jf Then z 1 z 2 z 3 = a + jb c + jd e + jf = ac bd + jbc + jad e + jf = z 1 z 2 z 3 = z 1 z 2 z 3 θ 1 + θ 2 + θ 3 z = z 1 z 2 z 3 e j θ 1+θ 2 +θ 3 6
Complex Number Review Let z 1 = a + jb z 2 = c + jd z 3 = e + jf Then 1 = 1 z 1 a + jb = (a jb) a + jb a jb = (a jb) a 2 + b 2 = a a 2 + b 2 j b a 2 + b 2 1 z 1 = 1 z 1 θ 1 1 z 1 = 1 z 1 e jθ 1 z 1 z 2 z 3 = 1 a + jb c + jd e + jf = 1 ac bd + jbc + jad e + jf = 1 z 1 z 2 z 3 = 1 z 1 z 2 z 3 (θ 1+θ 2 + θ 3 ) 1 z 1 z 2 z 3 = 1 z 1 z 2 z 3 e j(θ 1+θ 2 +θ 3 ) 7
Complex Number Review Let z 1 = a + jb z 2 = c + jd z 3 = e + jf Then z 1 z 2 = (a + jb) c + jd = (a + jb) (c jd ) c + jd (c jd) = z 1 z 2 = z 1 z 2 θ 1 θ 2 z 1 z 2 = z 1 z 2 e j θ 1 θ 2 z 1 n = a + jb n = a + jb 1 a + jb 2 a + jb n z 1 0.32 = a + jb 0.32 =? z 1 n = z n nθ z 1 0.32 = z 1 0.32 0.32 θ z n 1 = z n j nθ 1 e z 0.32 1 = z 0.32 j 0.32 θ 1 e 8
Concepts Review The Decibel (db) Decibel is defined as ratio of two powers P 1 and P 2 Ratio (db) = 10 log 10 P 1 /P 2 Note 10 and not 20 (that is why it is called deci-bell) Consider a signal source that delivers a power P i to an amplifier. The amplifier amplifies the signal and delivers a power P L to a load. The amplifier s power gain is: G = 10 log 10 P L P i V i R i R L V L In terms of the terminal voltages and resistances: V 2 2 L R i V L R L V L R L G = 10 log 10 V 2 = 10 log 10 2 = 20 log 10 i R L V i R i V i R i P i P L In the special case when R L = R i, the power and voltage gain is G = 20 log 10 V L V i Strictly speaking, even though the 20 should be used only when R i = R L, 20 is used universally when we express voltage ratios in db, regardless of the values of R i and R L. In microelectronics, voltages and currents are normally measured and calculated for, while in RF work, quite often power is the quantity of interest. Also, in RF work, in many instances R i = R L = 50 Ω. 9
Concepts Review Meaning of 3-dB We often encounter phrases such as 3-dB point, 3-dB frequency, 3-dB bandwidth, half power point, etc., and there is quite a bit of confusion surrounding these terms. First, the 3 comes from the fact that log 10 2 = 0.301 and 10log 10 2 3 Thus, when an amplifier has a power gain of 2, the power gain in db will be 3 db because: P L = 2P i G = 10 log P L P i = 10 log 2 = 3 db Consider the attenuator shown Devices such as this are frequently used in RF work. The 50 Ω indicates the input impedance of the device. In (a) below a sinusoidal signal source delivers 100 mw to a 50 Ω load. In (b), the attenuator reduces the power by 3 db, which is a factor 2 so the load now dissipates 50 mw. Signal Source R L = 50 Ω Signal Source 3 db Attenuator R L = 50 Ω (a) P L = 100 mw (b) P L = 50 mw Note that the load voltage in (a) is 0.1 50 = 2.236 V RMS and in (b) the load voltage is (0.05)(50) = 1.581 V RMS. While the power was reduced by a factor 2, the voltage was reduced by 1.581 2.236 = 1 2. 10
dbm, dbw, dbμ, etc. The decibel is related to the ratio of two powers. It is sometimes convenient to express a power relative to some reference power. One such reference is 1 mw, and this leads to the dbm: P dbm 10 log P 1 mw Example. Express the following power levels as dbm: 1 mw, 10 mw, 1 W, and 5 μw Solution 1 mw = 10 log 1 mw 1 mw = 0 dbm 10 mw = 10 log 10 mw 1 mw = 10 dbm 1 W = 10 log 1,000 mw 1 mw = 30 dbm 5 μw = 10 log 5 10 6 1 10 3 = 23 dbm 11
dbm, dbw, dbμ, etc. dbμ 10 log P 1 μw Power, reference is 1 μw. dbm 10 log P 1 mw Power, reference is 1 mw. dbw 10 log P 1 W Power, reference is 1 W. dbv 20 log V 1 V RMS Voltage, reference is 1 V RMS regardless of impedance. dbmv 20 log dbμv 20 log dbz, dba, dbi, V 1 mv RMS Voltage, reference is 1 mv RMS across 75 Ω. Used in cable TV. V 1 μv RMS Voltage, reference is 1 μv RMS. Used in radio sensitivity, amplifier and antenna specifications. Many others, used in radar, sound, etc. 12
dbm, dbw, dbμ, etc. Since log A B = log A + log B it follows that we know an amplifier s gain in db, we can easily perform power level calculations. Example. An amplifier with 15 db power gain amplifies a -10 dbm signal. What is the resulting output power, expressed in dbm? Solution 10 dbm is equivalent to 10 1 1 10 3 = 10 mw = 0.01 W. Further, 15 db gain is equivalent to a factor 10 1.5, so that the output power is P o = 10 1.5 0.01 = 316 mw = 0.316 W. Converting to dbm gives 10 log 0.316 1 10 3 = 25 dbm. However, using the logarithm property log A B = log A + log B we can write directly P o = P i + G = 10 dbm + 15 = 25 dbm This works for the other db units: dbμ, dbw, etc. 13
dbm, dbw, dbμ, etc. If an antenna delivers V g =10 dbμv to a receiver with input an impedance of 50 Ω, what is the signal level in dbm? Solution 10 = 20log V g 1 10 6 V RMS V g = 1 10 6 10 10 20 = 3.16 10 6 V RMS P = V g R 2 = 3.16 10 6 2 50 = 200 10 15 W = 10 log 200 10 15 1 10 3 dbm = 97 dbm A sine wave has an amplitude of V g = 20 V. Express this as dbv. Solution The definition of dbv requires that we use the RMS value of V g. Thus: V g = 20log 20 2 1 = 23 db 14
dbm, dbw, dbμ, etc. A spectrum analyzer with 50 Ω input impedance has a label next to the input warning the user to limit the input power to 10 dbm. What is maximum amplitude of a signal v t = A cos ωt that one can apply to the spectrum analyzer? Solution P 10 dbm = 10 log 1 10 3 = 10 mw 1 P = 1 10 3 10 10 10 P = A RMS 2 R 10 10 3 = A RMS 2 A RMS = 0.707 V 50 A = 2 A RMS A = 1 V 15
dbm, dbw, dbμ, etc. The (sine wave) voltage across a 50 Ω resistor is increased from 5 V to 7 V. (a) What power increase (in db) does this represent? (b) Repeat but now for a 75 Ω resistor. Solution Part (a) P 2 V 2 2 50 (db) 10 log P 10 1 V 2 1 50 = 10 log 10 7 2 = 2.923 db 52 Part (b) Since the same resistor is used in the calculation is used, it does not matter whether it is 50 Ω or 75 Ω. The values cancel and the increase in power is still 2.923 db 16
dbm, dbw, dbμ, etc. If at a certain frequency, a cable has a loss of 5 db per 100 feet, how much power will be delivered to an antenna from a transmitter that puts out 10 W? The length of cable between transmitter and cable is 40 feet. Assume the transmitter, cable, and antenna are impedance-matched. Solution The cable attenuation is = 40 100 5 db = 2 db This is a factor: 10 2 10 = 1.585 Thus, the power delivered is 10 1.585 = 6.31 W Alternative Solution The cable attenuation is = 40 100 5 db = 2 db 10 W is equivalent to 10 dbw Thus, the power delivered is 10 2 = 8 dbw = 10 8 10 = 6.31 W 17
Decoupling, Coupling, Bias Tees, etc. Decoupling and bypass capacitors provide capacitors provide a reliable signal or ac ground right at the point where it is needed in a circuit while blocking dc currents from flowing. The magnitude of the reactance of decoupling and bypass capacitors reactance magnitude, i.e., 1 ωc, must be small at the working frequency and they must low inductance and ESR. A radio frequency choke (RFC) provides a short circuit to dc currents but kills (choke to death ) the ac/signal. RFCs are inductors with large reactance ωl at the working frequency. They should have low capacitance and low ESR. Decoupling Capacitor Coupling Capacitor Coupling Capacitor Equivalent ac circuit. L and C are in parallel and form a resonant circuit Varactor provides voltagevariable capacitance 18
Decoupling, Coupling, Bias Tees, etc. A sub-circuit that frequently appears in RF circuits is a so-called Bias T. Image from Wikipedia One branch is a short to dc but an open to RF. The other branch is a short to both dc. The last branch is a open to dc and a short to RF. Bias Tee In the circuit shown there is a bias tee as highlighted. Often one can design the bias tee into the circuit, or make one on a small board. However, because of parasitics it is often non-trivial to get a good inductor/capacitor are very high frequencies. 19
Side Bar: Bias Tee However, because of parasitics it is often non-trivial to get a good inductor/capacitor at very high frequencies, so some companies market bias tees that will do the job. Self-Made Bias Tee Commercial ($$) Bias Tee. 20
American Wire Gauge (AWG) AWG numbers run from 0000 to 40. In the electronics industry, odd AWG numbers are generally not used. Shown is a partial table relevant to this course. 1 mil = 1/1,000 of an inch. 40.3 mil = 0.043 inch 21
AWG Thin part of IC (such as 555 timer) and an LED indicator lamp is ~0.55 mm, which corresponds to ~22 AWG diameter wire. The wire used on most solderless bread board is 22 AWG, sometimes 24. The leads on ¼ W carbon film resistor commonly used for bread boarding is 22 AWG 22
More on Wire The AWG refers to the conductor and does not include the insulation of the wire. Stranded wire have designations such as 7/32, 10/30, This means 7/32 7 strands, each 32 AWG 10/30 10 strands, each 30 AWG More strands means more flexible Strands 23
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