Radio Frequency Electronics

Transcription

1 Radio Frequency Electronics Preliminaries II Guglielmo Giovanni Maria Marconi Thought off by many people as the inventor of radio Pioneer in long-distance radio communications Shared Nobel Prize in 1909 Born in Italy Started Marconi Company in Britain and held many patents there. Competed with companies that had transatlantic cables Testified at inquiry on the sinking of the Titanic and was praised for his marvelous invention (radio) that helped save many lives. Image from Wikipedia 1

2 Skin Depth For alternating current, most (63.2%) of the electrical current flows between the surface and the skin depth, δ, which depends on the frequency of the current and the electrical and magnetic properties of the conductor. One can show that δ = 1 πfμσ δ is the skin depth in m f is the ac frequency in Hz σ is the conductivity of the conductor in S/m σ copper S/m μ is the permeability in H/m Distribution of current flow in a cylindrical conductor, shown in cross section. Graphic from Wikipedia μ = μ r μ 0 with μ 0 = 4π 10 7 value for vacuum, and μ r is the relative permeability μ 0 = 4π 10 7 H/m and μ r ~1 for copper 2

3 Skin Effect The skin effect is due to the circulating eddy currents cancelling the current flow in the center of a conductor and reinforcing it in the skin. The ac current in the conductor creates and changing H field which induces voltages, which causes the eddy currents. Since B = μh, the magnetic properties of the conductor greatly affect the skin depth. Magnetic materials such as iron and steel have small skin depths. Images and text from Wikipedia 3

4 Skin Depth Note the very small skin depth of iron even at 60 Hz. This means that at 60 Hz a cable with a diameter more than say 2 mm is a waste of steel, since the current will flow in the 0.65 mm skin of the conductor mm Thus, the power industry uses Aluminum Conductor, Steel Reinforced (ACSR) cable. The aluminum is non-magnetic, and steel provides mechanical strength. 2 mm 4

5 Skin Depth Consider a 22 AWG copper conductor. Calculate the skin depth and the % cross sectional area this represents at 100 khz, 1 MHz, and 100 MHz. At 100 khz δ = 1 πfμσ δ = 1 π π = mm At 1 MHz δ = = mm At 100 MHz δ = = mm % Cross sectional area = πa2 π a δ 2 πa 2 = δ 2a δ a AWG 2a = mm mm Frequency δ % A 100 khz mm 86.6% 1 MHz mm 36.8% 100 MHz 6.5 μm 4% Thus, at 100 MHz, most of the current flows through only 4% of the available cross section. 5

6 Skin Depth From RF Circuit Design: Theory and Applications, Ludwig & Bretchko 6

7 Resistance: ac and dc A conductor with cross sectional area A, length L, and resistivity ρ has resistance: R = ρ L A ρ = V m 1 A m 2 = Ω m Conductor with resistivity ρ Cross-sectional Area A Current The skin effect reduces the effective area through which current flows so that the ac resistance is different and higher than the dc resistance. For a circular wire with diameter 2a, length L at a frequency where the skin depth is δ has dc and ac resistances: R dc = ρ L πa 2 R L ac = ρ δπ 2a δ 7

8 Resistance: ac and dc Consider a 22 AWG copper conductor. Calculate the ratio R ac R dc 100 khz, 1 MHz, and 100 MHz. R dc = ρ L πa 2 L R ac = ρ δπ 2a δ R ac R dc = a 2 δ 2a δ Substituting the values for skin depth for the different frequencies give Since the resistance is proportional to the area, we could use the previously-calculated values for the area. For example at 1 MHz, the skin depth effect reduces the effective cross-sectional area of the conductor to 36.3% of its dc value. Thus, the ac resistance is = 2.72 times larger. Frequency δ R ac R dc 100 khz mm MHz mm MHz 6.5 μm 25 Frequency δ % Area 100 khz mm 86.6% 1 MHz mm 36.3% 100 MHz 6.5 μm 4% 8

9 Inductance The inductance of a straight piece of nonmagnetic wire with length l and diameter d is L = 0.002l 2.3 log 10 4l d 0.75 μh Note that both l and d are in cm. An equivalent formula in conventional units is L = μ 0 l ln 4l 2π d 3 4 H Here l and d are in m and the inductance is in H, and μ 0 = 4π 10 7 H m At high very high frequencies the inductance is smaller, a consequence of the skin effect. The change is small and we will ignore it in this course. 50 mm of 22 AWG wire => 50 nh 9

10 Self-Inductance Example Consider a ¼ -W, 10K metal film resistor with 5 mm leads. The leads are #22 AWG and assume the stray capacitance is 0.3 pf. Calculate the impedance at 200 MHz. 5 mm Solution Each lead has an inductance L = μ 0 l ln 4l 2π d 3 4 = 4π 10 7 ( ) ln 2π = 2.69 nh An equivalent model for the resistor is below. The impedance and 200 MHz is Z s = R + 2L s 2LC s 2 + RCs + 1 Z jω = 2.56K 75 ω=(2π)( ) 10

11 Proximity Effect Thus far, we have considered a straight wire in isolation. When a conductor carrying ac is brought near another conductor the first inductor s changing magnetic field excepts force on the second conductor charge carriers, and induces voltages in the second conductor. Since it is a conductor, currents flow. These are called eddy currents. Eddy currents generate heat and are problematic in transformers. One can minimize eddy currents with transformer laminations Changing B field induces currents in solid core 11

12 Proximity Effect B-field from 1 st conductor than back here More current flows here Eddy Currents 1 st Conductor 2 nd Conductor I The (changing) B-field from the 1 st conductor induces eddy currents in the second conductor, disturbing the current distribution. 12

13 Proximity Effect Proximity effect causes the ac resistance to be greater than if we were considering the skin effect only. The following graph gives a correction factor that one can apply. Graphic from Radio Engineers' Handbook, Frederick Emmons Terman 13

14 Skin and Proximity Effect In coaxial cables at frequencies where the skin effect and proximity effect are significant the current distribution is unusual. Most of the current flows on the outside of the inner conductor and on the inside of the outer conductor, typical for a waveguide. However, some of the current also flows on the outside of the outer conductor. Current Density 14

15 Inductors Various inductors Note that some look like ordinary through-hole resistors Some inductors have an air core. Others use some magnetic material as a core to boost the inductance. Inductors are available as small chip inductors. Some of these very small inductors are wound with copper wire and the very small ones use other techniques. 15

16 Chip Inductor Construction www. delevan. com www. murata. com 0402 = mill = mm 16

17 Inductors The self inductance of an inductor relates to the total number of magnetic flux lines produced and encircled by the inductor: L = Φ Total i Here L is the inductance and I is the current, and Φ Total is total flux encircle. The number of flux lines Φ depends on the normal component of the magnetic flux density B, and the area A and is: The magnetic flux density in an N-turn air-core solenoid with length l and current i is Φ = BA B = μ 0Ni l Consequently: Φ = BA = μ 0Ni l A Φ Total = NΦ = N μ 0Ni l A = μ 0N 2 i l A So that L = Φ Total i = μ 0N 2 A l l In general, the inductance is proportional to the area enclosed, and inversely proportional to the length. For magnetic materials μ 0 μ r μ 0 Area i i 17

18 Inductors It turns out (no pun intended) that is quite difficult find analytical solutions for the inductance of coils, except for a few cases such as the long solenoid. For example, from Wikipedia, here is the formula for a single-layer solenoid in air and then there is the skin and proximity effect, capacitance between the windings, etc. Because of this, various approximations and empirical formulas have been developed. 18

19 Some Air-Core Inductance Formulas Ideal, very long, air-core solenoid. Often derived in EM theory textbooks. l 2R L = μ 0N 2 R 2 l R in mm, l in cm, L in μh Shorter air-core solenoid. The approximations that are used in the derivation of ideal solenoids don t apply. Called long and short by various people. l 0.8 L = 10πr2 μ 0 N 2 9R + 10l R, l in cm, L in μh Very popular formula. You will encounter similarly-looking equations with different constants. The different constant take care of the different units of measure (mm, inch, etc.) Short (l) air-core inductor. K is the socalled Nagoaka factor. That can be found in various references. l < 0.8R L = πμ 0N 2 R 2 l K R, l in cm, L in μh 19

20 Reality-Check Inductance Formulas While calculating the inductance accurately is a worthwhile endeavor, in many practical situations it is less important. For example, assuming one has very accurate formula for the inductance of a solenoid. Then using this formula one calculates the number of turns to realize a particular inductance. Very rarely will the number of turns be an integer. One could make an inductor with, say 22½ turns, but how would one make a inductor with turns? It is common in RF work to wind a solenoid as accurately as possible, but assume one will not get to the exact value. Small deviation from the desired value can be accounted for in various ways. For example, air-core coils can be changed by slightly pressing the coil to adjust the pitch. It is important though, to have a solid understanding of how the inductor length, area, and number of turn affect the value of the inductance. Also, one should have a good understanding of skin effect, proximity effect, inductor resistance, frequency etc. affect the inductance. 20

21 Inductor Assuming the inductance L is known, we use the following equations for circuit analysis. v(t) = L di dt Time domain. Differential equation E = 1 2 Li2 Energy (Joule) X = jωl Frequency/phasor domain (steady state sinusoidal). Reactance (Ω) V = ωli m θ i + 90 Frequency/phasor domain (steady state sinusoidal). V leads by 90 X = sl τ = R L s-domain s = σ + jω Time constant for a single time constant circuit, L reactive element 21

22 Inductors Practical inductors have distributed resistance and inductance. Also, the conductors are subject to skin effect and the proximity effect. An equivalent circuit for an inductor at high frequencies. Note that at some frequency this inductor will resonate with its own capacitance. That frequency is called the self resonance frequency (SFR) At frequencies higher than the SRF, the inductor appears as a capacitor in the circuit. 22

23 Applications of Inductors in RF It is not very common to see inductor used in analog signal processing (filters), except in specialized audio applications. In RF, inductors are widely used in filters. We will see what later. At low frequencies, inductor are used in transformer to step up/down voltages levels. Not quite as common, they are also used for impedance-matching. In RF inductors, along with capacitors are extensively used to transform impedances in order to optimize power transfer. 1K jx c 1K jx C 100 Ω +jx C 100 Ω jx C 100 Ω We want 1K to appear as 100 Ω Add a shunt capacitor that has a reactance jx C Calculating the input impedance shows we have the correct R, but there is a reactance jx C in series No problem, add an inductor to cancel out the capacitor's reactance At the operating frequency the 1K resistor appears as a 100 Ω resistor 23

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