Radio Frequency Electronics

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Radio Frequency Electronics Preliminaries III Lee de Forest Born in Council Bluffs,

electronic amplifiers Vacuum tube started electronics age Patented Phonofilm, an

Transmitted first radio report on Presidential election Supposedly said: I came, I

1 Radio Frequency Electronics Preliminaries III Lee de Forest Born in Council Bluffs, Iowa in 1873 Had 180 patents Invented the vacuum tube that allows for building electronic amplifiers Vacuum tube started electronics age Patented Phonofilm, an improved method of adding a soundtrack to movies Transmitted first radio add Transmitted first radio report on Presidential election Supposedly said: I came, I saw, I invented it's that simple no need to sit and think it's all in your imagination. Image from Wikipedia 1

Litz Wire Due to the skin and proximity effects, current in ac circuits flows near the surface of the conductor.

To mitigate these effects, one can increase the surface area by bundling many small-diameter strands that are insulated from each

Litz wire find application in high frequency inductors and transformers, inverters, communication equipment, ultrasonic equipment,

2 Litz Wire Due to the skin and proximity effects, current in ac circuits flows near the surface of the conductor. This increases the ac resistance of the conductor. In transformers and inductors the proximity effect ma dominate. To mitigate these effects, one can increase the surface area by bundling many small-diameter strands that are insulated from each other. Litz wire find application in high frequency inductors and transformers, inverters, communication equipment, ultrasonic equipment, sonar equipment, television and radio equipment and induction heating equipment. Litz is derived from the German word Litzendraht meaning woven wire. The weaving patterns are designed to ensure the current in is all strands are equal. Images from Wikipedia Eight strands of insolated wire bundled together.

3 Litz Wire In RF work, Litz wire is used up to 1-3 MHz or so. It is used for RFC, inductors, and ferrite antennas. 0.5 H Inductor from Digikey Catalog. SRF = 5 khz. Portable radios use litz wire for LW and MW antennas RFC from a vintage radio. The wire is almost certainly litz. 3

The main and coupled winding acts as a transformer an can improve impedance matching with the

4 Litz Wire Applications Ferrite rod antenna with two windings. The main winding resonates will a tunic capacitor. Litz wire is used to reduce coils ac resistance. The main and coupled winding acts as a transformer an can improve impedance matching with the first RF stage. Ferrite rod antenna with litz wire windings for LW (< 300 khz) and MF (535 khz to 1705 khz) reception Image from Wikipedia 4

Inductor Quality Factor X = ωl R An ideal inductor has no resistance (or

A metric for how good an actual inductor is, is the so-called quality factor

using it extensively through the course.

cycle Assuming i L t I m sin ωt, the peak energy stored is E peak = 1 LI m

5 Inductor Quality Factor X = ωl R An ideal inductor has no resistance (or capacitance), only inductance. A metric for how good an actual inductor is, is the so-called quality factor Q. The quality factor is also used for capacitors and filters, and we will be using it extensively through the course. It is defined as peak enery stored per cycle Q = π energy dissipated per cycle Assuming i L t I m sin ωt, the peak energy stored is E peak = 1 LI m Energy dissipated/cycle is: E diss = P ave T = I rms R π ω = R I m π ω = R I m π ω peak enery stored per cycle Q = π energy dissipated per cycle = π 1 LI m 1 R I m ω π = ωl R 5

6 Inductor Quality Factor X = ωl R peak enery stored per cycle Q = π energy dissipated per cycle = ωl R Note that Q = ωl R = X s R s = Reactance Series Resistance This is also true for a capacitor with a series resistance: X = 1 ωc R s Q = Reactance Series Resistance Q = 1 ωc R s Q = 1 ωr s C Thus, we can write Q s = X s R s where s indicates a series connection One can show for the case where a resistor is in parallel with a reactance then Q p = R p X p Q p = R p ωl (inductor) Q p = R p 1 ωc = ωr PC (capacitor) 6

From this one can determine R. For inductors R is assumes to be R s unless otherwise noted. Often, R dc (i.e., the resistance at dc) is also specified.

7 Inductors Specifications Data sheets often assume the following model Simplified model at a specific frequency. Q L (or Q s ) is the component que Q s = ωl R s s series For inductors that are used in signal processing (i.e., filters), the indcutor iductance L and a number, the inductor Q, is normally specified along with the inductance. From this one can determine R. For inductors R is assumes to be R s unless otherwise noted. Often, R dc (i.e., the resistance at dc) is also specified. This is typically lower than the R s obtained from Q s Why? 7

8 Inductor Specifications Note that the manufacturer did not specify C or SRF 10 mh adjustable inductor Q s = ωl R s at the specified frequency 8

LR Series LR Parallel Transformation It is often convenient to work with a parallel RL network rather than a series RL network and there exists transformations between the two representations.

9 LR Series LR Parallel Transformation It is often convenient to work with a parallel RL network rather than a series RL network and there exists transformations between the two representations. Q s = ωl s R s L s = L p Q p 1 + Q p Q p = R p ωl p L p = L s 1 + Q s Q s Also since Q = R s = R p 1 + Q p π (peak enery stored) energy dissipated per cycle R p = R s 1 + Q s it follows from conservation of energy that Q p = Q s If the component Q is large then Q + 1 Q, so one can simplify as follows: R p Q s R s 9

10 LR Series LR Parallel Transformation Where do such transformations come from? Below we derive the inverse of the series parallel Parallel network Z p = jωl pr p = jωl pr p R p jωl p R p + jωl p R p + jωl p R p jωl p = ω L p R p R p + ω L + j ωl p R p p R p + ω L p Series network Z s = R s + jωl s For Z p = Z s the real parts must be equal and the imaginary parts must be equal Equate real parts R s = ω L p R p R p + ω L p = R p R p ω L p + 1 = R p 1 + Q p where Q p = R p ωl p Equate imaginary parts jωl s = j ωl p R p R p + ω L p L s = L p 1 + ω L p R p = L p Q p 1 + Q p 10

11 Example Example. Compute the Q of a 50 nh inductor with series resistance R s = 10 Ω at 100 MHz. Transform the circuit to an equivalent parallel RL network. Solution. The Q of the inductor is Q s = ωl = π R s 10 = 3.14 Since Q s is low, we will not use the approximation R p Q s R s. Rather R p = 1 + Q s R s = = 1,08.7 Ω L p = L s 1 + Q s Q s = = 55.1 nh 11

12 Capacitors C = ε 0 ε r A d ε 0 = permittivity of free space = F/m ε r = relative permittivity Q = CV Graphic from Wikipedia 1

Capacitor Assuming the capacitance C is known, we use the following

Differential equation E = 1 Cv X = 1 jωc Energy (Joule) Frequency/phasor

Reactance (Ω) V = I m ωc θ i 90 Frequency/phasor V lags by 90 X = 1 sc τ

13 Capacitor Assuming the capacitance C is known, we use the following equations for circuit analysis. i(t) = C dv dt Time domain. Differential equation E = 1 Cv X = 1 jωc Energy (Joule) Frequency/phasor domain (steady state sinusoidal). Reactance (Ω) V = I m ωc θ i 90 Frequency/phasor domain (steady state sinusoidal). V lags by 90 X = 1 sc τ = 1 RC s-domain s = σ + jω Time constant for a single time constant circuit, C reactive element 13

14 Capacitors Graphic from Wikipedia 14

resistance at a specific frequency, but in practice

15 Capacitor Models The Equivalent Series Resistance (ESR) is, strictly-speaking the equivalent series resistance at a specific frequency, but in practice people use the term as if it is frequency-independent 15

16 Capacitor Frequency Response Above ~ 0 MHz, this capacitor behaves as an inductor Self Resonance Frequency (SRF) 16

17 Decoupling Capacitors What is going on here and here? Question: Why place a 100 nf capacitor in parallel with a 100 μf capacitor? Answer: The 100 μf will begin to behave like an inductor as some frequency. At this frequency the 100 nf takes over and ensure low reactance. 17

Build a Good Capacitor One should carefully

capacitors in parallel so that the composite

18 Build a Good Capacitor One should carefully consider package sizes, since that determines inductance It is very common in RF and high-speed digital work to place a number of capacitors in parallel so that the composite capacitor has a low reactance where desired. Graphic from Planet Analog 18

19 De-queing 19

Ceramic Capacitor Construction Dipped ceramic capacitors are cut from

Electrodes are screen printed using silver, finely powdered glass, and

This evaporates the binder, and the melted glass binds the silver to

Next, hairpin wires are clipped onto the capacitor and it is dipped in

20 Ceramic Capacitor Construction Dipped ceramic capacitors are cut from large sheets of green ceramic material and fired. Electrodes are screen printed using silver, finely powdered glass, and a binder on both sides of the disk, then back to the oven. This evaporates the binder, and the melted glass binds the silver to the ceramic surface. Next, hairpin wires are clipped onto the capacitor and it is dipped in solder. These capacitors are inexpensive and available in values less that ~ 1 µf. These capacitors are inexpensive and available in values less that ~ 1 µf. 0

Monolithic/Multilayer Ceramic Capacitors A Surface Mount (SMT) MLC capacitor.

Electrodes are painted onto one side of the sheets using a paint that consists of a liquid binder with fine metal

The painted electrodes are arranged so that alternate electrodes exit from opposite ends.

21 Monolithic/Multilayer Ceramic Capacitors A Surface Mount (SMT) MLC capacitor. The base ceramic material is mixed with a binder and fashioned into sheets. Electrodes are painted onto one side of the sheets using a paint that consists of a liquid binder with fine metal particles in suspension. The sheets are stacked on top of each other. The painted electrodes are arranged so that alternate electrodes exit from opposite ends. The laminated layers are then compressed and fired, which sinters them into one monolithic structure. The ends are terminated, often using silver. For leaded capacitors, wires are attached, and finally the capacitor is encapsulated in plastic and marked 1

22 Capacitor Quality Factor Recall the definition of the component Q peak enery stored Q = π energy dissipated per cycle Q s = X s R s s series Q p = R p X p p parallel X = 1 ωc Q p = R p X p Q p = R p 1 ωc = ωr PC (capacitor) R p

RC Parallel Series RC Transformations Parallel network Z p = 1 jωc p R p 1 jωc p + R p = R p 1 + jωr P C p = R P 1 jωr P C P 1 + jωr P C P 1 jωrc p Series network = R p C p 1 + ωr p C j ωr p 1 + ωr p

23 RC Parallel Series RC Transformations Parallel network Z p = 1 jωc p R p 1 jωc p + R p = R p 1 + jωr P C p = R P 1 jωr P C P 1 + jωr P C P 1 jωrc p Series network = R p C p 1 + ωr p C j ωr p 1 + ωr p P C P Z s = R s + 1 jωc s = R s j ωc s For Z p = Z s the real parts must be equal and the imaginary parts must be equal R s = R p 1 + ωr P C p 1 ωc s = ωr p C p 1 + ωr P C P C s = C p 1 + ωr P C p 1 + ωr p C p ωc p ωr = C p p C p ωr p C p With Q p = ωr p C p the Q of the parallel network, the expressions become Q p = ωr p C p R s = R p 1 + Q C 1 + Q P s = C p p Q p 3

Capacitor Terminology Power Factor (PF) and Dissipation Factor (DF) Real capacitors have both resistive and reactive components (see the model for the capacitor) Z( j ) R( ) jx ( ) PF power loss

24 Capacitor Terminology Power Factor (PF) and Dissipation Factor (DF) Real capacitors have both resistive and reactive components (see the model for the capacitor) Z( j ) R( ) jx ( ) PF power loss apparent power I I R Z R Z cos DF power loss reactive power stored I I R X R X cot tan DF is normally expressed as a percentage. For example, if the DF of a capacitor is 13%, then: tan

25 5

ELECTROMAGNETIC OSCILLATIONS AND ALTERNATING CURRENT

Chapter 31: ELECTROMAGNETIC OSCILLATIONS AND ALTERNATING CURRENT 1 A charged capacitor and an inductor are connected in series At time t = 0 the current is zero, but the capacitor is charged If T is the