Introduction to RF Design. RF Electronics Spring, 2016 Robert R. Krchnavek Rowan University

Transcription

1 Introduction to RF Design RF Electronics Spring, 2016 Robert R. Krchnavek Rowan University

2 Objectives Understand why RF design is different from lowfrequency design. Develop RF models of passive components. Learn how to measure passive components (lab exercise).

3 Generic RF System

4 Circuit Diagram - 2 GHz Power Amp????????

5 Circuit Diagram - 2 GHz Power Amp

6 Printed Circuit Board - Power Amp

7 Why RF Design? DC Circuit Analysis (and low-frequency) Uses KCL and KVL. Assumes lumped components. As the frequency increases, or more precisely, when the circuit dimensions reach an appreciable percentage of the wavelength, KCL and KVL no longer apply and the wave nature must be considered. Components are no longer lumped but are in reality distributed networks.

8 Electromagnetic Waves Recall the Engineering Electromagnetics, the equations which describe a plane wave (TEM): E = E xˆx = E 0x e jβzˆx E 0x cos(ωt βz)ˆx H = H y ŷ = H 0y e jβz ŷ H 0y cos(ωt βz)ŷ What do these equations mean?

9 Important Relationships Propagation Constant: β = 2π λ = ω µϵ Phase Velocity: v p = ω β = 1 µϵ Wavelength: λ = 2π β = 2πv p ω = v p f

10 Conclusion #1 Because electromagnetic energy travels as waves, the electric (and magnetic) field will vary as a function of position in the material.therefore, the voltage (and current) will vary as a function of position. When a wire s length is a significant fraction of the wavelength of the EM waves in the conducting system, the voltage will not be constant along the wire. This contradicts KVL. In a similar fashion, the current will not be constant along the length of wire. This contradicts KCL. This leads to the concept of a distributed parameter network and will be considered next.

11 Components All components (passive, active, and even interconnects) need to be viewed as distributed parameter networks. We will consider resistors, capacitors, inductors, and the skin effect in conductors. Recall some fundamental principles: Resistance - occurs in any conducting medium (except superconductors) and limits the flow of current. Capacitance - occurs whenever two conductors are separated by a dielectric.

12 Components - continued Inductance - occurs whenever magnetic flux links a conductor. The physical dimensions and material properties of a component determines the equivalent distributed parameter network and we model the component as a network of discrete components.

13 Resistor What does a resistor look like at very high frequencies?

14 Capacitor What does a capacitor look like at very high frequencies?

15 Inductor What does an inductor look like at very high frequencies?

16 Skin Effect As frequency increases, the current density is greatest near the outer edges of the conductor. Only at DC is the current uniformly dense in the conductor. Skin Depth: δ = 1 πfµσcond AC Resistance: R = R DC a 2δ

17 Conclusion #2 All components have resistance, capacitance, and inductance. At low frequencies, the unintended device components are insignificant. At high frequencies, the unintended device components become significant. The unintended components are distributed throughout the device. We model these devices as consisting of a network of discrete devices.

18 Transmission Lines

19 Objectives Understand the distributed parameter model for a transmission line. Be able to DERIVE the general transmission line equations from the distributed parameter model for a transmission line. Be able to determine when transmission line modeling may be necessary in the analysis of an electronic circuit. Know how to calculate the characteristic impedance for the common transmission line structures. Know how to calculate the reflection coefficient, standing wave ratio, and input impedance for a terminated, lossless transmission line. Know how to do simple impedance matching using transmission lines.

20 Transmission Lines Most of what you need to know was covered in EEMAG. From earlier, we know passive components are more complex at RF frequencies than at low frequencies. The complexity is not only restricted to resistors, capacitors, and inductors, but even the interconnects on PWBs. PWBs are networks consisting of R, L, C, and G. Transmission line analysis provides a key method for designing and analyzing RF circuits.

21 Network Model of a Transmission Line R, L, G, and C are distributed parameters. In other words, their units are: Ω/m, H/m, mhos/m, and F/m respectively.

22 When do we have to use the distributed parameter network model? Rule of Thumb: When the average size of a discrete component is more than a tenth of the wavelength, the distributed parameter network model, i.e., transmission line theory, should be used. Use the wavelength in the medium, not the freespace wavelength. If the average size of a component is given by l A, then the frequency at which we should consider using transmission line theory is given by: v p f = 10l A

23 Two-wire Common Transmission Lines Microstrip Coaxial Triplate

24 Common Transmission Lines R, L, G, and C depend on the particular transmission line structure and the material properties. R, L, G, and C can be calculated using fundamental EEMAG techniques. Parameter Two-Wire Line Coaxial Line R L G C µ π 1 πaσcδ ( ) D arc cosh 2a πσ d arc cosh ( ) D 2a πϵ arc cosh ( ) D 2a 1 2πσcδ µ 2π ln ( 1 a + 1 ) b ( ) b a 2πσ d ln ( ) b a 2πϵ ln ( ) b a Parallel-Plate Line 2 wσ c δ µ d w σ d w d ϵ w d Unit Ω/m H/m S/m F/m

25 The Transmission Line Equations Using KVL: V (z) I(z)R z ȷωL zi(z) V (z + z) =0 V (z + z) V (z) z =(R + ȷωL)I(z) dv (z) dz =(R + ȷωL)I(z)

26 The Transmission Line Equations Using KCL: I(z + z) I(z)+V (z + z)(g + ȷωC) z =0 I(z + z) I(z) = V (z + z)(g + ȷωC) z di(z) =(G + ȷωC)V (z) dz

27 Solution V (z) =V + e kz + V e +kz I(z) =I + e kz I e +kz k = k r + ȷk i = (R + ȷωL)(G + ȷωC) k is the complex propagation constant. V + and I + are wavefronts propagating in the +z direction. V - and I - are wavefronts propagation in the -z direction.

28 Characteristic Impedance Consider a semi-infinite transmission line The voltage and current on this line (no reflections) is given by V (z) =V + e kz I(z) =I + e kz The voltage and current on the semi-infinite line are related by the characteristic impedance Z 0 = V (z) I(z) = V + I +

29 Recall Characteristic Impedance dv (z) dz =(R + ȷωL)I(z) V (z) =V + e kz I(z) =I + e kz kv + e kz =(R + ȷωL)I + e kz Z 0 = V + I + = R + ȷωL R + ȷωL = k G + ȷωC So, Z 0 is a function of the line parameters R, L, G, and C and the frequency. In a similar fashion, we can show Z 0 = V I

30 Terminated, Lossless Transmission Line The voltage on this line is given by V (z) =V + e kz + V e +kz ( V (z) =V + e kz + V Define the voltage reflection coefficient as Γ 0 = V V + V + e+kz )

31 Terminated, Lossless Transmission Line Then, V (z) =V + ( e kz +Γ 0 e +kz) Similarly, I(z) = V + Z 0 ( e kz Γ 0 e +kz) The impedance anywhere along the line is given by Z(z) = V (z) I(z) = Z 0 The impedance at the load end, Z L, is given by Z(0) = Z L = Z 0 1+Γ 0 1 Γ 0 e kz +Γ 0 e +kz e kz Γ 0 e +kz

32 Terminated, Lossless Transmission Line Then, Γ 0 = Z L Z 0 Z L + Z 0 CONCLUSION: The reflection coefficient is a function of the load impedance and the characteristic impedance. Recall k = k r + ȷk i = (R + ȷωL)(G + ȷωC) This is often written as k = α + ȷβ For the lossless case, α =0, and β = ω LC = k i Then, V (z) =V + ( e ȷβz +Γ 0 e +ȷβz) I(z) = V + Z 0 ( e ȷβz Γ 0 e +ȷβz)

33 Terminated, Lossless Transmission Line It is customary to change to a new coordinate system, d, at this point. Rewriting the expressions for voltage and current, we have V (d) =V + ( e +ȷβd +Γ 0 e ȷβd) Rearranging, I(d) = V + ( e +ȷβd Γ 0 e ȷβd) Z 0 V (d) =V + e +ȷβd ( 1+Γ 0 e 2ȷβd) I(d) = V + e +ȷβd ( 1 Γ 0 e 2ȷβd) Z 0

34 Impedance The impedance anywhere along the line is given by Z(d) = V (d) I(d) = Z 0 The reflection coefficient can be modified as follows Γ(d) =Γ 0 e 2ȷβd Then, the impedance can be written as 1+Γ 0 e 2ȷβd 1 Γ 0 e 2ȷβd Z(d) =Z 0 1+Γ(d) 1 Γ(d) After some algebra, an alternative expression for the impedance is given by Z(d) =Z 0 Z L + ȷZ 0 tan βd Z 0 + ȷZ L tan βd CONCLUSION: The load impedance is transformed as we move away from the load.

35 Miscellaneous - But Important! The (Voltage) Standing Wave Ratio - SWR (or VSWR) is defined as SWR = V max V min = I max I min SWR = 1+ Γ 0 1 Γ 0

36 Closing Comments RF circuit design requires impedance transformations/matching to maximize the transfer of power. Components (passive, active, PWB interconnects) do not have the idealized impedances seen at low frequency. Techniques are needed that determine the impedance of a component and then how to transform its impedance as necessary.