# 2 Port Parameters I 1. ECE145A/218A Notes Set #4 1. Two-ways of describing device: A. Equivalent - Circuit-Model

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1 ECE45A/8A Notes et #4 Port Parameters Two-ways of describing device: A. Equivalent - Circuit-Model Physically based Includes bias dependence Includes frequency dependence Includes size dependence - scalability Ideal for IC design Weakness: Model necessarily simplified; some errors. Thus, weak for highly resonant designs B. Port Model Matrix of tabular data vs. frequency Need one matrix for each bias point and device size Clumsy huge data sets required Traditional microwave method Exact Port descriptions These are black box (mathematical) descriptions. I I + port port + Inside might be a transistor, a FET, a transmission line, or just about anything. The terminal characteristics are I & I there are degrees of freedom. Rev./7

2 ECE45A/8A Notes et #4 Admittance Parameters I I Y Y Y Y Example: imple FET Model C gd g m gs C gs + gs Rds By inspection: Y jωc gs + jωc gd jωc gd g m jωc gd G ds + jωc gd Easy! I Y Y I Rev./7

3 ECE45A/8A Notes et #4 3 Impedance Parameters I I Example R R R 3 By inspection R + R 3 R 3 R 3 R + R 3 I I I I But, y, z, and h parameters are not suitable for high frequency measurement. Problem: How can you get a true open or short at the circuit terminals? Any real short is inductive. Any real open is capacitive. To make matters worse, if you are trying to measure a high freq. active device, a short or open can make it oscillate! olution: Use termination in instead! Broadband. Not very sensitive to parasitic L,C Kills reflections. Redefine parameters to use fwd. and rev. voltage waves. Measurement can use directional couplers. Rev./7

4 ECE45A/8A Notes et #4 4 Parameters o a a b b z z o input reflection coeff a rev. transm. gain a b b a a fwd transm. gain output a Γ a Note that must be defined. We don t really need transmission lines. Our objective now is to de-mystify -parameters they are easy! Recall (x) + (x) + (x) I(x) + (x) (x) phasor quantities. amplitude, not rms values. We can normalize the amplitude of waves to : a(x) + (x) b(x) (x) forward wave reverse wave Why? o that a(x)a* (x) power in forward wave. if a.44 then power in wave is watt. (or a rms ) Rev./7

5 ECE45A/8A Notes et #4 5 likewise, b(x)b * (x)/ is the power in the reverse wave o, in terms of total voltage (x) and current I(x), or, a( x) b( x) () x ( ) vx ax ()+ bx () ix () Ix () ax () bx () [ v( x) + i( x) ] [ ( x) + I( x) ] [ v( x) i( x) ] [ ( x) I( x) ] Reflection o, how is Γ defined in terms of the parameters? At port, But, Γ b a b a+ a We need to eliminate a. How? If L o, a Γ L so, therefore a if port is terminated in o. b ame with at port with : b Γ a a b Γ a a Rev./7

6 ECE45A/8A Notes et #4 6 Transmission b a+ a o, the forward transmission can be found by setting a (terminate output) b a a Reverse transmission, similarly, is found by setting a (terminate input in o) b a+ a b a a Rev./7

7 ECE45A/8A Notes et #4 7 ome comments on power measurement: Power can vary over a large range, therefore it is often specified on a logarithmic scale. There must be a point of reference on the scale; the power measurements are usually with reference to mw. The unit is called dbm meaning db relative to mw of power. Thus, dbm mw dbm mw - dbm. mw etc. To convert mw to dbm: dbm log (P) To convert dbm to mw: P dbm/ What is the difference between db and dbm? db is a power ratio used to describe a gain or loss for example. G log (P out /P in ) db Return Loss - log Γ db But, db says nothing about the absolute power level. Don t confuse their usage! Rev./7

8 ECE45A/8A Notes et #4 8 Now, define available power: P A max power output from a source with impedance s that can be absorbed into a load. let, L * (in this case) because maximum power transfer occurs when we have a conjugate match gen gen / generator load Or, in terms of a and b: P load P A 8 gen a + + gen ~ b b + + gen a and b ; gen and + o, gen * Pload PA aa 8 Rev./7

9 ECE45A/8A Notes et #4 9 We see that the available power is independent of load impedance. Even if the load is not matched, available power remains constant. Actual power in the load is reduced however. Generator output power is calibrated and displayed as available power. Actual Load Power P Re or * Load a b I PLoad PA ( ) Reflected Power b a imilarly, PR b a P Power reflected from input b Power incident on input a a A Power reflected from network output Power incident on output Power incident on output Reflected power from load b a b b Power reflected from input port Power incident on load from the network Rev./7

10 ECE45A/8A Notes et #4 a b a b Also, by definition, transducer gain P load P avs G T even if. load isn t matched to network and. input of network not matched to generator Here, P Load L b ( Γ ) is defined in terms of transducer gain for the special case of where L : b a a b power incident on load (and is absorbed since Γ L) a source available power imilarly, transducer gain with source and load reverse transducer power gain Rev./7

11 ECE45A/8A Notes et #4 Reference Planes E B C Microwave transistor in package E On board: B C [] connection to instruments here Define defining x at z both ports here. Defining the reference planes differently changes the -parameters. Rev./7

12 ECE45A/8A Notes et #4 phase shifts! 5Ω microstrip transmission lines b a e j θ e j ( θ +θ ) e j ( θ +θ ) e j θ a x θ π x x x λ b connections to instruments here π θ βx λ π θ βx λ θ π λ ' θ e j( θ+ θ ) e j( θ+ θ ) e jθ e The reflection parameters are shifted in phase by twice the electrical length because the incident wave travels twice over this length upon reflection. The transmission parameters have the sum of the electrical lengths, since the transmitted wave must pass through both lengths. Rev./7

13 ECE45A/8A Notes et #4 3 Comment on electrical length: The microwave literature will say a line is 43 long at 5 GHz. What does this mean? Electrical length E 36 λ ref Recall f λ v so f ref λ ref υ E v / f ref 36 v f ref 36 f ref E T f ref 36 a line which is ns long has an electrical length E 36 at f ref GHz and an electrical length E 36 at F ref MHz Why not just say Τ ns? you should be conversant with both terminologies. Converting to physical length f λ λ ref ref v v f p p thus: physical length E(deg) λ ref 36 Electrical length (in wavelengths) λ ref or: Rev./7

14 ECE45A/8A Notes et #4 4 How to Calculate -Parameters Quickly First Comment b a a b + a a (We must kill a in order to measure or calculate ) Γ L L b a if L, then Γ L is zero and so a Γ L b. o b a L o if we say that in L is the input impedance with L then in L in L + Γ in or: in L + The same comment clearly applies for. The mith Chart is often used to plot,. Rev./7

15 ECE45A/8A Notes et #4 5 Example: 4Ω Given 5Ω, what is? 4Ω 5Ω imilar arguments give 4 4. in L 54Ω Find b a a Γ a a ~ gen b b L Rev./7

16 ECE45A/8A Notes et #4 6 What is a in this case? We know that: a + and + o gen o, a gen o Consider the load: b out Why? b a + L out _ a Γ L b But, Γ L because L, so a. out + + a + b b Now, calculate out / gen : ( a ) out + b a But, a because the load impedance, so ubstitute for a : out a so, gen a out gen Rev./7

17 ECE45A/8A Notes et #4 7 thus, out when L gen Why the factor of? gen gen / generator load We see that the generator voltage is split between the source and load in the matched case. Here, we see that out/gen ½, but the transducer gain must be equal to. (P LOAD /P A ). is the transducer gain in this situation. If we insert an amplifier into the network, the signal has been increased by an amount. gen out gen / generator load Rev./7

18 ECE45A/8A Notes et #4 8 o, is the FORWARD INERTION GAIN or FORWARD TRANDUCER GAIN in a system of impedance. EXAMPLE: Find 5 4 gen 5 out / gen out/gen 5/ OR, we could let gen. Then, out. What about a reference plane extension? X - l X X X - l 5 4 gen 5 out / gen Θ i π l i /λ e jθ Γ IN () Γ OUT () e jθ and π π θ β θ β λ λ ' j( θ + θ ) π j( + )/ λ e e Rev./7

19 ECE45A/8A Notes et #4 9 EXAMPLE: Find the 4 parameters of the following circuit: gen C : Find in (with L ), then calculate input reflection coefficient. IN L / ( sc + / ) IN IN + IN IN + turning the crank, jωc / + jωc / will be the same due to symmetry. Note that we calculated IN with port terminated in. This is part of the definition of so is essential. Rev./7

20 ECE45A/8A Notes et #4 Now find : first use Thevenin Norton transformation: out gen / C out gen + sc I/Y out gen + jωc / Rev./7

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