By Ananjan Basu

"This ebook begins with a quick assessment, after which offers a quick precis of the fundamental concept and the construction blocks required to appreciate and use microwave size ideas. After a short description of older dimension strategies (slotted wave advisor etc.) the e-book supplies a reasonably designated description of the vector community analyzer and the spectrum analyzer, which shape the spine of contemporary microwaveRead more...

summary: "This e-book starts off with a short evaluation, after which supplies a brief precis of the elemental thought and the development blocks required to appreciate and use microwave size options. After a quick description of older size options (slotted wave advisor etc.) the ebook supplies a pretty certain description of the vector community analyzer and the spectrum analyzer, which shape the spine of recent microwave measurements. complicated themes akin to pulsed measurements and non-linear community research are just pointed out in passing"

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**Example text**

Also, S11 in case I is Zm – Z1 Rm + j X m – (R1 + j X 1 ) = Zm + Z1 Rm + j X m + R1 + j X 1 = (Rm − R1 ) + j(X m – X 1 ) (Rm + R1 ) + j(X m + X 1 ) So, |S11|2 = (Rm − R1 )2 + (X m – X 1 )2 (Rm + R1 )2 + (X m + X 1 )2 Suppose Rm = 10, R1 = 1, Xm = 10, and X1 = –10. Then |S11|2 = though these are realistic values for passive circuits. In case II, 81+ 400 121 > 1, even 2 |S11|2 = Zm – Z1* (R − R1 )2 + (X m + X1 )2 = m Zm + Z1 (Rm + R1 )2 + (X m + X1 )2 This is always <1 for passive circuits, because Rm and R1 are positive for all passive circuits.

7. Let Zm be the impedance seen by the current source. Zm = Zc R + jZc tan βL 10 + j50 tan βL = 50 Zc + jR tan βL 50 + j10 tan βL So, V1 = I0Zm and I1 = I0. S11 can be evaluated from these relations, but for S21 we need V2 and I2 as well. For this, we use the ABCD matrix of the transmission line. 7 Circuit for calculating S-parameters of a transmission line. 15 Background Information So, V2 = cosβL V1 –jZc sinβL I1 and I2 = V1 (j/Zc)sinβL – I1 cosβL. Using the expressions for V1 and I1 above, V2 = I 0 [Zm cos βL – jZc sin βL] I2 = I0 j I 0 Zm + RI 0 I Z – RI 0 , b1 = 0 m 2√R 2√R ∴ a1 = Zm sin βL – cos βL Zc which gives S11 = Zm – R Zm + R b2 = V2 – RI 2 I RZ = 0 Zm cos βL – jZc sin βL – j m sin βL + R cos βL Zc 2√R 2√R Zc2 + RZm sin βL Zc Zm + R ( Zm + R ) cos βL – j ∴ S21 = = cos βL – j Zc2 + RZm sin βL RZc + Zc Zm R + jZc tan βL Zc + jR tan βL sin βL = cos βL – j R + jZc tan βL RZc + Zc2 Zc + jR tan βL Zc2 + RZc = cos βL – j Zc3 + jRZc2 tan βL + R 2 Zc + jRZc2 tan βL sin βL RZc2 + jR 2 Zc tan βL + RZc2 + jZc3 tan βL = cos βL – j Zc2 + R 2 + 2 jRZc tan βL sin βL 2 RZc + j R 2 + Zc2 tan βL ( ( ) ) ( ) 2 RZc cos βL + j R 2 + Zc2 sin βL – j R 2 + Zc2 sin βL + 2 RZc = ( ) 2 RZc + j R 2 + Zc2 tan βL sin 2 βL cos βL 16 An Introduction to Microwave Measurements = = 2 RZc/cos βL 2 RZc + j R 2 + Zc2 tan βL ( ) 2 R Zc 2 cos βL + j sin βL + Zc R Also, S11 can be simplified further: R + jZc tan βL –R Zc + jR tan βL S11 = R + jZc tan βL +R Zc Zc + jR tan βL Zc = Zc R + jZc2 tan βL – Zc R – jR 2 tan βL Zc R + jZc2 tan βL + Zc R + jR 2 tan βL = ( ) j Zc2 – R 2 tan βL ( ) 2 Zc R + j Zc2 + R 2 tan βL Zc R – sin βL R Zc = Z R 2 cos βL + j c + sin βL R Zc j By symmetry, S11 = S22 and S21 = S12.

Ideally, this should be ∞, but its minimum value in practice (specially over the band of operation) should be significantly higher (in dB) than the coupling. Directivity: This is isolation coupling. In practice this should be at least 10 dB, and most vendors offer products with better than 20 dB directivity. Usually, the phases of C and T are not important, except of course in the special case of C = T. This category is called the hybrid coupler or 3 dB coupler. Further details regarding directional couplers can be found in Pozar (1998).