AWR eBooks

3.1 Introduction 123 3 Foundations of RF & Microwave Circuit Characterisation 3.1 Introduction Let us briefly recap our work on transmission lines. Although, in an ideal world, such lines may be seen as large ladders of series inductors and shunt capacitors (Figure 2.3-4), in real life we must include the loss elements shown in Figure 3.1-1 where R and G represent series losses and dielectric losses respectively. Figure 3.1-1 Lumped element equivalent circuit for an incremental length of transmission line The expression for the characteristic impedance of our transmission line therefore becomes √ Equation (3.1-1) clearly shows the characteristic impedance of lossy lines is frequency dependent. This is different from the case of lossless lines for which the characteristic impedance is not dependent on frequency (equation (3.1-2)) √ The other complication that the introduction of loss elements in our model introduces is that, instead of using a purely imaginary exponent ( ) for our complex exponential (equations (2.14-13) and (2.14-14)) we must use , which comprises of a real part to account for losses. Now let us take a look at Figure 3.1-2. Notice how, in this figure, we have chosen the z-axis to point in the opposite direction to the one which we used in section 2.14.2, whilst keeping the origin at the load end. This is done so that we can use positive values for our transmission line lengths ( ) instead of negative ones (- ). We will use this direction for the z-axiz from here on. On a mathematical level, this has the effect of turning every into a – in the equations, as you may see by comparing (3.1-4)and (2.14-19). v(z,t) i(z,t) v(z+ z, t) i(z+ z,t) z z z z z (3.1-1) (3.1-2) (3.1-3) Conquer Radio Frequency 123 www.cadence.com/go/awr

• Cover