Davis W.A.Radio frequency circuit design.2001

302 TRANSISTOR AND AMPLIFIER FORMULAS

Junction Field Effect Transistor Parameters (JFET)

ð 1 VDS

VA

VDS < VGS Vt

304 TRANSISTOR AND AMPLIFIER FORMULAS

Small Signal Single-Transistor Amplifier Configurations

306 TRANSFORMED FREQUENCY DOMAIN MEASUREMENTS USING SPICE

First of all, the SPICE S-parameter simulation can be expanded to include unequal input and output impedance levels. This is useful in analyzing impedance steps and transformers with two different resistance levels. Second, rather than using ac steady state voltage sources, a time domain pulse can be used. The automatic network analyzer is made to simulate a time domain reflectometer by mathematically producing an “impulse” that can be replicated in SPICE approximately by using the PULSE function or more accurately by the piecewise linear (PWL) transient function. This SPICE replica of the network analyzer “impulse” can be used to show how various circuit elements react to this “impulse.” Simple formulas are given that can be used to obtain the value of the inductance, capacitance, or impedance step directly from time domain “impulse” data.

G.2 FREQUENCY DOMAIN S-PARAMETERS

In [4] the S parameters were obtained for a given circuit from SPICE for a twoport circuit in which the input and output resistances were both 50 . However, the conversion circuit used to obtain the S parameters can be modified so that the two ports of the circuit are at different impedance levels (Fig. G.1). This is done by the addition of an ideal transformer whose turns ratio is

where R01 and R02 are the characteristic impedance levels of the input and output sides, respectively. The input impedance, Z1, looking into the transformer from the port-1 side is

which is equal to R01 when ZL D R02. The voltage drop caused by the independent current source of value 1/R01 is

The voltage at node 11 in Fig. G.1 is numerically the same as S11, since

For the output side, the secondary current in the transformer is I2 D I1/n. The portion of current going into the transformer primary from the current source side is

308 TRANSFORMED FREQUENCY DOMAIN MEASUREMENTS USING SPICE

The voltage at node 2 is

The voltage across R21 with the marked polarity is V 21 D 2V2/n which, upon replacing V2 with the value above, gives a value for V 21 that is numerically equal to S21:

The right-hand side is obtained using Eqs. (G.1) and (G.2). For a matched load when ZL D R02, S21 D 1 as expected. Finding the S22 and S12 for a circuit is achieved by direct analogy. A suggested SPICE listing for finding the S parameters with unequal source and load impedance levels is shown in Section G.6.

G.3 TIME DOMAIN REFLECTOMETRY ANALYSIS

Time domain reflectometer (TDR) measurements from an automatic network analyzer can be directly compared with a theoretical circuit model in the time domain in SPICE. All that is required is to make the two modifications described in Section G.7. Of course, a near-ideal impulse can be implemented in SPICE, but the point of view taken here is to calculate time domain data that would actually be measured using the time domain feature on a network analyzer. Two actual time domain “impulses” were measured on a network analyzer under the conditions in which the maximum frequency was 18 and 26 GHz, respectively. A third “impulse” with a maximum frequency of 50 GHz was calculated from Hewlett Packard’s Microwave Development System (MDS) program. This was justified on the basis that (1) the program and the network analyzer use the same algorithm for the chirp-Z transform and (2) the measured and calculated 18 GHz “impulses” are nearly identical. Circuit responses to “impulses” with different maximum frequency content will of course differ, so the time domain response must be always coupled to the frequencies used in producing the “impulse.”

Two options for representing the network analyzer “impulse” are illustrated in Fig. G.2. The approximate impulse is modeled in SPICE as a PULSE, while the more accurate approach is achieved by using the piecewise linear (PWL) SPICE function. In the present case, the PWL function uses 77 different x, y coordinate pairs to represent the 26 GHz “impulse.” The result is indistinguishable from the measured “impulse” within the resolution of the graph in Fig. G.2. A similar fit was made for the 18 and 50 GHz “impulses.” The piecewise linear fit of the network analyzer “impulses” for use in SPICE are found in Section G.7. The approximate trapezoidal pulse for a 50 source impedance has the advantage of providing results similar to the PWL approximation with a lot less data entry. However, to closely represent the actual time domain response of the network analyzer, the piecewise linear approach should be used.