# HP ADS 1.5 User-defined Models User Manual

Brand: HP, Pages: 216, PDF Size: 1.33 MB

### Page 51 from 216

Deriving S-Parameter Equations 2-1

Chapter 2: Creating Linear Circuit Elements

This chapter describes creating linear elements through the use of examples. A linear

element differs from a nonlinear element in that a linear element contains only linear

elements while a nonlinear element can contain both linear and nonlinear elements.

Deriving S-Parameter Equations

One way to characterize a circuit element is by its S-parameters. To help you derive

the S-parameters, refer to the following book: Microwave Transistor Amplifiers by

Guillermo Gonzalez (Englewood Cliffs: Prentice-Hall, Inc., 1984).

Begin the process of deriving the S-parameters by examining the circuit

configurations shown in Figure 2-1. Although this example shows 2-port

S-parameters, the technique is the same for elements with a greater number of ports.

Figure 2-1. 2-port network

Alternative, but equivalent, expressions for S

11 and S22 are

S

11 = (Z1 - ZO)/(Z1 + ZO)

S

22 = (Z2 - ZO)/(Z2 + ZO)

where

ZO is the normalizing impedance for the circuit (usually 50 ohms)

Z1 is the impedance looking into port 1 when port 2 is terminated with ZO

Z2 is the impedance looking into port 2 when port 1 is terminated with ZO

S11 = V1 - 1

S

21 = V2 S22 = V2´ - 1

S

12 = V1´

V1´V2´

### Page 52 from 216

2-2 Deriving S-Parameter Equations

Creating Linear Circuit Elements

For example, consider a grounded pi-section resistive attenuator as shown in

Figure 2-2. Inserting Figure 2-2 into the two-port network in Figure 2-1 results in

Figure 2-3.

Figure 2-2. Schematic for pi-section resistive attenuator

Figure 2-3. Resulting circuit schematic

### Page 53 from 216

Deriving S-Parameter Equations 2-3

Using Figure 2-3, the following relations are defined:

YA1 = 1.0/R3 + 1.0/ZO

ZA1 = 1.0/YA1

ZB1 = R2 + ZA1

Because the network is symmetrical, the following relations also hold:

YA2 = 1.0/R1 + 1.0/ZO

ZA2 = 1.0/YA2

ZB2 = R2 + ZA2

From the definition of Z1 and Z2:

Z1 = (R1· ZB1)/(R1 + ZB1)

Z2 = (R3 · ZB2)/(R3 + ZB2)

The S-parameters are obtained from the following equations:

S

11 = (Z1 - ZO)/(Z1 + ZO)

S

22 = (Z2 - ZO)/(Z2 + ZO)

S

12 = S21 = (2.0 / ZO)/(1.0 / ZA1+1.0 / ZA2 + R2 / (ZA1· ZA2))

These basic equations are sufficient to write the C function for the element.

### Page 54 from 216

2-4 Deriving Y-Parameter Equations

Creating Linear Circuit Elements

Deriving Y-Parameter Equations

Y-parameters equations can be used to describe a user-defined element as an

alternative to S-parameter equations. Figure 2-4 shows Y-parameters for a resistor

connected between two ports; Y-parameter definitions follow the figure.

Figure 2-4. Y-parameters for a 2-port resistor connection

In general,

With V

2 equal to zero, V1 = I1R, which is also equal to -I2R. Y11 reduces to 1/R and

Y

21 to -1/R. Setting V1 to zero, V2 = I2R =-I1R. The expressions for Y22 and Y12 are

1/R and -1/R, respectively. The resultant Y-parameter matrix is:

Y11

I1

V1

-------

2V0 =

=Y21

I2

V1

-------

2V0 =

=

Y12

I1

V2

-------

1V0 =

=Y22

I2

V2

-------

1V0 =

=

Y []1R ¤1R ¤ –

1R ¤ –1R ¤ =

### Page 55 from 216

Deriving Y-Parameter Equations 2-5

The following code is a portion of the example file:

/***************************************************************************

/

# define EPS 1.0e-8

/*

* This example shows direct Y-parameter loading, instead of S-parameters.

* For some elements, admittance parameters are easier to derive than

* scattering parameters. For a series resistor, the admittance matrix is

* as follows:

* | g -g|

* |Y| = |-g g| where g = 1 / R

*

* ELEMENT U2PD Id n1 n2 R=#

*/

static boolean u2pd_y(

UserInstDef *userInst,

double omega,

COMPLEX *yPar)

{

double res, cond;

res = userInst->pData->value.dVal * get_runit(userInst->eeElemInst);

if (res < EPS)

res = EPS;

cond = 1.0 / res;

yPar[0].real = yPar[3].real = cond;

yPar[1].real = yPar[2].real = -cond;

yPar[0].imag = yPar[1].imag = yPar[2].imag = yPar[3].imag = 0.0;

return TRUE;

}

#undef EPS

/***************************************************************************

/