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Fundamentals of RF Circuit Design with Low Noise Oscillators. Jeremy Everard Copyright © 2001 John Wiley & Sons Ltd
ISBNs: 0-471-49793-2 (Hardback); 0-470-84175-3 (Electronic)
2
Two Port Network Parameters
2.1 Introduction
This chapter will describe the important linear parameters which are currently used to characterise two port networks. These parameters enable manipulation and optimisation of RF circuits and lead to a number of figures of merit for devices and circuits. Commonly used figures of merit include hFE, the short circuit low frequency current gain, fT, the transition frequency at which the modulus of the short circuit current gain equals one, GUM (Maximum Unilateral Gain), the gain when the device is matched at the input and the output and the internal feedback has been assumed to be zero. All of these figures of merit give some information of device performance but the true worth of them can only be appreciated through an understanding of the boundary conditions defined by the parameter sets.
The most commonly used parameters are the z, y, h, ABCD and S parameters. These parameters are used to describe linear networks fully and are interchangeable. Conversion between them is often used as an aid to circuit design when, for example, conversion enables easy deconvolution of certain parts of an equivalent circuit. This is because the terminating impedance’s and driving sources vary. Further if components are added in parallel the admittance parameters can be directly added; similarly if they are added in series impedance parameters can be used. Matrix manipulation also enables easy conversion between, for example, common base, common emitter and common collector configurations.
For RF design the most commonly quoted parameters are the y, h and S parameters and within this book familiarity with all three parameters will be required for circuit design. For low frequency devices the h and y parameters are quoted. At higher frequencies the S parameters hFE and fT are usually quoted. It is often easier to obtain equivalent circuit information more directly from the h and y
64 Fundamentals of RF Circuit Design
parameters, however, the later part of the chapter will describe how S parameters can be deconvolved.
All these parameters are based on voltages, currents and travelling waves applied to a network. Each of them can be used to characterise linear networks fully and all show a generic form. This chapter will concentrate on two port networks though all the rules described can be extended to N port devices.
The z, y, h and ABCD parameters cannot be accurately measured at higher frequencies because the required short and open circuit tests are difficult to achieve over a broad range of frequencies. The scattering (S) parameters are currently the easiest parameters to measure at frequencies above a few tens of MHz as they are measured with 50Ω or 75Ω network analysers. The network analyser is the basic measurement tool required for most RF and microwave circuit design and the modern instrument offers rapid measurement and high accuracy through a set of basic calibrations. The principle of operation will be described in the Chapter 3 on amplifier design (measurements section).
Note that all these parameters are linear parameters and are therefore regarded as being independent of signal power level. They can be used for large signal design over small perturbations but care must be taken. This will be illustrated in the Chapter 6 on power amplifier design.
A two port network is shown in Figure 2.1.
Port
+ I1 1 2 I2
V V2
_
ki ko (k11 ) (k22 )
kf (k21 )
kr(k12 )
Figure.2.1 General representation of a two port network.
Two Port Network Parameters 65
The first point to note is the direction of the currents. The direction of the current is into both ports of the networks. There is therefore symmetry about a central line. This is important as inversion of a symmetrical network must not change the answer. For a two port network there are four parameters which are measured:
k = the input (port 1) parameter 11
k = the output (port 2) parameter 22
k = the forward transfer function 21
k = the reverse transfer function 12
As mentioned earlier there is a generic form to all the parameters. This is most easily illustrated by taking the matrix form of the two port network and expressing it in terms of the dependent and independent variables.
Dependent variables
Parameters Independent variables
Φd1 Φd2
ki kr Φi1 (2.1) f o 2
In more normal notation:
Φd1 k11 k12 Φi1 Φd2 k21 k22 Φi2
(2.2)
Therefore:
jd1 =k11jii1 +k12ji2 (2.3)
jd2 =k21jii1 +k22ji2 (2.4)
One or other of the independent variables can be set to zero by placing a S/C on a port for the parameters using voltages as the independent variables, an O/C for the parameters using current as the independent variable and by placing Z0 as a termination when dealing with travelling waves.
66 Fundamentals of RF Circuit Design
Therefore in summary:
CURRENTS SET TO ZERO BY TERMINATING IN AN O/C
VOLTAGES SET TO ZERO BY TERMINATING IN A S/C
REFLECTED WAVES SET TO ZERO BY TERMINATION IN Z0
Now let us examine each of the parameters in turn.
2.2 Impedance Parameters
The current is the independent variable which is set to zero by using O/C terminations. These parameters are therefore called the O/C impedance parameters. These parameters are shown in the following equations:
V1 = z11 z12 I1 (2.5) 2 21 22 2
V1 = Z11I1 + Z12 I2 (2.6)
V2 = Z21I1 + Z22I2 (2.7)
z11 = V1 (I2 = 0) (2.8) 1
z12 = V1 (I1 = 0) (2.9) 2
z21 = V2 (I2 = 0) (2.10) 1
z22 = V2 (I1 = 0) (2.11) 2
Two Port Network Parameters 67
z11 is the input impedance with the output port terminated in an O/C (I2 = 0). This may be measured, for example, by placing a voltage V1 across port 1 and measuring I1.
Similarly z22 is the output impedance with the input terminals open circuited. z21 is the forward transfer impedance with the output terminal open circuited and z12 is the reverse transfer impedance with the input port terminated in an O/C.
Open circuits are not very easy to implement at higher frequencies owing to fringing capacitances and therefore these parameters were only ever measured at low frequencies. When measuring an active device a bias network was required. This should still present an O/C at the signal frequencies but of course should be a short circuit to the bias voltage. This would usually consist of a large inductor with a low series resistance.
A Thévenin equivalent circuit for the z parameters is shown in Figure 2.2. This is an abstract representation for a generic two port.
I1 I2
z11 z22
V V2 z12I2 z21I1
Figure 2.2 Thévenin equivalent circuit for z parameter model
The effect of a non-ideal O/C means that these parameters would produce most accurate results for measurements of fairly low impedances. Thus for example these parameters would be more accurate for the forward biased base emitter junction rather than the reverse biased collector base junction.
The open circuit parameters were used to some extent in the early days of transistor development at signal frequencies up to a few megahertz but with advances in technology they are now very rarely used in specification sheets. They are, however, useful for circuit manipulation and have a historical significance.
Now let us look at the S/C y parameters where the voltages are the independent variables. These are therefore called the S/C admittance parameters and describe the input, output, forward and reverse admittances with the opposite port terminated in a S/C. These parameters are regularly used to describe FETs and dual gate MOSFETs up to 1 GHz and we shall use them in the design of VHF
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