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S (scattering) Parameters
Scattering, or S, parameters are another extremely useful design aid that most manufacturers provide for their higher frequency transistors. While Y-parameters utilize input and output voltages and currents to characterize the operation of a two-port network, S parameters use normalized incident and reflected traveling waves in each network port. There is no need to present a short circuit to the two-port device. This termination often causes an active device, such as a transistor, to become unstable, thus making measurements impossible. Instead, the network is always terminated in the characteristic impedance of the measuring system and this is, in most cases, purely resistive 50 Ohms. In addition, the 50-Ohm source and load seen by the two-port network forces the device under test, if active, to be stable and not oscillate.
I. Transmission line theory.
To understand the concept of S-parameters, it is necessary to first have a working knowledge of transmission line theory. Voltage, current , or power emanating from a source impedance Zs and delivered to a load Zl can be considered to be in the form of incident and reflected waves traveling in opposite directions along a transmission line of characteristic impedance Zo.
If Zl = Zo then the incident wave is totally absorbed in the load. If Zl is not equal to Zo then some of the incident wave is not absorbed in the load but is reflected back toward the source. If Zs = Zo then the reflected wave would be absorbed by the source entirely. For Zs not eqaul to Zo the reflected from the load wave is re-reflected from the source back toward the load and the entire process repeats itself. The degree of mismatch between Zo and Zl, or Zs, detemines the amount of incident wave that is reflected. The ratio of the reflected wave to the incident wave is known as the reflection coefficient. The reflection coefficient, ΓΓΓΓ is a complex number often expressed as a magnitude and an angle in a polar form.
ΓΓΓΓ (^) = V (^) reflected/V (^) incident
ΓΓΓΓ = 0 if a line is terminated with a resistance equal to Zo. A perfect match exists and therefore, there is no reflected wave. ΓΓΓΓ (^) = 1 if the load impedance is an open , short circuit or purely reactive. All of the incident wave will be reflected back toward the source and therefore, a perfect mismatch exists. So, the normal range of Γ is between 0 and 1. For Γ > 1 the magnitude of the reflected wave from a load must be greater than the magnitude of the incident wave to that load. The load must be a source of power in this case. This concept is useful in the design of oscillators. However, it is highly undesired in the design of input networks of amplifiers. Reflection coefficients can be plotted directly on the Smith Chart and the corresponding load impedance read off of the chart without a need of any calculations. As well, given a specific characteristic impedance and a load impedance, a reflection coefficient can be read directly from the chart.
II. S-parameters and the two-port network
The following is true for any travelling wave that originates at the source:
- A portion of the wave from a source and incident wave upon the two port device (a1) will be reflected (b1) and another portion will be transmitted through the two-port device.
- A fraction of the transmitted signal is then reflected from the load and becomes incident upon the output of the two-port device (a2).
- A portion of a signal (a2) is then reflected from the output port back toward the load (b2), while a fraction is transmitted through the two-port device back to the source.
Concluding the above, any travelling wave is made up of two components. For example, the output of the two-port device to the load consists of the portion that is reflected from the output of the two-port device (a2) and the portion that is transmitted through the two-port device (a1). Similarly, the total wave flowing from the input of the two-port device toward the source consists of the portion that is reflected from the input port (a1) and the fraction of (a2) that is transmitted through the two-port device.
These observations can be illustrated in the equation form as:
b1 = S 11 a1 + S (^) 12a2 and b2 = S 21 a1 + S (^) 22a
where,
S 11 -^ the input reflection coefficient S 12 - the reverse transmission coefficient (reverse gain or loss) S 21 - the forward transmission coefficient (forward gain) S 22 - the output reflection coefficient
S 11 = b1/a1 for a2= 0 or Zl=Zo. This is an input reflection coefficient. S 11 is equal to the ratio of a reflected wave and an incident wave with Zl=Zo. Thus, S 11 can be plotted on a Smith chart and the input impedance of the two-port device can be found immediately. Similarly, S 22 = b2/a2 for a1 = 0 or Zs = Zo. This is an output reflection coefficient that can be plotted on a Smith chart and the output impedance of the two-port device can be found immediately. The other two S parameters are found as follows: S 21 = b2/a1 for a2 = 0 or Zl=Zo S 12 = b1/a2 for a1 = 0 or Zs = Zo Notice that in order to measure the individual S parameters, a1 and a2 must be set to zero. This is easily done by terminating a network (source and load) or forcing Zs and Zl to be equal to the characteristic impedance of the measuring system, thus eliminating all reflections from the termination. S-parameters are expressed in Re/Im, Mag/Phase or dBMag/Phase. Note that dBMag is calculated as 20log 10 √√√√ (R 2 +X 2 ) for all s-parameters.
III. SMALL SIGNAL AMPLIFIER DESIGN:
Transistors can be completely characterized by their S parameters that vary with frequency and bias level. With these parameters, it is possible to calculate potential instabilities (tendency to oscillation), maximum available gain, input and output impedances, and transducer gain. It is also possible to calculate optimum source and load impedances either for simultaneous conjugate matching or simply to help choose specific source and load impedances for a specified transducer gain.
STABILITY:
To calculate the stability of a transistor with S parameters, the intermediate quantity, Ds must be obtained first:
(Delta) Ds = S (^) 11S 22 – S 12 S 21
Then, Rollet Stability Factor, K is derived as follows:
Gain Circles
Unilateral Input/Output Gain Circle
Unilateral Figure of Merit, U.
All two-port models are bilateral, so both the forward and reverse signal flow must be considered. If the signal flow in the reverse direction is much smaller than the flow in the forward direction then it’s possible to make the simplification that the reverese flow is zero. The Unilateral Figure of Merit, U is a quick calculation that can be used to determine where this simplification can be made without significantly affecting the accuracy of the model.
A transistor is said to be unilateral if the signal/power bounced back from the output to the source is equal to 0. This occurs if the reverse transmission coefficient, S 12 or the reverse transducer power gain, |S 12 |^2 is equal to 0. In order to determine how close to ideal unilateral a network is, the Unilateral Figure of Merit , U has been defined as follows:
U = ( |S12||S 21 ||S11||S22|)/[(1-|S11|^2 )(1-|S22|^2 )]
Error limits on Unilateral calculations:
1/(1 + U) 2 < G (^) T /GTU < 1/(1-U) 2
where, GT – transducer power gain , defined as the ratio of the output power delivered to a load by a source and the maximum power available from the source. It includes the effects of input and output impedance matching. Component resistive losses are neglected.
GT = |S21|^2 (1-|Γs|^2 )(1-|ΓL|^2 )/|(1-S 11 ΓS)(1-S 22 ΓL)-S 12 S 21 ΓLΓS|^2
GTU – transducer power gain assuming unilateral property, S 12 = 0.
The unilateral gain circles are developed to identify the best load and source impedances to minimize the error due to unilateral assumption.
Input Gain Circle:
- Calculate gain Gs: Gs = (1- |Γs| 2 )/(|1 –S 11 Γs| 2 ) Note: Multisim RF simulator allows to enter Gs manually. Gs is expressed in dB.
- Calculate gs: g (^) s = G (^) s (1-|S 11 | 2 ). The magnitude of Gs is used in the above mentioned equation.
- Calculate the center location of the circle: Cs = gsS 11 */(1-(1-gs)|S 11 | 2 )
- Calculate the radius of the circle: Rs =√ 1-gs (1-|S 11 |2)/(1-(1-gs)|S 11 |^2 ) Note: All points on the circle represent the same gain for different input impedances.
Output Gain Circle:
- Calculate gain GL: G (^) L = (1- |ΓL| 2 )/(|1 –S 22 ΓL| 2 ) Note: Multisim RF simulator allows to enter GL manualy. It is expressed in dB.
- Calculate gl: g l = G (^) L (1-|S 22 | 2 ).
The magnitude of GL is used in the above mentioned equation.
- Calculate the center location of the circle: CL = glS22*/(1-(1-gl)|S 22 | 2 )
- Calculate the radius of the circle: R (^) L =√ 1-gl (1-|S 22 |2)/(1-(1-gl)|S 22 |^2 ) Note: All points on the circle represent the same gain for different output impedances.
For maximum gain the circles turn into a dot on Smith Chart. Gsmax = 1/(1-|S 11 |^2 ) and G lmax = 1/(1-|S 22 |^2 ). The circles would pass through the center of the Smith Chart if Gs = GL = 1 or 0 dB gain. The input and ouput impedances are equal to 50 Ohms. So, the complete matching has been achieved.
Matching impedances would provide maximum power transfer from the input to the ouput for a very narrow frequency range. Sometimes, designers intentionally introduce mismatching at either input or output to maintain the same power gain (which is less than the maximum power gain) for a wider frequency range loosing some power.
The steps to design amplifier using Input and Output Gain Circles:
- Identify the gain of the mismatched network (a transistor) , |S21|2.^ This gain is a constant value.
- Calculate Gsmax and Glmax
- Select Gs and G (^) L so that Gs < Gsmax and GL < Glmax and the desired gain is achieved. Note that the desired gain in dB is: G (^) overall = Gs + G (^) L + 10log 10 |S 21 |^2
- Select the closest point on the circles to the center of the Smith Chart to minimize the mismatch and maximize the bandwidth. These points will identify the input and output matching impedance for the desired power gain.
POWER GAINS:
Multisim RF Simulator calculates Power Gain (PG), Available Power Gain (APG) and Transducer Power Gain (TPG) for Zo = 50Ohm at a given frequency. The dBMag is derived as 10log 10 |PG|.
PG is defined as the ratio of the power delivered to the load and the average power delivered to the network from the input. RF simulator assumes Zl = Zo and Zs = Zo or Γs = Γl = 0. Therefore,
PG = |S21|^2 /(1-|S11| 2 ).
The Transducer Power Gain, TPG is the ratio of the power delivered to the load to the power available from the source. For Γs = ΓL= 0, TPG = |S21| 2.
The Available Power Gain, APG is the ratio of the power available from the output port of the network to the power available from the source and it is expressed as
APG = |S 21 |^2 / (1- |S 22 | 2 )
VOLTAGE GAIN:
Voltage Gain, VG is obtained for Γs = Γl = 0 and is expressed as
