Linear two-port (and multi-port) networks are characterized by a number of equivalent circuit parameters, such as their transfer matrix, impedance matrix, admittance matrix, and scattering matrix. The transfer matrix, also known as the ABCD matrix, relates the voltage and current at port 1 to those at port 2, whereas the impedance matrix relates the two voltages V1,V2 to the two currents I1, I2
:† _ V1 I1 _ = _ A B C D __ V2 I2 _ (transfer matrix) _ V1 V2 _ = _ Z11 Z12 Z21 Z22 __ I1 −I2 _ (impedance matrix) (14.1.1) Thus, the transfer and impedance matrices are the 2×2 matrices: T = _ A B C D _ , Z= _ Z11 Z12 Z21 Z22 _ (14.1.2) The admittance matrix is simply the inverse of the impedance matrix, Y = Z−1. The scattering matrix relates the outgoing waves b1, b2 to the incoming waves a1, a2 that are incident on the two-port: †In the figure, I2 flows out of port 2, and hence −I2 flows into it. In the usual convention, both currents I1, I2 are taken to flow into their respective ports. 664 14. S-Parameters _ b1 b2 _ = _ S11 S12 S21 S22 __ a1 a2 _ , S= _ S11 S12 S21 S22 _ (scattering matrix) (14.1.3) The matrix elements S11, S12, S21, S22 are referred to as the scattering parameters or the S-parameters. The parameters S11, S22 have the meaning of reflection coefficients, and S21, S12, the meaning of transmission coefficients. The many properties and uses of the S-parameters in applications are discussed in [1135–1174]. One particularly nice overview is the HP application note AN-95-1 by Anderson  and is available on the web . We have already seen several examples of transfer, impedance, and scattering matrices. Eq. (11.7.6) or (11.7.7) is an example of a transfer matrix and (11.8.1) is the corresponding impedance matrix. The transfer and scattering matrices of multilayer structures, Eqs. (6.6.23) and (6.6.37), are more complicated examples. The traveling wave variables a1, b1 at port 1 and a2, b2 at port 2 are defined in terms of V1, I1 and V2, I2 and a real-valued positive reference impedance Z0 as follows: a1 = V1 + Z0I1 2 _ Z0 b1 = V1 − Z0I1 2 _ Z0 a2 = V2 − Z0I2 2 _ Z0 b2 = V2 + Z0I2 2 _ Z0 (traveling waves) (14.1.4) The definitions at port 2 appear different from those at port 1, but they are really the same if expressed in terms of the incoming current −I2: a2 = V2 − Z0I2 2 _ Z0 = V2 + Z0(−I2) 2 _ Z0 b2 = V2 + Z0I2 2 _ Z0 = V2 − Z0(−I2) 2 _ Z0 The term traveling waves is justified below. Eqs. (14.1.4) may be inverted to express the voltages and currents in terms of the wave variables: V1 = _ Z0(a1 + b1) I1 = 1 _ Z0 (a1 − b1) V2 = _ Z0(a2 + b2) I2 = 1 _ Z0 (b2 − a2) (14.1.5) In practice, the reference impedance is chosen to be Z0 = 50 ohm. At lower frequencies the transfer and impedance matrices are commonly used, but at microwave frequencies they become difficult to measure and therefore, the scattering matrix description is preferred. The S-parameters can be measured by embedding the two-port network (the deviceunder- test, or, DUT) in a transmission line whose ends are connected to a network analyzer. Fig. 14.1.2 shows the experimental setup. A typical network analyzer can measure S-parameters over a large frequency range, for example, the HP 8720D vector network analyzer covers the range from 50 MHz to 14.1. Scattering Parameters 665 40 GHz. Frequency resolution is typically 1 Hz and the results can be displayed either on a Smith chart or as a conventional gain versus frequency graph. Fig. 14.1.2 Device under test connected to network analyzer. Fig. 14.1.3 shows more details of the connection. The generator and load impedances are configured by the network analyzer. The connections can be reversed, with the generator connected to port 2 and the load to port 1. Fig. 14.1.3 Two-port network under test. The two line segments of lengths l1, l2 are assumed to have characteristic impedance equal to the reference impedance Z0. Then, the wave variables a1, b1 and a2, b2 are recognized as normalized versions of forward and backward traveling waves. Indeed, according to Eq. (11.7.8), we have: a1 = V1 + Z0I1 2 _ Z0 = 1 _ Z0 V1+ b1 = V1 − Z0I1 2 _ Z0 = 1 _ Z0 V1− a2 = V2 − Z0I2 2 _ Z0 = 1 _ Z0 V2− b2 = V2 + Z0I2 2 _ Z0 = 1 _ Z0 V2+ (14.1.6) Thus, a1 is essentially the incident wave at port 1 and b1 the corresponding reflected wave. Similarly, a2 is incident from the right onto port 2 and b2 is the reflected wave from port 2. The network analyzer measures the waves a_ 1, b_ 1 and a_ 2, b_ 2 at the generator and load ends of the line segments, as shown in Fig. 14.1.3. From these, the waves at the inputs of the two-port can be determined. Assuming lossless segments and using the propagation matrices (11.7.7), we have: 666 14. S-Parameters _ a1 b1 _ = _ e−jδ1 0 0 ejδ1 __ a_ 1 b_ 1 _ , _ a2 b2 _ = _ e−jδ2 0 0 ejδ2 __ a_ 2 b_ 2 _ (14.1.7) where δ1 = βll and δ2 = βl2 are the phase lengths of the segments. Eqs. (14.1.7) can be rearranged into the forms: _ b1 b2 _ = D _ b_ 1 b_ 2 _ , _ a_ 1 a_ 2 _ = D _ a1 a2 _ , D= _ ejδ1 0 0 ejδ2 _ The network analyzer measures the corresponding S-parameters of the primed variables, that is, _ b_ 1 b_ 2 _ = _ S_ 11 S_ 12 S_ 21 S_ 22 __ a_ 1 a_ 2 _ , S_ = _ S_ 11 S_ 12 S_ 21 S_ 22 _ (measured S-matrix) (14.1.8) The S-matrix of the two-port can be obtained then from: _ b1 b2 _ = D _ b_ 1 b_ 2 _ = DS_ _ a_ 1 a_ 2 _ = DS_D _ a1 a2 _ ⇒ S = DS_D or, more explicitly, _ S11 S12 S21 S22 _ = _ ejδ1 0 0 ejδ2 __ S_ 11 S_ 12 S_ 21 S_ 22 __ ejδ1 0 0 ejδ2 _ = _ S_ 11e2jδ1 S_ 12ej(δ1+δ2) S_ 21ej(δ1+δ2) S_ 22e2jδ2 _ (14.1.9) Thus, changing the points along the transmission lines at which the S-parameters are measured introduces only phase changes in the parameters. Without loss of generality, we may replace the extended circuit of Fig. 14.1.3 with the one shown in Fig. 14.1.4 with the understanding that either we are using the extended two-port parameters S_, or, equivalently, the generator and segment l1 have been replaced by their Th´evenin equivalents, and the load impedance has been replaced by its propagated version to distance l2. Fig. 14.1.4 Two-port network connected to generator and load. 14.2. Power Flow 667 The actual measurements of the S-parameters are made by connecting to a matched load, ZL = Z0. Then, there will be no reflected waves from the load, a2 = 0, and the S-matrix equations will give: b1 = S11a1 + S12a2 = S11a1 ⇒ S11 = b1 a1 ____ ZL=Z0 = reflection coefficient b2 = S21a1 + S22a2 = S21a1 ⇒ S21 = b2 a1 ____ ZL=Z0 = transmission coefficient Reversing the roles of the generator and load, one can measure in the same way the parameters S12 and S22. 14.2 Power Flow Power flow into and out of the two-port is expressed very simply in terms of the traveling wave amplitudes. Using the inverse relationships (14.1.5), we find: 1 2 Re[V∗ 1 I1] = 1 2 |a1|2 − 1 2 |b1|2 −1 2 Re[V∗ 2 I2] = 1 2 |a2|2 − 1 2 |b2|2 (14.2.1) The left-hand sides represent the power flow into ports 1 and 2. The right-hand sides represent the difference between the power incident on a port and the power reflected from it. The quantity Re[V∗ 2 I2]/2 represents the power transferred to the load. Another way of phrasing these is to say that part of the incident power on a port gets reflected and part enters the port: 1 2 |a1|2 = 1 2 |b1|2 + 1 2 Re[V∗ 1 I1] 1 2 |a2|2 = 1 2 |b2|2 + 1 2 Re[V∗ 2 (−I2)] (14.2.2) One of the reasons for normalizing the traveling wave amplitudes by _ Z0 in the definitions (14.1.4) was precisely this simple way of expressing the incident and reflected powers from a port. If the two-port is lossy, the power lost in it will be the difference between the power entering port 1 and the power leaving port 2, that is, Ploss = 1 2 Re[V∗ 1 I1]−1 2 Re[V∗ 2 I2]= 1 2 |a1|2 + 1 2 |a2|2 − 1 2 |b1|2 − 1 2 |b2|2 Noting that a†a = |a1|2 + |a2|2 and b†b = |b1|2 + |b2|2, and writing b†b = a†S†Sa, we may express this relationship in terms of the scattering matrix: Ploss = 1 2 a†a − 1 2 b†b = 1 2 a†a − 1 2 a†S†Sa = 1 2 a†(I − S†S)a (14.2.3) 668 14. S-Parameters For a lossy two-port, the power loss is positive, which implies that the matrix I−S†S must be positive definite. If the two-port is lossless, Ploss = 0, the S-matrix will be unitary, that is, S†S = I. We already saw examples of such unitary scattering matrices in the cases of the equal travel-time multilayer dielectric structures and their equivalent quarter wavelength multisection transformers. 14.3 Parameter Conversions It is straightforward to derive the relationships that allow one to pass from one parameter set to another. For example, starting with the transfer matrix, we have: V1 = AV2 + BI2 I1 = CV2 + DI2 ⇒ V1 = A _ 1 C I1 − D C I2 _ + BI2 = A C I1 − AD − BC C I2 V2 = 1 C I1 − D C I2 The coefficients of I1, I2 are the impedance matrix elements. The steps are reversible, and we summarize the final relationships below: Z = _ Z11 Z12 Z21 Z22 _ = 1 C _ A AD− BC 1 D _ T = _ A B C D _ = 1 Z21 _ Z11 Z11Z22 − Z12Z21 1 Z22 _ (14.3.1) We note the determinants det(T)= AD − BC and det(Z)= Z11Z22 − Z12Z21. The relationship between the scattering and impedance matrices is also straightforward to derive. We define the 2×1 vectors: V = _ V1 V2 _ , I = _ I1 −I2 _ , a = _ a1 a2 _ , b = _ b1 b2 _ (14.3.2) Then, the definitions (14.1.4) can be written compactly as: a = 1 2 _ Z0 (V + Z0I)= 1 2 _ Z0 (Z + Z0I)I b = 1 2 _ Z0 (V − Z0I)= 1 2 _ Z0 (Z − Z0I)I (14.3.3) where we used the impedance matrix relationship V = ZI and defined the 2×2 unit matrix I. It follows then, 1 2 _ Z0 I = (Z + Z0I)−1a ⇒ b = 1 2 _ Z0 (Z − Z0I)I = (Z − Z0I)(Z + Z0I)−1a Thus, the scattering matrix S will be related to the impedance matrix Z by S = (Z − Z0I)(Z + Z0I)−1 _ Z = (I − S)−1(I + S)Z0 (14.3.4) 14.4. Input and Output Reflection Coefficients 669 Explicitly, we have: S = _ Z11 − Z0 Z12 Z21 Z22 − Z0 __ Z11 + Z0 Z12 Z21 Z22 + Z0 _−1 = _ Z11 − Z0 Z12 Z21 Z22 − Z0 _ 1 Dz _ Z22 + Z0 −Z12 −Z21 Z11 + Z0 _ where Dz = det(Z + Z0I)= (Z11 + Z0)(Z22 + Z0)−Z12Z21. Multiplying the matrix factors, we obtain: S = 1 Dz _ (Z11 − Z0)(Z22 + Z0)−Z12Z21 2Z12Z0 2Z21Z0 (Z11 + Z0)(Z22 − Z0)−Z12Z21 _ (14.3.5) Similarly, the inverse relationship gives: Z = Z0 Ds _ (1 + S11)(1 − S22)+S12S21 2S12 2S21 (1 − S11)(1 + S22)+S12S21 _ (14.3.6) where Ds = det(I −S)= (1−S11)(1−S22)−S12S21. Expressing the impedance parameters in terms of the transfer matrix parameters, we also find: S = 1 Da ⎡ ⎢⎢⎣ A + B Z0 − CZ0 − D 2(AD − BC) 2 −A + B Z0 − CZ0 + D ⎤ ⎥⎥⎦ (14.3.7) where Da = A + B Z0 + CZ0 + D. 14.4 Input and Output Reflection Coefficients When the two-port is connected to a generator and load as in Fig. 14.1.4, the impedance and scattering matrix equations take the simpler forms: V1 = ZinI1 V2 = ZLI2 _ b1 = Γina1 a2 = ΓLb2 (14.4.1) where Zin is the input impedance at port 1, and Γin, ΓL are the reflection coefficients at port 1 and at the load: Γin = Zin − Z0 Zin + Z0 , ΓL = ZL − Z0 ZL + Z0 (14.4.2) The input impedance and input reflection coefficient can be expressed in terms of the Z- and S-parameters, as follows: 670 14. S-Parameters Zin = Z11 − Z12Z21 Z22 + ZL _ Γin = S11 + S12S21ΓL 1 − S22ΓL (14.4.3) The equivalence of these two expressions can be shown by using the parameter conversion formulas of Eqs. (14.3.5) and (14.3.6), or they can be shown indirectly, as follows. Starting with V2 = ZLI2 and using the second impedance matrix equation, we can solve for I2 in terms of I1: V2 = Z21I1 − Z22I2 = ZLI2 ⇒ I2 = Z21 Z22 + ZL I1 (14.4.4) Then, the first impedance matrix equation implies: V1 = Z11I1 − Z12I2 = _ Z11 − Z12Z21 Z22 + ZL _ I1 = ZinI1 Starting again with V2 = ZLI2 we find for the traveling waves at port 2: a2 = V2 − Z0I2 2 _ Z0 = ZL − Z0 2 _ Z0 I2 b2 = V2 + Z0I2 2 _ Z0 = ZL + Z0 2 _ Z0 I2 ⇒ a2 = ZL − Z0 ZL + Z0 b2 = ΓLb2 Using V1 = ZinI1, a similar argument implies for the waves at port 1: a1 = V1 + Z0I1 2 _ Z0 = Zin + Z0 2 _ Z0 I1 b1 = V1 − Z0I1 2 _ Z0 = Zin − Z0 2 _ Z0 I1 ⇒ b1 = Zin − Z0 Zin + Z0 a1 = Γina1 It follows then from the scattering matrix equations that: b2 = S21a1 + S22a2 = S22a1 + S22ΓLb2 ⇒ b2 = S21 1 − S22ΓL a1 (14.4.5) which implies for b1: b1 = S11a1 + S12a2 = S11a1 + S12ΓLb2 = _ S11 + S12S21ΓL 1 − S22ΓL _ a1 = Γina1 Reversing the roles of generator and load, we obtain the impedance and reflection coefficients from the output side of the two-port: Zout = Z22 − Z12Z21 Z11 + ZG _ Γout = S22 + S12S21ΓG 1 − S11ΓG (14.4.6) where Γout = Zout − Z0 Zout + Z0 , ΓG = ZG − Z0 ZG + Z0 (14.4.7) 14.5. Stability Circles 671 Fig. 14.4.1 Input and output equivalent circuits. The input and output impedances allow one to replace the original two-port circuit of Fig. 14.1.4 by simpler equivalent circuits. For example, the two-port and the load can be replaced by the input impedance Zin connected at port 1, as shown in Fig. 14.4.1. Similarly, the generator and the two-port can be replaced by a Th´evenin equivalent circuit connected at port 2. By determining the open-circuit voltage and short-circuit current at port 2, we find the corresponding Th´evenin parameters in terms of the impedance parameters: Vth = Z21VG Z11 + ZG , Zth = Zout = Z22 − Z12Z21 Z11 + ZG (14.4.8) 14.5 Stability Circles In discussing the stability conditions of a two-port in terms of S-parameters, the following definitions of constants are often used: Δ = det(S)= S11S22 − S12S21 K = 1 − |S11|2 − |S22|2 + |Δ|2 2|S12S21| (Rollett stability factor) μ1 = 1 − |S11|2 |S22 − ΔS∗ 11| + |S12S21| (Edwards-Sinsky stability parameter) μ2 = 1 − |S22|2 |S11 − ΔS∗ 22| + |S12S21| B1 = 1 + |S11|2 − |S22|2 − |Δ|2 B2 = 1 + |S22|2 − |S11|2 − |Δ|2 C1 = S11 − ΔS∗ 22, D1 = |S11|2 − |Δ|2 C2 = S22 − ΔS∗ 11, D2 = |S22|2 − |Δ|2 (14.5.1) The quantity K is the Rollett stability factor , and μ1, μ2, the Edwards-Sinsky stability parameters . The following identities hold among these constants: 672 14. S-Parameters B21 − 4|C1|2 = B22 − 4|C2|2 = 4|S12S21|2(K2 − 1) |C1|2 = |S12S21|2 + _ 1 − |S22|2_ D1 |C2|2 = |S12S21|2 + _ 1 − |S11|2_ D2 (14.5.2) For example, noting that S12S21 = S11S22 − Δ, the last of Eqs. (14.5.2) is a direct consequence of the identity: |A − BC|2 − |B − AC∗|2 = _ 1 − |C|2__ |A|2 − |B|2_ (14.5.3) We define also the following parameters, which will be recognized as the centers and radii of the source and load stability circles: cG = C∗ 1 D1 , rG = |S12S21| |D1| (source stability circle) (14.5.4) cL = C∗ 2 D2 , rL = |S12S21| |D2| (load stability circle) (14.5.5) They satisfy the following relationships, which are consequences of the last two of Eqs. (14.5.2) and the definitions (14.5.4) and (14.5.5): 1 − |S11|2 = _ |cL|2 − r2L _ D2 1 − |S22|2 = _ |cG|2 − r2G _ D1 (14.5.6) We note also that using Eqs. (14.5.6), the stability parameters μ1, μ2 can be written as: μ1 = _ |cL| − rL _ sign(D2) μ2 = _ |cG| − rG _ sign(D1) (14.5.7) For example, we have: μ1 = 1 − |S11|2 |C2| + |S12S21| = D2 _ |cL|2 − r2L _ |D2||cL| + |D2|rL = D2 _ |cL|2 − r2L _ |D2| _ |cL| + rL _ = D2 |D2| _ |cL| − rL _ We finally note that the input and output reflection coefficients can be written in the alternative forms: Γin = S11 + S12S21ΓL 1 − S22ΓL = S11 − ΔΓL 1 − S22ΓL Γout = S22 + S12S21ΓG 1 − S22ΓG = S22 − ΔΓG 1 − S11ΓG (14.5.8) Next, we discuss the stability conditions. The two-port is unconditionally stable if any generator and load impedances with positive resistive parts RG,RL, will always lead to input and output impedances with positive resistive parts Rin,Rout. Equivalently, unconditional stability requires that any load and generator with |ΓL| < 1 and |ΓG| < 1 will result into |Γin| < 1 and |Γout| < 1. The two-port is termed potentially or conditionally unstable if there are |ΓL| < 1 and |ΓG| < 1 resulting into |Γin| ≥ 1 and/or |Γout| ≥ 1. 14.5. Stability Circles 673 The load stability region is the set of all ΓL that result into |Γin| < 1, and the source stability region, the set of all ΓG that result into |Γout| < 1. In the unconditionally stable case, the load and source stability regions contain the entire unit-circles |ΓL| < 1 or |ΓG| < 1. However, in the potentially unstable case, only portions of the unit-circles may lie within the stability regions and such ΓG, ΓL will lead to a stable input and output impedances. The connection of the stability regions to the stability circles is brought about by the following identities, which can be proved easily using Eqs. (14.5.1)–(14.5.8): 1 − |Γin|2 = |ΓL − cL|2 − r2L |1 − S22ΓL|2 D2 1 − |Γout|2 = |ΓG − cG|2 − r2G |1 − S11ΓG|2 D1 (14.5.9) For example, the first can be shown starting with Eq. (14.5.8) and using the definitions (14.5.5) and the relationship (14.5.6): 1 − |Γin|2 = 1 − ____ S11 − ΔΓL 1 − S22ΓL ____ 2 = |S11 − ΔΓL|2 − |1 − S22ΓL|2 |1 − S22ΓL|2 = _ |S22|2 − |Δ|2_ |ΓL|2 − (S22 − ΔS∗ 11)ΓL − (S∗ 22 − Δ∗S11)Γ∗ L + 1 − |S11|2 |1 − S22ΓL|2 = D2|ΓL|2 − C2ΓL − C∗ 2 Γ∗ L + 1 − |S11|2 |1 − S22ΓL|2 = D2 _ |ΓL|2 − c∗ L ΓL − c∗ L Γ∗ L + |cL|2 − r2L _ |1 − S22ΓL|2 = D2 _ |ΓL − cL|2 − r2L _ |1 − S22ΓL|2 It follows from Eq. (14.5.9) that the load stability region is defined by the conditions: 1 − |Γin|2 > 0 _ _ |ΓL − cL|2 − r2L _ D2 > 0 Depending on the sign of D2, these are equivalent to the outside or the inside of the load stability circle of center cL and radius rL: |ΓL − cL| > rL , if D2 > 0 |ΓL − cL| < rL , if D2 < 0 (load stability region) (14.5.10) The boundary of the circle |ΓL−cL| = rL corresponds to |Γin| = 1. The complement of these regions corresponds to the unstable region with |Γin| > 1. Similarly, we find for the source stability region: |ΓG − cG| > rG , if D1 > 0 |ΓG − cG| < rG , if D1 < 0 (source stability region) (14.5.11) In order to have unconditional stability, the stability regions must contain the unitcircle in its entirety. If D2 > 0, the unit-circle and load stability circle must not overlap 674 14. S-Parameters at all, as shown in Fig. 14.5.1. Geometrically, the distance between the points O and A in the figure is (OA)= |cL|−rL. The non-overlapping of the circles requires the condition (OA)> 1, or, |cL| − rL > 1. If D2 < 0, the stability region is the inside of the stability circle, and therefore, the unit-circle must lie within that circle. This requires that (OA)= rL −|cL| > 1, as shown in Fig. 14.5.1. Fig. 14.5.1 Load stability regions in the unconditionally stable case. These two conditions can be combined into sign(D2) _ |cL| − rL _ > 1. But, that is equivalent to μ1 > 1 according to Eq. (14.5.7). Geometrically, the parameter μ1 represents the distance (OA). Thus, the condition for the unconditional stability of the input is equivalent to: μ1 > 1 (unconditional stability condition) (14.5.12) It has been shown by Edwards and Sinsky  that this single condition (or, alternatively, the single condition μ2 > 1) is necessary and sufficient for the unconditional stability of both the input and output impedances of the two-port. Clearly, the source stability regions will be similar to those of Fig. 14.5.1. If the stability condition is not satisfied, that is, μ1 < 1, then only that portion of the unit-circle that lies within the stability region will be stable and will lead to stable input and output impedances. Fig. 14.5.2 illustrates such a potentially unstable case. If D2 > 0, then μ1 < 1 is equivalent to |cL| − rL < 1, and if D2 < 0, it is equivalent to rL − |cL| < 1. In either case, the unit-circle is partially overlapping with the stability circle, as shown in Fig. 14.5.2. The portion of the unit-circle that does not lie within the stability region will correspond to an unstable Zin. There exist several other unconditional stability criteria that are equivalent to the single criterion μ1 > 1. They all require that the Rollett stability factor K be greater than unity, K > 1, as well as one other condition. Any one of the following criteria are necessary and sufficient for unconditional stability : 14.5. Stability Circles 675 Fig. 14.5.2 Load stability regions in potentially unstable case. K > 1 and |Δ| < 1 K > 1 and B1 > 0 K > 1 and B2 > 0 K > 1 and |S12S21| < 1 − |S11|2 K > 1 and |S12S21| < 1 − |S22|2 (stability conditions) (14.5.13) Their equivalence to μ1 > 1 has been shown in . In particular, it follows from the last two conditions that unconditional stability requires |S11| < 1 and |S22| < 1. These are necessary but not sufficient for stability. A very common circumstance in practice is to have a potentially unstable two-port, but with |S11| < 1 and |S22| < 1. In such cases, Eq. (14.5.6) implies D2 _ |cL|2 − r2L)> 0, and the lack of stability requires μ1 = sign(D2) _ |cL|2 − r2L )< 1. Therefore, if D2 > 0, then we must have |cL|2 − r2L > 0 and |cL| − rL < 1, which combine into the inequality rL < |cL| < rL + 1. This is depicted in the left picture of Fig. 14.5.2. The geometrical distance (OA)= |cL| − rL satisfies 0 < (OA)< 1, so that stability circle partially overlaps with the unit-circle but does not enclose its center. On the other hand, ifD2 < 0, the two conditions require |cL|2−r2L < 0 and rL−|cL| < 1, which imply |cL| < rL < |cL| + 1. This is depicted in the right Fig. 14.5.2. The geometrical distance (OA)= rL −|cL| again satisfies 0 < (OA)< 1, but now the center of the unit-circle lies within the stability circle, which is also the stability region. We have written a number of MATLAB functions that facilitate working with Sparameters. They are described in detail later on: smat reshape S-parameters into S-matrix sparam calculate stability parameters sgain calculate transducer, available, operating, and unilateral power gains smatch calculate simultaneous conjugate match for generator and load gin,gout calculate input and output reflection coefficients smith draw a basic Smith chart smithcir draw a stability or gain circle on Smith chart sgcirc determine stability and gain circles nfcirc determine noise figure circles nfig calculate noise figure 676 14. S-Parameters The MATLAB function sparam calculates the stability parameters μ1, K, |Δ|, B1, B2, as well as the parameters C1,C2,D1,D2. It has usage: [K,mu,D,B1,B2,C1,C2,D1,D2] = sparam(S); % stability parameters The function sgcirc calculates the centers and radii of the source and load stability circles. It also calculates gain circles to be discussed later on. Its usage is: [cL,rL] = sgcirc(S,’l’); % load or Zin stability circle [cG,rG] = sgcirc(S,’s’); % source or Zout stability circle The MATLAB function smith draws a basic Smith chart, and the function smithcir draws the stability circles: smith(n); % draw four basic types of Smith charts, n = 1, 2, 3, 4 smith; % default Smith chart corresponding to n = 3 smithcir(c,r,max,width); % draw circle of center c and radius r smithcir(c,r,max); % equivalent to linewidth width=1 smithcir(c,r); % draw full circle with linewidth width=1 The parameter max controls the portion of the stability circle that is visible outside the Smith chart. For example, max = 1.1 will display only that portion of the circle that has |Γ| < 1.1. Example 14.5.1: The Hewlett-Packard AT-41511 NPN bipolar transistor has the following Sparameters at 1 GHz and 2 GHz : S11 = 0.48∠−149o, S21 = 5.189∠89o, S12 = 0.073∠43o, S22 = 0.49∠−39o S11 = 0.46∠162o, S21 = 2.774∠59o, S12 = 0.103∠45o, S22 = 0.42∠−47o Determine the stability parameters, stability circles, and stability regions. Solution: The transistor is potentially unstable at 1 GHz, but unconditionally stable at 2 GHz. The source and load stability circles at 1 GHz are shown in Fig. 14.5.3. Fig. 14.5.3 Load and source stability circles at 1 GHz. The MATLAB code used to generate this graph was: 14.6. Power Gains 677 S = smat([0.48 -149 5.189 89 0.073 43 0.49 -39]); % form S-matrix [K,mu,D,B1,B2,C1,C2,D1,D2] = sparam(S); % stability parameters [cL,rL] = sgcirc(S,’l’); % stability circles [cG,rG] = sgcirc(S,’s’); smith; % draw basic Smith chart smithcir(cL, rL, 1.1, 1.5); % draw stability circles smithcir(cG, rG, 1.1, 1.5); The computed stability parameters at 1 GHz were: [K,μ1, |Δ|, B1, B2,D1,D2]= [0.781, 0.847, 0.250, 0.928, 0.947, 0.168, 0.178] The transistor is potentially unstable because K < 1 even though |Δ| < 1, B1 > 0, and B2 > 0. The load and source stability circle centers and radii were: cL = 2.978∠51.75o, rL = 2.131 cG = 3.098∠162.24o, rG = 2.254 Because both D1 and D2 are positive, both stability regions will be the portion of the Smith chart that lies outside the stability circles. For 2 GHz, we find: [K,μ1, |Δ|, B1, B2,D1,D2]= [1.089, 1.056, 0.103, 1.025, 0.954, 0.201, 0.166] cL = 2.779∠50.12o, rL = 1.723 cG = 2.473∠−159.36o, rG = 1.421 The transistor is stable at 2 GHz, with both load and source stability circles being completely outside the unit-circle. __ Problem 14.2 presents an example for which the D2 parameter is negative, so that the stability regions will be the insides of the stability circles. At one frequency, the unit-circle is partially overlapping with the stability circle, while at another frequency, it lies entirely within the stability circle. 14.6 Power Gains The amplification (or attenuation) properties of the two-port can be deduced by comparing the power Pin going into the two-port to the power PL coming out of the two-port and going into the load. These were given in Eq. (14.2.1) and we rewrite them as: Pin = 1 2 Re[V∗ 1 I1]= 1 2 Rin|I1|2 (power into two-port) PL = 1 2 Re[V∗ 2 I2]= 1 2 RL|I2|2 (power out of two-port and into load) (14.6.1) 678 14. S-Parameters where we used V1 = ZinI1, V2 = ZLI2, and defined the real parts of the input and load impedances by Rin = Re(Zin) and RL = Re(ZL). Using the equivalent circuits of Fig. 14.4.1, we may write I1, I2 in terms of the generator voltage VG and obtain: Pin = 1 2 |VG|2Rin |Zin + ZG|2 PL = 1 2 |Vth|2RL |Zout + ZL|2 = 1 2 |VG|2RL|Z21|2 __ (Z11 + ZG)(Zout + ZL) __ 2 (14.6.2) Using the identities of Problem 14.1, PL can also be written in the alternative forms: PL = 1 2 |VG|2RL|Z21|2 __ (Z22 + ZL)(Zin + ZG) _ _ 2 = 1 2 |VG|2RL|Z21|2 __ (Z11 + ZG)(Z22 + ZL)−Z12Z21 __ 2 (14.6.3) The maximum power that can be delivered by the generator to a connected load is called the available power of the generator, PavG, and is obtained when the load is conjugate-matched to the generator, that is, PavG = Pin when Zin = Z∗ G. Similarly, the available power from the two-port network, PavN, is the maximum power that can be delivered by the Th´evenin-equivalent circuit of Fig. 14.4.1 to a connected load, that is, PavN = PL when ZL = Z∗ th = Z∗ out. It follows then from Eq. (14.6.2) that the available powers will be: PavG = max Pin = |VG|2 8RG (available power from generator) PavN = max PL = |Vth|2 8Rout (available power from network) (14.6.4) Using Eq. (14.4.8), PavN can also be written as: PavN = |VG|2 8Rout ____ Z21 Z11 + ZG ____ 2 (14.6.5) The powers can be expressed completely in terms of the S-parameters of the twoport and the input and output reflection coefficients. With the help of the identities of Problem 14.1, we find the alternative expressions for Pin and PL: Pin = |VG|2 8Z0 _ 1 − |Γin|2_ |1 − ΓG|2 |1 − ΓinΓG|2 PL = |VG|2 8Z0 _ 1 − |ΓL|2_ |1 − ΓG|2|S21|2 __ (1 − ΓinΓG)(1 − S22ΓL) _ _ 2 = |VG|2 8Z0 _ 1 − |ΓL|2_ |1 − ΓG|2|S21|2 __ ( 1 − ΓoutΓL)(1 − S11ΓG) _ _ 2 = |VG|2 8Z0 _ 1 − |ΓL|2_ |1 − ΓG|2|S21|2 __ (1 − S11ΓG)(1 − S22ΓL)−S12S21ΓGΓL _ _ 2 (14.6.6) Similarly, we have for PavG and PavN: 14.6. Power Gains 679 PavG = |VG|2 8Z0 |1 − ΓG|2 1 − |ΓG|2 PavN = |VG|2 8Z0 |1 − ΓG|2|S21|2 _ 1 − |Γout|2 _ |1 − S11ΓG|2 (14.6.7) It is evident that PavG, PavN are obtained from Pin, PL by setting Γin = Γ∗ G and ΓL = Γ∗ out, which are equivalent to the conjugate-match conditions. Three widely used definitions for the power gain of the two-port network are the transducer power gain GT, the available power gain Ga, and the power gain Gp, also called the operating gain. They are defined as follows: GT = power out of network maximum power in = PL PavG (transducer power gain) Ga = maximum power out maximum power in = PavN PavG (available power gain) Gp = power out of network power into network = PL Pin (operating power gain) (14.6.8) Each gain is expressible either in terms of the Z-parameters of the two-port, or in terms of its S-parameters. In terms of Z-parameters, the transducer gain is given by the following forms, obtained from the three forms of PL in Eqs. (14.6.2) and (14.6.3): GT = 4RGRL|Z21|2 __ (Z22 + ZL)(Zin + ZG) _ _ 2 = 4RGRL|Z21|2 __ (Z11 + ZG)(Zout + ZL) __ 2 = 4RGRL|Z21|2 __ (Z11 + ZG)(Z22 + ZL)−Z12Z21 _ _ 2 (14.6.9) And, in terms of the S-parameters: GT = 1 − |ΓG|2 |1 − ΓinΓG|2 |S21|2 1 − |ΓL|2 |1 − S22ΓL|2 = 1 − |ΓG|2 |1 − S11ΓG|2 |S21|2 1 − |ΓL|2 |1 − ΓoutΓL|2 = (1 − |ΓG|2)|S21|2(1 − |ΓL|2) __ (1 − S11ΓG)(1 − S22ΓL)−S12S21ΓGΓL _ _ 2 (14.6.10) Similarly, we have for Ga and Gp: Ga = RG Rout ____ Z21 Z11 + ZG ____ 2 = 1 − |ΓG|2 |1 − S11ΓG|2 |S21|2 1 1 − |Γout|2 Gp = RL Rin ____ Z21 Z22 + ZL ____ 2 = 1 1 − |Γin|2 |S21|2 1 − |ΓL|2 |1 − S22ΓL|2 (14.6.11) 680 14. S-Parameters The transducer gain GT is, perhaps, the most representative measure of gain for the two-port because it incorporates the effects of both the load and generator impedances, whereas Ga depends only on the generator impedance and Gp only on the load impedance. If the generator and load impedances are matched to the reference impedance Z0, so that ZG = ZL = Z0 and ΓG = ΓL = 0, and Γin = S11, Γout = S22, then the power gains reduce to: GT = |S21|2, Ga = |S21|2 1 − |S22|2, Gp = |S21|2 1 − |S11|2 (14.6.12) A unilateral two-port has by definition zero reverse transmission coefficient, that is, S12 = 0. In this case, the input and output reflection coefficients simplify into: Γin = S11, Γout = S22 (unilateral two-port) (14.6.13) The expressions of the power gains simplify somewhat in this case: GTu = 1 − |ΓG|2 |1 − S11ΓG|2 |S21|2 1 − |ΓL|2 |1 − S22ΓL|2 Gau = 1 − |ΓG|2 |1 − S11ΓG|2 |S21|2 1 1 − |S22|2 Gpu = 1 1 − |S11|2 |S21|2 1 − |ΓL|2 |1 − S22ΓL|2 (unilateral gains) (14.6.14) For both the bilateral and unilateral cases, the gains Ga,Gp are obtainable from GT by setting ΓL = Γ∗ out and Γin = Γ∗ G, respectively, as was the case for PavN and PavG. The relative power ratios Pin/PavG and PL/PavN measure the mismatching between the generator and the two-port and between the load and the two-port. Using the definitions for the power gains, we obtain the input and output mismatch factors: Min = Pin PavG = GT Gp = 4RinRG |Zin + ZG|2 = _ 1 − |Γin|2__ 1 − |ΓG|2_ |1 − ΓinΓG|2 (14.6.15) Mout = PL PavN = GT Ga = 4RoutRL |Zout + ZL|2 = _ 1 − |Γout|2__ 1 − |ΓL|2_ |1 − ΓoutΓL|2 (14.6.16) The mismatch factors are always less than or equal to unity (for positive Rin and Rout.) Clearly, Min = 1 under the conjugate-match condition Zin = Z∗ G or Γin = Γ∗ G, and Mout = 1 if ZL = Z∗ out or ΓL = Γ∗ out. The mismatch factors can also be written in the following forms, which show more explicitly the mismatch properties: Min = 1 − _____Γin − Γ ∗ G 1 − ΓinΓG _____ 2 , Mout = 1 − _____ Γout − Γ∗ L 1 − ΓoutΓL _____ 2 (14.6.17) These follow from the identity: |1 − Γ1Γ2|2 − |Γ1 − Γ∗ 2 |2 = _ 1 − |Γ1|2__ 1 − |Γ2|2_ (14.6.18) 14.6. Power Gains 681 The transducer gain is maximized when the two-port is simultaneously conjugate matched, that is, when Γin = Γ∗ G and ΓL = Γ∗ out. Then, Min = Mout = 1 and the three gains become equal. The common maximum gain achieved by simultaneous matching is called the maximum available gain (MAG): GT,max = Ga,max = Gp,max = GMAG (14.6.19) Simultaneous matching is discussed in Sec. 14.8. The necessary and sufficient condition for simultaneous matching is K ≥ 1, where K is the Rollett stability factor. It can be shown that the MAG can be expressed as: GMAG = |S21| |S12| _ K − _ K2 − 1 _ (maximum available gain) (14.6.20) The maximum stable gain (MSG) is the maximum value GMAG can have, which is achievable when K = 1: GMSG = |S21| |S12| (maximum stable gain) (14.6.21) In the unilateral case, the MAG is obtained either by setting ΓG = Γ∗ in = S∗ 11 and ΓL = Γ∗ out = S∗ 22 in Eq. (14.6.14), or by a careful limiting process in Eq. (14.6.20), in which K→∞so that both the numerator factor K− √ K2 − 1 and the denominator factor |S12| tend to zero. With either method, we find the unilateral MAG: GMAG,u = |S21|2 _ 1 − |S11|2 __ 1 − |S22|2 _ = G1|S21|2G2 (unilateral MAG) (14.6.22) The maximum unilateral input and output gain factors are: G1 = 1 1 − |S11|2, G2 = 1 1 − |S22|2 (14.6.23) They are the maxima of the input and output gain factors in Eq. (14.6.14) realized with conjugate matching, that is, with ΓG = S∗ 11 and ΓL = S∗ 22. For any other values of the reflection coefficients (such that |ΓG| < 1 and ΓL| < 1), we have the following inequalities, which follow from the identity (14.6.18): 1 − |ΓG|2 |1 − S11ΓG|2 ≤ 1 1 − |S11|2 , 1 − |ΓL|2 |1 − S22ΓL|2 ≤ 1 1 − |S22|2 (14.6.24) Often two-ports, such as most microwave transistor amplifiers, are approximately unilateral, that is, the measured S-parameters satisfy |S12| |S21|. To decide whether the two-port should be treated as unilateral, a figure of merit is used, which is essentially the comparison of the maximum unilateral gain to the transducer gain of the actual device under the same matching conditions, that is, ΓG = S∗ 11 and ΓL = S∗ 22. For these matched values of ΓG, ΓL, the ratio of the bilateral and unilateral transducer gains can be shown to have the form: gu = GT GTu = 1 |1 − U|2, U= S12S21S∗ 11S∗ _ 22 1 − |S11|2 __ 1 − |S22|2 _ (14.6.25) 682 14. S-Parameters The quantity |U| is known as the unilateral figure of merit. If the relative gain ratio gu is near unity (typically, within 10 percent of unity), the two-port may be treated as unilateral. The MATLAB function sgain computes the transducer, available, and operating power gains, given the S-parameters and the reflection coefficients ΓG, ΓL. In addition, it computes the unilateral gains, the maximum available gain, and the maximum stable gain. It also computes the unilateral figure of merit ratio (14.6.25). It has usage: Gt = sgain(S,gG,gL); transducer power gain at given ΓG, ΓL Ga = sgain(S,gG,’a’); available power gain at given ΓG with ΓL = Γ∗ out Gp = sgain(S,gL,’p’); operating power gain at given ΓL with ΓG = Γ∗ in Gmag = sgain(S); maximum available gain (MAG) Gmsg = sgain(S,’msg’); maximum stable gain (MSG) Gu = sgain(S,’u’); maximum unilateral gain, Eq. (14.6.22) G1 = sgain(S,’ui’); maximum unilateral input gain, Eq. (14.6.23) G2 = sgain(S,’uo’); maximum unilateral output gain, Eq. (14.6.23) gu = sgain(S,’ufm’); unilateral figure of merit gain ratio, Eq. (14.6.25) The MATLAB functions gin and gout compute the input and output reflection coefficients from S and ΓG, ΓL. They have usage: Gin = gin(S,gL); input reflection coefficient, Eq. (14.4.3) Gout = gout(S,gG); output reflection coefficient, Eq. (14.4.6) Example 14.6.1: A microwave transistor amplifier uses the Hewlett-Packard AT-41410 NPN bipolar transistor with the following S-parameters at 2 GHz : S11 = 0.61∠165o, S21 = 3.72∠59o, S12 = 0.05∠42o, S22 = 0.45∠−48o Calculate the input and output reflection coefficients and the various power gains, if the amplifier is connected to a generator and load with impedances ZG = 10 − 20j and ZL = 30 + 40j ohm. Solution: The following MATLAB code will calculate all the required gains: Z0 = 50; % normalization impedance ZG = 10+20j; gG = z2g(ZG,Z0); % ΓG = −0.50 + 0.50j = 0.71∠135o ZL = 30-40j; gL = z2g(ZL,Z0); % ΓL = −0.41 − 0.43j = 0.59∠−133.15o S = smat([0.61 165 3.72 59 0.05 42 0.45 -48]); % reshape S into matrix Gin = gin(S,gL); % Γin = 0.54∠162.30o Gout = gout(S,gG); % Γout = 0.45∠−67.46o Gt = sgain(S,gG,gL); % GT = 4.71, or, 6.73 dB Ga = sgain(S,gG,’a’); % Ga = 11.44, or, 10.58 dB Gp = sgain(S,gL,’p’); % Gp = 10.51, or, 10.22 dB Gu = sgain(S,’u’); % Gu = 27.64, or, 14.41 dB G1 = sgain(S,’ui’); % G1 = 1.59, or, 2.02 dB G2 = sgain(S,’uo’); % G2 = 1.25, or, 0.98 dB 14.7. Generalized S-Parameters and Power Waves 683 gu = sgain(S,’ufm’); % gu = 1.23, or, 0.89 dB Gmag = sgain(S); % GMAG = 41.50, or, 16.18 dB Gmsg = sgain(S,’msg’); % GMSG = 74.40, or, 18.72 dB The amplifier cannot be considered to be unilateral as the unilateral figure of merit ratio gu = 1.23 is fairly large (larger than 10 percent from unity.) The amplifier is operating at a gain of GT = 6.73 dB, which is far from the maximum value of GMAG = 16.18 dB. This is because it is mismatched with the given generator and load impedances. To realize the optimum gain GMAG the amplifier must ‘see’ certain optimum generator and load impedances or reflection coefficients. These can be calculated by the MATLAB function smatch and are found to be: ΓG = 0.82∠−162.67o ⇒ ZG = g2z(ZG,Z0)= 5.12 − 7.54j Ω ΓL = 0.75∠52.57o ⇒ ZL = g2z(ZL,Z0)= 33.66 + 91.48j Ω The design of such optimum matching terminations and the function smatch are discussed in Sec. 14.8. The functions g2z and z2g were discussed in Sec. 11.7 . __ 14.7 Generalized S-Parameters and Power Waves The practical usefulness of the S-parameters lies in the fact that the definitions (14.1.4) represent forward and backward traveling waves, which can be measured remotely by connecting a network analyzer to the two-port with transmission lines of characteristic impedance equal to the normalization impedance Z0. This was depicted in Fig. 14.1.3. A generalized definition of S-parameters and wave variables can be given by using in Eq. (14.1.4) two different normalization impedances for the input and output ports. Anticipating that the two-port will be connected to a generator and load of impedances ZG and ZL, a particularly convenient choice is to use ZG for the input normalization impedance and ZL for the output one, leading to the definition of the power waves (as opposed to traveling waves) [1137–1139,1141]: a_ 1 = V1 + ZGI1 2 _ RG b_ 1 = V1 − Z∗ GI1 2 _ RG a_ 2 = V2 − ZLI2 2 _ RL b_ 2 = V2 + Z∗ L I2 2 _ RL (power waves) (14.7.1) We note that the b-waves involve the complex-conjugates of the impedances. The quantities RG,RL are the resistive parts of ZG,ZL and are assumed to be positive. These definitions reduce to the conventional traveling ones if ZG = ZL = Z0. These “wave” variables can no longer be interpreted as incoming and outgoing waves from the two sides of the two-port. However, as we see below, they have a nice interpretation in terms of power transfer to and from the two-port and simplify the expressions for the power gains. Inverting Eqs. (14.7.1), we have: 684 14. S-Parameters V1 = 1 _ RG (Z∗ Ga_ 1 + ZGb_ 1) I1 = 1 _ RG (a_ 1 − b_ 1) V2 = 1 _ RL (Z∗ L a_ 2 + ZLb_ 2) I2 = 1 _ RL (b_ 2 − a_ 2) (14.7.2) The power waves can be related directly to the traveling waves. For example, expressing Eqs. (14.7.1) and (14.1.5) in matrix form, we have for port-1: _ a_ 1 b_ 1 _ = 1 2 _ RG _ 1 ZG 1 −Z∗ G __ V1 I1 _ , _ V1 I1 _ = 1 _ Z0 _ Z0 Z0 1 −1 __ a1 b1 _ It follows that: _ a_ 1 b_ 1 _ = 1 2 _ RGZ0 _ 1 ZG 1 −Z∗ G __ Z0 Z0 1 −1 __ a1 b1 _ or, _ a_ 1 b_ 1 _ = 1 2 _ RGZ0 _ Z0 + ZG Z0 − ZG Z0 − Z∗ G Z0 + Z∗ G __ a1 b1 _ (14.7.3) The entries of this matrix can be expressed directly in terms of the reflection coefficient ΓG. Using the identities of Problem 14.3, we may rewrite Eq. (14.7.3) and its inverse as follows:: _ a_ 1 b_ 1 _ = 1 _ 1 − |ΓG|2 _ ejφG −ΓGejφG −Γ∗ Ge−jφG e−jφG __ a1 b1 _ _ a1 b1 _ = 1 _ 1 − |ΓG|2 _ e−jφG ΓGejφG Γ∗ Ge−jφG ejφG __ a_ 1 b_ 1 _ (14.7.4) where, noting that the quantity |1 − ΓG|/(1 − ΓG) is a pure phase factor, we defined: ΓG = ZG − Z0 ZG + Z0 , ejφG = |1 − ΓG| 1 − ΓG = 1 − Γ∗ G |1 − ΓG| (14.7.5) Similarly, we have for the power and traveling waves at port-2: _ a_ 2 b_ 2 _ = 1 _ 1 − |ΓL|2 _ ejφL −ΓLejφL −Γ∗ L e−jφL e−jφL __ a2 b2 _ _ a2 b2 _ = 1 _ 1 − |ΓL|2 _ e−jφL ΓLejφL Γ∗ L e−jφL ejφL __ a_ 2 b_ 2 _ (14.7.6) where ΓL = ZL − Z0 ZL + Z0 , ejφL = |1 − ΓL| 1 − ΓL = 1 − Γ∗ L |1 − ΓL| (14.7.7) The generalized S-parameters are the scattering parameters with respect to the power wave variables, that is, 14.7. Generalized S-Parameters and Power Waves 685 _ b_ 1 b_ 2 _ = _ S_ 11 S_ 12 S_ 21 S_ 22 __ a_ 1 a_ 2 _ ⇒ b_ = S_a_ (14.7.8) To relate S_ to the conventional scattering matrix S, we define the following diagonal matrices: Γ = _ ΓG 0 0 ΓL _ , F= ⎡ ⎢⎢⎢⎣ ejφG _ 1 − |ΓG|2 0 0 ejφL _ 1 − |ΓL|2 ⎤ ⎥⎥⎥⎦ = _ FG 0 0 FL _ (14.7.9) Using these matrices, it follows from Eqs. (14.7.4) and (14.7.6): a_ 1 = FG(a1 − ΓGb1) a_ 2 = FL(a2 − ΓLb2) ⇒ a_ = F(a − Γ b) (14.7.10) b_ 1 = F∗ G(b1 − Γ∗ Ga1) b_ 2 = F∗ L (b2 − Γ∗ L a2) ⇒ b_ = F∗(b − Γ∗a) (14.7.11) Using b = Sa, we find a_ = F(a − Γ b)= F(I − ΓS)a ⇒ a = (I − ΓS)−1F−1a_ b_ = F∗(S − Γ∗)a = F∗(S − Γ∗)(I − ΓS)−1F−1a_ = S_a_ where I is the 2×2 unit matrix. Thus, the generalized S-matrix is: S_ = F∗(S − Γ∗)(I − ΓS)−1F−1 (14.7.12) We note that S_ = S when ZG = ZL = Z0, that is, when ΓG = ΓL = 0. The explicit expressions for the matrix elements of S_ can be derived as follows: S_ 11 = (S11 − Γ∗ G)(1 − S22ΓL)+S21S12ΓL (1 − S11ΓG)(1 − S22ΓL)−S12S21ΓGΓL e−2jφG S_ 22 = (S22 − Γ∗ L )(1 − S11ΓG)+S21S12ΓG (1 − S11ΓG)(1 − S22ΓL)−S12S21ΓGΓL e−2jφL (14.7.13a) S_ 21 = _ 1 − |ΓG|2 S21 _ 1 − |ΓL|2 (1 − S11ΓG)(1 − S22ΓL)−S12S21ΓGΓL e−j(φG+φL) S_ 12 = _ 1 − |ΓL|2 S12 _ 1 − |ΓG|2 (1 − S11ΓG)(1 − S22ΓL)−S12S21ΓGΓL e−j(φL+φG) (14.7.13b) The S_ 11, S_ 22 parameters can be rewritten in terms of the input and output reflection coefficients by using Eq. (14.13.2) and the following factorization identities: (S11 − Γ∗ G)(1 − S22ΓL)+S21S12ΓL = (Γin − Γ∗ G)(1 − S22ΓL) (S22 − Γ∗ L )(1 − S11ΓG)+S21S12ΓG = (Γout − Γ∗ L )(1 − S11ΓG) It then follows from Eq. (14.7.13) that: 686 14. S-Parameters S_ 11 = Γin − Γ∗ G 1 − ΓinΓG e−2jφG, S_ 22 = Γout − Γ∗ L 1 − ΓoutΓL e−2jφL (14.7.14) Therefore, the mismatch factors (14.6.17) are recognized to be: MG = 1 − |S_ 11|2, ML = 1 − |S_ 22|2 (14.7.15) The power flow relations (14.2.1) into and out of the two-port are also valid in terms of the power wave variables. Using Eq. (14.7.2), it can be shown that: Pin = 1 2 Re[V∗ 1 I1]= 1 2 |a_ 1|2 − 1 2 |b_ 1|2 PL = 1 2 Re[V∗ 2 I2]= 1 2 |b_ 2|2 − 1 2 |a_ 2|2 (14.7.16) In the definitions (14.7.1), the impedances ZG,ZL are arbitrary normalization parameters. However, if the two-port is actually connected to a generator VG with impedance ZG and a load ZL, then the power waves take particularly simple forms. It follows from Fig. 14.1.4 that VG = V1+ZGI1 and V2 = ZLI2. Therefore, definitions Eq. (14.7.1) give: a_ 1 = V1 + ZGI1 2 _ RG = VG 2 _ RG a_ 2 = V2 − ZLI2 2 _ RL = 0 b_ 2 = V2 + Z∗ L I2 2 _ RL = ZL + Z∗ L 2 _ RL I2 = 2RL 2 _ RL I2 = _ RL I2 (14.7.17) It follows that the available power from the generator and the power delivered to the load are given simply by: PavG = |VG|2 8RG = 1 2 |a_ 1|2 PL = 1 2 RL|I2|2 = 1 2 |b_ 2|2 (14.7.18) Because a_ 2 = 0, the generalized scattering matrix gives, b_ 1 = S_ 11a_ 1 and b_ 2 = S_ 21a_ 1. The power expressions (14.7.16) then become: Pin = 1 2 |a_ 1|2 − 1 2 |b_ 1|2 = _ 1 − |S_ 11|2_1 2 |a_ 1|2 = _ 1 − |S_ 11|2_ PavG PL = 1 2 |b_ 2|2 − 1 2 |a_ 2|2 = 1 2 |b_ 2|2 = |S_ 21|2 1 2 |a_ 1|2 = |S_ 21|2PavG (14.7.19) It follows that the transducer and operating power gains are: GT = PL PavG = |S_ 21|2, Gp = PL Pin = |S_ 21|2 1 − |S_ 11|2 (14.7.20) 14.8. Simultaneous Conjugate Matching 687 These also follow from the explicit expressions (14.7.13) and Eqs. (14.6.10) and (14.6.11). We can also express the available power gain in terms of the generalized S-parameters, that is, Ga = |S_ 21|2/ _ 1 − |S_ 22|2_ . Thus, we summarize: GT = |S_ 21|2, Ga = |S_ 21|2 1 − |S_ 22|2, Gp = |S_ 21|2 1 − |S_ 11|2 (14.7.21) When the load and generator are matched to the network, that is, Γin = Γ∗ G and ΓL = Γ∗ out, the generalized reflections coefficients vanish, S_ 11 = S_ 22 = 0, making all the gains equal to each other. 14.8 Simultaneous Conjugate Matching We saw that the transducer, available, and operating power gains become equal to the maximum available gain GMAG when both the generator and the load are conjugately matched to the two-port, that is, Γin = Γ∗ G and ΓL = Γ∗ out. Using Eq. (14.5.8), these conditions read explicitly: Γ∗ G = S11 + S12S21ΓL 1 − S22ΓL = S11 − ΔΓL 1 − S22ΓL Γ∗ L = S22 + S12S21ΓG 1 − S22ΓG = S22 − ΔΓG 1 − S11ΓG (14.8.1) Assuming a bilateral two-port, Eqs. (14.8.1) can be solved in the two unknowns ΓG, ΓL (eliminating one of the unknowns gives a quadratic equation for the other.) The resulting solutions can be expressed in terms of the parameters (14.5.1): ΓG = B1 ∓ _ B21 − 4|C1|2 2C1 ΓL = B2 ∓ _ B22 − 4|C2|2 2C2 (simultaneous conjugate match) (14.8.2) where the minus signs are used when B1 > 0 and B2 > 0, and the plus signs, otherwise. A necessary and sufficient condition for these solutions to have magnitudes |ΓG| < 1 and |ΓL| < 1 is that the Rollett stability factor be greater than unity, K > 1. This is satisfied when the two-port is unconditionally stable, which implies that K > 1 and B1 > 0, B2 > 0. A conjugate match exists also when the two-port is potentially unstable, but with K > 1. Necessarily, this means that B1 < 0, B2 < 0, and also |Δ| > 1. Such cases are rare in practice. For example, most microwave transistors have either K > 1 and are stable, or, they are potentially unstable with K < 1 and |Δ| < 1. If the two-port is unilateral, S12 = 0, then the two equations (14.8.1) decouple, so that the optimum conjugately matched terminations are: ΓG = S∗ 11, ΓL = S∗ 22 (unilateral conjugate match) (14.8.3) 688 14. S-Parameters The MATLAB function smatch implements Eqs. (14.8.2). It works only if K > 1. Its usage is as follows: [gG,gL] = smatch(S); % conjugate matched terminations ΓG, ΓL To realize such optimum conjugately matched terminations, matching networks must be used at the input and output of the two-port as shown in Fig. 14.8.1. The input matching network can be thought as being effectively connected to the impedance Zin = Z∗ G at its output terminals. It must transform Zin into the actual impedance of the connected generator, typically, Z0 = 50 ohm. The output matching network must transform the actual load impedance, here Z0, into the optimum load impedance ZL = Z∗ out. Fig. 14.8.1 Input and output matching networks. The matching networks may be realized in several possible ways, as discussed in Chap. 13. Stub matching, quarter-wavelength matching, or lumped L-section or Π- section networks may be used. In designing the matching networks, it proves convenient to first design the reverse network as mentioned in Sec. 13.13. Fig. 14.8.2 shows the procedure for designing the output matching network using a reversed stub matching transformer or a reversed quarter-wave transformer with a parallel stub. In both cases the reversed network is designed to transform the load impedance Z∗ L into Z0. Example 14.8.1: A microwave transistor amplifier uses the Hewlett-Packard AT-41410 NPN bipolar transistor having S-parameters at 2 GHz : S11 = 0.61∠165o, S21 = 3.72∠59o, S12 = 0.05∠42o, S22 = 0.45∠−48o Determine the optimum conjugately matched source and load terminations, and design appropriate input and output matching networks. Solution: This is the continuation of Example 14.6.1. The transistor is stable with K = 1.1752 and |Δ| = 0.1086. The function smatch gives: [ΓG, ΓL]= smatch(S) ⇒ ΓG = 0.8179∠−162.6697o, ΓL = 0.7495∠52.5658o The corresponding source, load, input, and output impedances are (with Z0 = 50): ZG = Z∗ in = 5.1241 − 7.5417j Ω, ZL = Z∗ out = 33.6758 + 91.4816j Ω 14.8. Simultaneous Conjugate Matching 689 Fig. 14.8.2 Two types of output matching networks and their reversed networks.. Fig. 14.8.3 Optimum load and source reflection coefficients. The locations of the optimum reflection coefficients on the Smith chart are shown in Fig. 14.8.3. For comparison, the unilateral solutions of Eq. (14.8.3) are also shown. We consider three types of matching networks: (a) microstrip single-stub matching networks with open shunt stubs, shown in Fig. 14.8.4, (b) microstrip quarter-wavelength matching networks with open λ/8 or 3λ/8 stubs, shown in Fig. 14.8.5, and (c) L-section matching networks, shown in 14.8.6. Fig. 14.8.4 Input and output stub matching networks. 690 14. S-Parameters In Fig. 14.8.4, the input stub must transform Zin to Z0. It can be designed with the help of the function stub1, which gives the two solutions: dl = stub1(Zin/Z0, ’po’)= _ 0.3038 0.4271 0.1962 0.0247 _ We choose the lower one, which has the shortest lengths. Thus, the stub length is d = 0.1962λ and the segment length l = 0.0247λ. Both segments can be realized with microstrips of characteristic impedance Z0 = 50 ohm. Similarly, the output matching network can be designed by: dl = stub1(Zout/Z0, ’po’)= _ 0.3162 0.1194 0.1838 0.2346 _ Again, we choose the lower solutions, d = 0.1838λ and l = 0.2346λ. The solutions using shorted shunt stubs are: stub1(Zin/Z0)= _ 0.0538 0.4271 0.4462 0.0247 _ , stub1(Zout/Z0)= _ 0.0662 0.1194 0.4338 0.2346 _ Using microstrip lines with alumina substrate ( r = 9.8), we obtain the following values for the width-to-height ratio, effective permittivity, and wavelength: u = w h = mstripr( r,Z0)= 0.9711 eff = mstripa( r, u)= 6.5630 λ = √λ0 eff = 5.8552 cm where λ0 = 15 cm is the free-space wavelength at 2 GHz. It follows that the actual segment lengths are d = 1.1486 cm, l = 0.1447 cm for the input network, and d = 1.0763 cm, l = 1.3734 cm for the output network. In the quarter-wavelength method shown in Fig. 14.8.5, we use the function qwt2 to carry out the design of the required impedances of the microstrip segments. We have for the input and output networks: [Z1,Z2]= qwt2(Zin,Z0)= [28.4817,−11.0232] Ω [Z1,Z2]= qwt2(Zout,Z0)= [118.7832, 103.8782] Ω For the input case, we find Z2 = −11.0232 Ω, which means that we should use either a 3λ/8-shorted stub or a λ/8-opened one. We choose the latter. Similarly, for the output case, we have Z2 = 103.8782 Ω, and we choose a 3λ/8-opened stub. The parameters of each microstrip segment are: Z1 = 28.4817 Ω, u= 2.5832, eff = 7.2325, λ= 5.578 cm, λ/4 = 1.394 cm Z2 = 11.0232 Ω, u= 8.9424, eff = 8.2974, λ= 5.207 cm, λ/8 = 0.651 cm Z1 = 118.7832 Ω, u= 0.0656, eff = 5.8790, λ= 6.186 cm, λ/4 = 1.547 cm Z2 = 103.8782 Ω, u= 0.1169, eff = 7.9503, λ= 6.149 cm, 3λ/8 = 2.306 cm 14.8. Simultaneous Conjugate Matching 691 Fig. 14.8.5 Quarter-wavelength matching networks with λ/8-stubs. Fig. 14.8.6 Input and output matching with L-sections. Finally, the designs using L-sections shown in Fig. 14.8.6, can be carried out with the help of the function lmatch. We have the dual solutions for the input and output networks: [X1,X2]= lmatch(Z0,Zin, ’n’)= _ 16.8955 −22.7058 −16.8955 7.6223 _ [X1,X2]= lmatch(Zout,Z0, ’n’)= _ 57.9268 −107.7472 502.4796 7.6223 _ According to the usage of lmatch, the output network transforms Z0 into Z∗ out, but that is equal to ZL as required. Choosing the first rows as the solutions in both cases, the shunt part X1 will be inductive and the series part X2, capacitive. At 2 GHz, we find the element values: L1 = X1 ω = 1.3445 nH, C1 = − 1 ωX2 = 3.5047 pF L2 = X1 ω = 4.6097 nH, C2 = − 1 ωX2 = 0.7386 pF The output network, but not the input one, also admits a reversed L-section solution: [X1,X2]= lmatch(Zout,Z0, ’r’)= _ 71.8148 68.0353 −71.8148 114.9280 _ The essential MATLAB code used to generate the above results was as follows: Z0 = 50; f = 2; w=2*pi*f; la0 = 30/f; er = 9.8; % f in GHz S = smat([0.61 165 3.72 59 0.05 42 0.45 -48]); % S-matrix 692 14. S-Parameters [gG,gL] = smatch(S); % simultaneous conjugate match smith; % draw Fig. 14.8.3 plot(gG, ’.’); plot(conj(S(1,1)), ’o’); plot(gL, ’.’); plot(conj(S(2,2)), ’o’); ZG = g2z(gG,Z0); Zin = conj(ZG); ZL = g2z(gL,Z0); Zout = conj(ZL); dl = stub1(Zin/Z0, ’po’); % single-stub design dl = stub1(Zout/Z0, ’po’); u = mstripr(er,Z0); % microstrip w/h ratio eff = mstripa(er,u); % effective permittivity la = la0/sqrt(eff); % wavelength within microstrip [Z1,Z2] = qwt2(Zin, Z0); % quarter-wavelength with λ/8 stub [Z1,Z2] = qwt2(Zout, Z0); X12 = lmatch(Z0,Zin,’n’); L1 = X12(1,1)/w; C1 = -1/(w * X12(1,2))*1e3; X12 = lmatch(Zout,Z0,’n’); L2 = X12(1,1)/w; C2 = -1/(w * X12(1,2))*1e3; X12 = lmatch(Zout,Z0,’r’); % L,C in units of nH and pF One could replace the stubs with balanced stubs, as discussed in Sec. 13.9, or use Π- or T-sections instead of L-sections. __ 14.9 Power Gain Circles For a stable two-port, the maximum transducer gain is achieved at single pair of points ΓG, ΓL. When the gain G is required to be less than GMAG, there will be many possible pairs ΓG, ΓL at which the gain G is realized. The locus of such points ΓG and ΓL on the Γ-plane is typically a circle of the form: |Γ − c| = r (14.9.1) where c, r are the center and radius of the circle and depend on the desired value of the gain G. In practice, several types of such circles are used, such as unilateral, operating, and available power gain circles, as well as constant noise figure circles, constant SWR circles, and others. The gain circles allow one to select appropriate values for ΓG, ΓL that, in addition to providing the desired gain, also satisfy other requirements, such as striking a balance between minimizing the noise figure and maximizing the gain. The MATLAB function sgcirc calculates the stability circles as well as the operating, available, and unilateral gain circles. Its complete usage is: [c,r] = sgcirc(S,’s’); % source stability circle [c,r] = sgcirc(S,’l’); % load stability circle [c,r] = sgcirc(S,’p’,G); % operating power gain circle [c,r] = sgcirc(S,’a’,G); % available power gain circle 14.10. Unilateral Gain Circles 693 [c,r] = sgcirc(S,’ui’,G); % unilateral input gain circle [c,r] = sgcirc(S,’uo’,G); % unilateral output gain circle where in the last four cases G is the desired gain in dB. 14.10 Unilateral Gain Circles We consider only the unconditionally stable unilateral case, which has |S11| < 1 and |S22| < 1. The dependence of the transducer power gain on ΓG and ΓL decouples and the value of the gain may be adjusted by separately choosing ΓG and ΓL. We have from Eq. (14.6.14): GT = 1 − |ΓG|2 |1 − S11ΓG|2 |S21|2 1 − |ΓL|2 |1 − S22ΓL|2 = GG |S21|2 GL (14.10.1) The input and output gain factors GG,GL satisfy the inequalities (14.6.24). Concentrating on the output gain factor, the corresponding gain circle is obtained as the locus of points ΓL that will lead to a fixed value, say GL = G, which necessarily must be less than the maximum G2 given in Eq. (14.6.23), that is, 1 − |ΓL|2 |1 − S22ΓL|2 = G ≤ G2 = 1 1 − |S22|2 (14.10.2) Normalizing the gain G to its maximum value g = G/G2 = G _ 1 − |S22|2_ , we may rewrite (14.10.2) in the form: _ 1 − |ΓL|2__ 1 − |S22|2_ |1 − S22ΓL|2 = g ≤ 1 (14.10.3) This equation can easily be rearranged into the equation of a circle |ΓL−c| = r, with center and radius given by: c = gS∗ 22 1 − (1 − g)|S22|2, r= _ 1 − g _ 1 − |S22|2_ 1 − (1 − g)|S22|2 (14.10.4) When g = 1 or G = G2, the gain circle collapses onto a single point, that is, the optimum point ΓL = S∗ 22. Similarly, we find for the constant gain circles of the input gain factor: c = gS∗ 11 1 − (1 − g)|S11|2, r= _ 1 − g _ 1 − |S11|2_ 1 − (1 − g)|S11|2 (14.10.5) where here, g = G/G1 = G _ 1 − |S11|2_ and the circles are |ΓG − c| = r. Both sets of c, r satisfy the conditions |c| < 1 and |c| +r < 1, the latter implying that the circles lie entirely within the unit circle |Γ| < 1, that is, within the Smith chart. Example 14.10.1: A unilateral microwave transistor has S-parameters: S11 = 0.8∠120o, S21 = 4∠60o, S12 = 0, S22 = 0.2∠−30o 694 14. S-Parameters The unilateral MAG and the maximum input and output gains are obtained as follows: GMAG,u = sgain(S, ’u’)= 16.66 dB G1 = sgain(S, ’ui’)= 4.44 dB G2 = sgain(S, ’uo’)= 0.18 dB Most of the gain is accounted for by the factor |S21|2, which is 12.04 dB. The constant input gain circles for GG = 1, 2, 3 dB are shown in Fig. 14.10.1. Their centers lie along the ray to S∗ 11. For example, the center and radius of the 3-dB case were computed by [c3, r3]= sgcirc(S, ’ui’, 3) ⇒ c3 = 0.701∠−120o, r3 = 0.233 Fig. 14.10.1 Unilateral input gain circles. Because the output does not provide much gain, we may choose the optimum value ΓL = S∗ 22 = 0.2∠30o. Then, with any point ΓG along the 3-dB input gain circle the total transducer gain will be in dB: GT = GG + |S21|2 + GL = 3 + 12.04 + 0.18 = 15.22 dB Points along the 3-dB circle are parametrized as ΓG = c3 + r3ejφ, where φ is any angle. Choosing φ = arg(S∗ 11)−π will correspond to the point on the circle that lies closest to the origin, that is, ΓG = 0.468∠−120o, as shown in Fig. 14.10.1. The corresponding generator and load impedances will be: ZG = 69.21 + 14.42j Ω, ZL = 23.15 − 24.02j Ω The MATLAB code used to generate these circles was: S = smat([0.8, 120, 4, 60, 0, 0, 0.2, -30]); [c1,r1] = sgcirc(S,’ui’,1); [c2,r2] = sgcirc(S,’ui’,2); [c3,r3] = sgcirc(S,’ui’,3); smith; smithcir(c1,r1); smithcir(c2,r2); smithcir(c3,r3); c = exp(-j*angle(S(1,1))); line([0,real(c)], [0,imag(c)]); 14.11. Operating and Available Power Gain Circles 695 gG = c3 – r3*exp(j*angle(c3)); plot(conj(S(1,1)),’.’); plot(conj(S(2,2)),’.’); plot(gG,’.’); The input and output matching networks can be designed using open shunt stubs as in Fig. 14.8.4. The stub lengths are found to be (with Z0 = 50 Ω): dl = stub1(Z∗ G/Z0, ’po’)= _ 0.3704 0.3304 0.1296 0.0029 _ dl = stub1(Z∗ L /Z0, ’po’)= _ 0.4383 0.0994 0.0617 0.3173 _ Choosing the shortest lengths, we have for the input network d = 0.1296λ, l = 0.0029λ, and for the output network, d = 0.0617λ, l = 0.3173λ. Fig. 14.10.2 depicts the complete matching circuit. __ Fig. 14.10.2 Input and output stub matching networks. 14.11 Operating and Available Power Gain Circles Because the transducer power gain GT depends on two independent parameters—the source and load reflection coefficients—it is difficult to find the simultaneous locus of points for ΓG, ΓL that will result in a given value for the gain. If the generator is matched, Γin = Γ∗ G, then the transducer gain becomes equal to the operating gain GT = Gp and depends only on the load reflection coefficient ΓL. The locus of points ΓL that result in fixed values of Gp are the operating power gain circles. Similarly, the available power gain circles are obtained by matching the load end, ΓL = Γ∗ out, and varying ΓG to achieve fixed values of the available power gain. Using Eqs. (14.6.11) and (14.5.8), the conditions for achieving a constant value, say G, for the operating or the available power gains are: Gp = 1 1 − |Γin|2 |S21|2 1 − |ΓL|2 |1 − S22ΓL|2 = G, Γ∗ G = Γin = S11 − ΔΓL 1 − S22ΓL Ga = 1 − |ΓG|2 |1 − S11ΓG|2 |S21|2 1 1 − |Γout|2 = G, Γ∗ L = Γout = S22 − ΔΓG 1 − S11ΓG (14.11.1) 696 14. S-Parameters We consider the operating gain first. Defining the normalized gain g = G/|S21|2, substituting Γin, and using the definitions (14.5.1), we obtain the condition: g = 1 − |ΓL|2 |1 − S22ΓL|2 − |S11 − ΔΓL|2 = 1 − |ΓL|2 _ |S22|2 − |Δ|2 _ |ΓL|2 − (S22 − ΔS∗ 11)ΓL − (S∗ 22 − Δ∗S11)Γ∗ L + 1 − |S11|2 = 1 − |ΓL|2 D2|ΓL|2 − C2ΓL − C∗ 2 Γ∗ L + 1 − |S11|2 This can be rearranged into the form: |ΓL|2 − gC2 1 + gD2 ΓL − gC∗ 2 1 + gD2 Γ∗ L = 1 − g _ 1 − |S11|2_ 1 + gD2 and then into the circle form: _____ ΓL − gC∗ 2 1 + gD2 _____ 2 = g2|C2|2 (1 + gD2)2 + 1 − g _ 1 − |S11|2_ 1 + gD2 Using the identities (14.5.2) and 1 − |S11|2 = 2K|S12S21| + D2, which follows from (14.5.1), the right-hand side of the above circle form can be written as: g2|C2|2 (1 + gD2)2 + 1 − g _ 1 − |S11|2_ 1 + gD2 = g2|S12S21|2 − 2gK|S12S21| + 1 (1 + gD2)2 (14.11.2) Thus, the operating power gain circle will be |ΓL − c|2 = r2 with center and radius: c = gC∗ 2 1 + gD2 , r= _ g2|S12S21|2 − 2gK|S12S21| + 1 |1 + gD2| (14.11.3) The points ΓL on this circle result into the value Gp = G for the operating gain. Such points can be parametrized as ΓL = c + rejφ, where 0 ≤ φ ≤ 2π. As ΓL traces this circle, the conjugately matched source coefficient ΓG = Γ∗ in will also trace a circle because Γin is related to ΓL by the bilinear transformation (14.5.8). In a similar fashion, we find the available power gain circles to be |ΓG − c|2 = r2, where g = G/|S21|2 and: c = gC∗ 1 1 + gD1 , r= _ g2|S12S21|2 − 2gK|S12S21| + 1 |1 + gD1| (14.11.4) We recall from Sec. 14.5 that the centers of the load and source stability circles were cL = C∗ 2 /D2 and cG = C∗ 1 /D1. It follows that the centers of the operating power gain circles are along the same ray as cL, and the centers of the available gain circles are along the same ray as cG. 14.11. Operating and Available Power Gain Circles 697 For an unconditionally stable two-port, the gain G must be 0 ≤ G ≤ GMAG, with GMAG given by Eq. (14.6.20). It can be shown easily that the quantities under the square roots in the definitions of the radii r in Eqs. (14.11.3) and (14.11.4) are non-negative. The gain circles lie inside the unit circle for all such values of G. The radii r vanish when G = GMAG, that is, the circles collapse into single points corresponding to the simultaneous conjugate matched solutions of Eq. (14.8.2). The MATLAB function sgcirc calculates the center and radii c, r of the operating and available power gain circles. It has usage, where G must be entered in dB: [c,r] = sgcirc(S,’p’,G); operating power gain circle [c,r] = sgcirc(S,’a’,G); available power gain circle Example 14.11.1: A microwave transistor amplifier uses the Hewlett-Packard AT-41410 NPN bipolar transistor with the following S-parameters at 2 GHz : S11 = 0.61∠165o, S21 = 3.72∠59o, S12 = 0.05∠42o, S22 = 0.45∠−48o Calculate GMAG and plot the operating and available power gain circles for G = 13, 14, 15 dB. Then, design source and load matching circuits for the case G = 15 dB by choosing the reflection coefficient that has the smallest magnitude. Solution: The MAG was calculated in Example 14.6.1, GMAG = 16.18 dB. The gain circles and the corresponding load and source stability circles are shown in Fig. 14.11.1. The operating gain and load stability circles were computed and plotted by the MATLAB statements: [c1,r1] = sgcirc(S,’p’,13); % c1 = 0.4443∠52.56o, r1 = 0.5212 [c2,r2] = sgcirc(S,’p’,14); % c2 = 0.5297∠52.56o, r2 = 0.4205 [c3,r3] = sgcirc(S,’p’,15); % c3 = 0.6253∠52.56o, r3 = 0.2968 [cL,rL] = sgcirc(S,’l’); % cL = 2.0600∠52.56o, rL = 0.9753 smith; smithcir(cL,rL,1.7); % display portion of circle with |ΓL| ≤ 1.7 smithcir(c1,r1); smithcir(c2,r2); smithcir(c3,r3); Fig. 14.11.1 Operating and available power gain circles. 698 14. S-Parameters The gain circles lie entirely within the unit circle, for example, we have r3+|c3| = 0.9221 < 1, and their centers lie along the ray of cL. As ΓL traces the 15-dB circle, the corresponding ΓG = Γ∗ in traces its own circle, also lying within the unit circle. The following MATLAB code computes and adds that circle to the above Smith chart plots: phi = linspace(0,2*pi,361); % equally spaced angles at 1o intervals gammaL = c3 + r3 * exp(j*phi); % points on 15-dB operating gain circle gammaG = conj(gin(S,gammaL)); % circle of conjugate matched source points plot(gammaG); In particular, the point ΓL on the 15-dB circle that lies closest to the origin is ΓL = c3 − r3ej arg c3 = 0.3285∠52.56o. The corresponding matched load will be ΓG = Γ∗ in = 0.6805∠−163.88o. These and the corresponding source and load impedances were computed by the MATLAB statements: gL = c3 – r3*exp(j*angle(c3)); zL = g2z(gL); gG = conj(gin(S,gL)); zG = g2z(gG); The source and load impedances normalized to Z0 = 50 ohm are: zG = ZG Z0 = 0.1938 − 0.1363j , zL = ZL Z0 = 1.2590 + 0.7361j The matching circuits can be designed in a variety of ways as in Example 14.8.1. Using open shunt stubs, we can determine the stub and line segment lengths with the help of the function stub1: dl = stub1(z∗ G, ’po’)= _ 0.3286 0.4122 0.1714 0.0431 _ dl = stub1(z∗ L , ’po’)= _ 0.4033 0.0786 0.0967 0.2754 _ In both cases, we may choose the lower solutions as they have shorter total length d + l. The available power gain circles can be determined in a similar fashion with the help of the MATLAB statements: [c1,r1] = sgcirc(S,’a’,13); % c1 = 0.5384∠−162.67o, r1 = 0.4373 [c2,r2] = sgcirc(S,’a’,14); % c2 = 0.6227∠−162.67o, r2 = 0.3422 [c3,r3] = sgcirc(S,’a’,15); % c3 = 0.7111∠−162.67o, r3 = 0.2337 [cG,rG] = sgcirc(S,’s’); % cG = 1.5748∠−162.67o, rG = 0.5162 smith; smithcir(cG,rG); % plot entire source stability circle smithcir(c1,r1); smithcir(c2,r2); smithcir(c3,r3); Again, the circles lie entirely within the unit circle. As ΓG traces the 15-dB circle, the corresponding matched load ΓL = Γ∗ out traces its own circle on the Γ-plane. It can be plotted with: phi = linspace(0,2*pi,361); % equally spaced angles at 1o intervals gammaG = c3 + r3 * exp(j*phi); % points on 15-dB available gain circle gammaL = conj(gout(S,gammaG)); % circle of conjugate matched loads plot(gammaL); 14.11. Operating and Available Power Gain Circles 699 In particular, the point ΓG = c3 − r3ej arg c3 = 0.4774∠−162.67o lies closest to the origin. The corresponding matched load will have ΓL = Γ∗ out = 0.5728∠50.76o. The resulting normalized impedances are: zG = ZG Z0 = 0.3609 − 0.1329j , zL = ZL Z0 = 1.1135 + 1.4704j and the corresponding stub matching networks will have lengths: stub1(z∗ G, ’po’)= _ 0.3684 0.3905 0.1316 0.0613 _ , stub1(z∗ L , ’po’)= _ 0.3488 0.1030 0.1512 0.2560 _ The lower solutions have the shortest lengths. For both the operating and available gain cases, the stub matching circuits will be similar to those in Fig. 14.8.4. __ When the two-port is potentially unstable (but with |S11| < 1 and |S22| < 1,) the stability circles intersect with the unit-circle, as shown in Fig. 14.5.2. In this case, the operating and available power gain circles also intersect the unit-circle and at the same points as the stability circles. We demonstrate this in the specific case of K < 1, |S11| < 1, |S22| < 1, but with D2 > 0, an example of which is shown in Fig. 14.11.2. The intersection of an operating gain circle with the unit-circle is obtained by setting |ΓL| = 1 in the circle equation |ΓL − c| = r. Writing ΓL = ejθL and c = |c|ejθc , we have: r2 = |ΓL − c|2 = 1 − 2|c| cos(θL − θc)+|c|2 ⇒ cos(θL − θc)= 1 + |c|2 − r2 2|c| Similarly, the intersection of the load stability circle with the unit-circle leads to the relationship: r2L = |ΓL − cL|2 = 1 − 2|cL| cos(θL − θcL)+|cL|2 ⇒ cos(θL − θcL)= 1 + |cL|2 − r2L 2|cL| Because c = gC∗ 2 /(1 + gD2), cL = C∗ 2 /D2, and D2 > 0, it follows that the phase angles of c and cL will be equal, θc = θcL . Therefore, in order for the load stability circle and the gain circle to intersect the unit-circle at the same ΓL = ejθL , the following condition must be satisfied: cos(θL − θc)= 1 + |c|2 − r2 2|c| = 1 + |cL|2 − r2L 2|cL| (14.11.5) Using the identities 1 − |S11|2 = B2 − D2 and 1 − |S11|2 = _ |cL|2 − r2L _ D2, which follow from Eqs. (14.5.1) and (14.5.6), we obtain: 1 + |cL|2 − r2L 2|cL| = 1 + (B2 − D2)/D2 2|C2|/|D2| = B2 2|C2| where we used D2 > 0. Similarly, Eq. (14.11.2) can be written in the form: r2 = |c|2 + 1 − g _ 1 − |S11|2_ 1 + gD2 ⇒ |c|2 − r2 = g _ 1 − |S11|2_ − 1 1 + gD2 = g(B2 − D2)−1 1 + gD2 700 14. S-Parameters Therefore, we have: 1 + |c|2 − r2 2|c| = 1 + _ g(B2 − D2)−1 _ /(1 + gD2) 2g|C2|/|1 + gD2| = B2 2|C2| Thus, Eq. (14.11.5) is satisfied. This condition has two solutions for θL that correspond to the two points of intersection with the unit-circle. When D2 > 0, we have arg c = argC∗ 2 = −argC2. Therefore, the two solutions for ΓL = ejθL will be: ΓL = ejθL, θL = −arg(C2)±acos _ B2 2|C2| _ (14.11.6) Similarly, the points of intersection of the unit-circle and the available gain circles and source stability circle are: ΓG = ejθG, θG = −arg(C1)±acos _ B1 2|C1| _ (14.11.7) Actually, these expressions work also when D2 < 0 or D1 < 0. Example 14.11.2: The microwave transistor Hewlett-Packard AT-41410 NPN is potentially unstable at 1 GHz with the following S-parameters : S11 = 0.6∠−163o, S21 = 7.12∠86o, S12 = 0.039∠35o, S22 = 0.50∠−38o Calculate GMSG and plot the operating and available power gain circles for G = 20, 21, 22 dB. Then, design source and load matching circuits for the 22-dB case by choosing the reflection coefficients that have the smallest magnitudes. Solution: The MSG computed from Eq. (14.6.21) is GMSG = 22.61 dB. Fig. 14.11.2 depicts the operating and available power gain circles as well as the load and source stability circles. The stability parameters are: K = 0.7667, μ1 = 0.8643, |Δ| = 0.1893,D1 = 0.3242,D2 = 0.2142. The computations and plots are done with the following MATLAB code:† S = smat([0.60, -163, 7.12, 86, 0.039, 35, 0.50, -38]); % S-parameters [K,mu,D,B1,B2,C1,C2,D1,D2] = sparam(S); % stability parameters Gmsg = db(sgain(S,’msg’)); % GMSG = 22.61 dB % operating power gain circles: [c1,r1] = sgcirc(S,’p’,20); % c1 = 0.6418∠50.80o, r1 = 0.4768 [c2,r2] = sgcirc(S,’p’,21); % c2 = 0.7502∠50.80o, r2 = 0.4221 [c3,r3] = sgcirc(S,’p’,22); % c3 = 0.8666∠50.80o, r3 = 0.3893 % load and source stability circles: [cL,rL] = sgcirc(S,’l’); % cL = 2.1608∠50.80o, rL = 1.2965 [cG,rG] = sgcirc(S,’s’); % cG = 1.7456∠171.69o, rG = 0.8566 smith; smithcir(cL,rL,1.5); smithcir(cG,rG,1.5); % plot Smith charts smithcir(c1,r1); smithcir(c2,r2); smithcir(c3,r3); % plot gain circles gL = c3 – r3*exp(j*angle(c3)); % ΓL of smallest magnitude gG = conj(gin(S,gL)); % corresponding matched ΓG plot(gL,’.’); plot(gG,’.’); †The function db converts absolute scales to dB. The function ab converts from dB to absolute units. 14.12. Noise Figure Circles 701 Fig. 14.11.2 Operating and available power gain circles. % available power gain circles: [c1,r1] = sgcirc(S,’a’,20); % c1 = 0.6809∠171.69o, r1 = 0.4137 [c2,r2] = sgcirc(S,’a’,21); % c2 = 0.7786∠171.69o, r2 = 0.3582 [c3,r3] = sgcirc(S,’a’,22); % c3 = 0.8787∠171.69o, r3 = 0.3228 figure; smith; smithcir(cL,rL,1.5); smithcir(cG,rG,1.5); smithcir(c1,r1); smithcir(c2,r2); smithcir(c3,r3); gG = c3 – r3*exp(j*angle(c3)); % ΓG of smallest magnitude gL = conj(gout(S,gG)); % corresponding matched ΓL plot(gL,’.’); plot(gG,’.’); Because D1 > 0 and D2 > 0, the stability regions are the portions of the unit-circle that lie outside the source and load stability circles. We note that the operating gain circles intersect the unit-circle at exactly the same points as the load stability circle, and the available gain circles intersect it at the same points as the source stability circle. The value of ΓL on the 22-dB operating gain circle that lies closest to the origin is ΓL = c3 − r3ej arg c3 = 0.4773∠50.80o and the corresponding matched source is ΓG = Γ∗ in = 0.7632∠167.69o. We note that both ΓL and ΓG lie in their respective stability regions. For the 22-dB available gain circle (also denoted by c3, r3), the closest ΓG to the origin will be ΓG = c3 −r3ej arg c3 = 0.5559∠171.69o with a corresponding matched load ΓL = Γ∗ out = 0.7147∠45.81o. Again, both ΓL, ΓG lie in their stable regions. Once the ΓG, ΓL have been determined, the corresponding matching input and output networks can be designed with the methods of Example 14.8.1. __ 14.12 Noise Figure Circles Every device is a source of internally generated noise. The noise entering the device and the internal noise must be added to obtain the total input system noise. If the device is an amplifier, the total system noise power will be amplified at the output by the gain of the device. If the output load is matched, this gain will be the available gain. 702 14. S-Parameters The internally generated noise is quantified in practice either by the effective noise temperature Te, or by the noise figure F of the device. The internal noise power is given by Pn = kTeB, where k is the Boltzmann constant and B the bandwidth in Hz. These concepts are discussed further in Sec. 16.8. The relationship between Te and F is defined in terms of a standard reference temperature T0 = 290 K (degrees Kelvin): F = 1 + Te T0 (14.12.1) The noise figure is usually quoted in dB, FdB = 10 log10 F. Because the available gain of a two-port depends on the source impedance ZG, or the source reflection coefficient ΓG, so will the noise figure. The optimum source impedance ZGopt corresponds to the minimum noise figure Fmin that can be achieved by the two-port. For other values of ZG, the noise figure F is greater than Fmin and is given by [117–120]: F = Fmin + Rn RG|ZGopt|2 |ZG − ZGopt|2 (14.12.2) where RG = Re(ZG) and Rn is an equivalent noise resistance. We note that F = Fmin when ZG = ZGopt. Defining the normalized noise resistance rn = Rn/Z0, where Z0 = 50 ohm, we may write Eq. (14.12.2) in terms of the corresponding source reflection coefficients: F = Fmin + 4rn |ΓG − ΓGopt|2 |1 + ΓGopt|2 _ 1 − |ΓG|2 _ (14.12.3) The parameters Fmin, rn, and ΓGopt characterize the noise properties of the two-port and are usually known. In designing low-noise microwave amplifiers, one would want to achieve the minimum noise figure and the maximum gain. Unfortunately, the optimum source reflection coefficient ΓGopt does not necessarily correspond to the maximum available gain. The noise figure circles and the available gain circles are useful tools that allow one to obtain a balance between low noise and high gain designs. The noise figure circles are the locus of points ΓG that correspond to fixed values of F. They are obtained by rewriting Eq. (14.12.3) as the equation of a circle |ΓG − c|2 = r2. We write Eq. (14.12.3) in the form: |ΓG − ΓGopt|2 1 − |ΓG|2 = N, where N = (F − Fmin)|1 + ΓGopt|2 4rn (14.12.4) which can be rearranged into the circle equation: ____ ΓG − ΓGopt N + 1 ____ 2 = N2 + N _ 1 − |ΓGopt|2_ (N + 1)2 Thus, the center and radius of the noise figure circle are: c = ΓGopt N + 1 , r= _ N2 + N _ 1 − |ΓGopt|2 _ N + 1 (14.12.5) 14.12. Noise Figure Circles 703 The MATLAB function nfcirc implements Eq. (14.12.5). Its inputs are the noise parameters Fmin, rn, ΓGopt, and the desired value of F in dB, and its outputs are c, r: [c,r] = nfcirc(F,Fmin,rn,gGopt); % noise figure circles The function nfig implements Eq. (14.12.3). Its inputs are Fmin, rn, ΓGopt, and a vector of values of ΓG, and its output is the corresponding vector of values of F: F = nfig(Fmin, rn, gGopt, gG); % calculate noise figure F in dB Example 14.12.1: The microwave transistor of Example 14.11.1 has the following noise parameters at 2 GHz : Fmin = 1.6 dB, rn = 0.16, and ΓGopt = 0.26∠172o. Determine the matched load ΓLopt corresponding to ΓGopt and calculate the available gain. Then, plot the noise figure circles for F = 1.7, 1.8, 1.9, 2.0 dB. For the 1.8-dB noise figure circle, determine ΓG, ΓL that correspond to the maximum possible available gain and design appropriate input and output matching networks. Solution: The conjugate matched load corresponding to ΓGopt is: ΓLopt = Γ∗ out = _ S22 − ΔΓGopt 1 − S11ΓGopt _∗ = 0.4927∠52.50o The value of the available gain at ΓGopt is Ga,opt = 13.66 dB. This is to be compared with the MAG of 16.18 dB determined in Example 14.11.1. To increase the available gain, we must also increase the noise figure. Fig. 14.12.1 shows the locations of the optimum reflection coefficients, as well as several noise figure circles. The MATLAB code for generating this graph was:† Fig. 14.12.1 Noise figure circles. S = smat([0.61, 165, 3.72, 59, 0.05, 42, 0.45, -48]); Fmin = 1.6; rn = 0.16; gGopt = p2c(0.26, 172); †The function p2c converts from phasor form to cartesian complex form, and the function c2p, from cartesian to phasor form. 704 14. S-Parameters Gmag = db(sgain(S,’mag’)); % maximum available gain Gaopt = db(sgain(S,gGopt,’a’)) % available gain at ΓGopt gLopt = conj(gout(S,gGopt)); % matched load [c1,r1] = nfcirc(1.7,Fmin,rn,gGopt); % noise figure circles [c2,r2] = nfcirc(1.8,Fmin,rn,gGopt); [c3,r3] = nfcirc(1.9,Fmin,rn,gGopt); [c4,r4] = nfcirc(2.0,Fmin,rn,gGopt); smith; plot([gGopt, gLopt],’.’); smithcir(c1,r1); smithcir(c2,r2); smithcir(c3,r3); smithcir(c4,r4); The larger the noise figure F, the larger the radius of its circle. As F increases, so does the available gain. But as the gain increases, the radius of its circle decreases. Thus, for a fixed value of F, there will be a maximum value of the available gain corresponding to that gain circle that has the smallest radius and is tangent to the noise figure circle. In the extreme case of the maximum available gain, the available gain circle collapses to a point—the simultaneous conjugate matched point ΓG = 0.8179∠−162.67o— with a corresponding noise figure of F = 4.28 dB. These results can be calculated by the MATLAB statements: gG = smatch(S); F = nfig(Fmin, rn, gopt, gG); Thus, we see that increasing the gain comes at the price of increasing the noise figure. As ΓG traces the F = 1.8 dB circle, the available gain Ga varies as shown in Fig. 14.12.2. Points around this circle can be parametrized as ΓG = c2 + r2ejφ, with 0 ≤ φ ≤ 2π. Fig. 14.12.2 plots Ga versus the angle φ. We note that the gain varies between the limits 12.22 ≤ Ga ≤ 14.81 dB. 0 90 180 270 360 12 13 14 15 Ga (dB) φ (degrees) Available Gain for F = 1.8 dB Fig. 14.12.2 Variation of available gain around the noise figure circle F = 1.8 dB. The maximum value, Ga = 14.81 dB, is reached when ΓG = 0.4478∠−__________169.73o, with a resulting matched load ΓL = Γ∗ out = 0.5574∠52.50o. The two points ΓG, ΓL, as well as the 14.12. Noise Figure Circles 705 Fig. 14.12.3 Maximum available gain for given noise figure. Ga = 14.81 dB gain circle, which is tangential to the 1.8-dB noise figure circle, are shown in Fig. 14.12.3. The following MATLAB code performs these calculations and plots: phi = linspace(0,2*pi,721); % angle in 1/2o increments gG = c2 + r2*exp(j*phi); % ΓG around the c2, r2 circle G = db(sgain(S,gG,’a’)); % available gain in dB plot(phi*180/pi, G); [Ga,i] = max(G); % maximum available gain gammaG = gG(i); % ΓG for maximum gain gammaL = conj(gout(S,gammaG)); % matched load ΓL [ca,ra] = sgcirc(S,’a’,Ga); % available gain circle smith; smithcir(c2,r2); smithcir(ca,ra); plot([gammaG,gammaL],’.’); The maximum gain and the point of tangency with the noise figure circle are determined by direct search, that is, evaluating the gain around the 1.8-dB noise figure circle and finding where it reaches a maximum. The input and output stub matching networks can be designed with the help of the function stub1. The normalized source and load impedances are: zG = 1 + ΓG 1 − ΓG = 0.3840 − 0.0767j , zL = 1 + ΓL 1 − ΓL = 1.0904 + 1.3993j The stub matching networks have lengths: stub1(z∗ G, ’po’)= _ 0.3749 0.3977 0.1251 0.0738 _ , stub1(z∗ L , ’po’)= _ 0.3519 0.0991 0.1481 0.2250 _ The lower solutions have shorter total lengths d + l. The implementation of the matching networks with microstrip lines will be similar to that in Fig. 14.8.4. __ 706 14. S-Parameters If the two-port is potentially unstable, one must be check that the resulting solutions for ΓG, ΓL both lie in their respective stability regions. Problems 14.6 and 14.7 illustrate the design of such potentially unstable low noise microwave amplifiers. 14.13 Problems 14.1 Using the relationships (14.4.3) and (14.4.6), derive the following identities: (Z11 + ZG)(Z22 + ZL)−Z12Z21 = (Z22 + ZL)(Zin + ZG)= (Z11 + ZG)(Zout + ZL) (14.13.1) (1 − S11ΓG)(1 − S22ΓL)−S12S21ΓGΓL = (1 − S22ΓL)(1 − ΓinΓG)= (1 − S11ΓG)(1 − ΓoutΓL) (14.13.2) Using Eqs. (14.4.4) and (14.4.5), show that: Z21 Z22 + ZL = S21 1 − S22ΓL 1 − ΓL 1 − Γin , Z21 Z11 + ZG = S21 1 − S11ΓG 1 − ΓG 1 − Γout (14.13.3) 2Z0 Zin + ZG = (1 − Γin)(1 − ΓG) 1 − ΓinΓG , 2Z0 Zout + ZL = (1 − Γout)(1 − ΓL) 1 − ΓoutΓL (14.13.4) Finally, for the real part RL = Re(ZL), show that: ZL = Z0 1 + ΓL 1 − ΓL ⇒ RL = Z0 1 − |ΓL|2 |1 − ΓL|2 (14.13.5) 14.2 Computer Experiment. The Hewlett-Packard ATF-10136 GaAs FET transistor has the following S-parameters at 4 GHz and 8 GHz : S11 = 0.54∠−120o, S21 = 3.60∠61o, S12 = 0.137∠31o, S22 = 0.22∠−49o S11 = 0.60∠87o, S21 = 2.09∠−32o, S12 = 0.21∠−36o, S22 = 0.32∠−48o Determine the stability parameters, stability circles, and stability regions at the two frequencies. 14.3 Derive the following relationships, where RG = Re(ZG): Z0 + ZG 2 _ RGZ0 = 1 _ 1 − |ΓG|2 |1 − ΓG| 1 − ΓG , Z0 − ZG 2 _ RGZ0 = −_ ΓG 1 − |ΓG|2 |1 − ΓG| 1 − ΓG 14.4 Derive Eqs. (14.7.13) relating the generalized S-parameters of power waves to the conventional S-parameters. 14.5 Derive the expression Eq. (14.6.20) for the maximum available gain GMAG, and show that it is the maximum of all three gains, that is, transducer, available, and operating gains. 14.6 Computer Experiment. The microwave transistor of Example 14.11.2 has the following noise parameters at a frequency of 1 GHz : Fmin = 1.3 dB, rn = 0.16, and ΓGopt = 0.06∠49o. Determine the matched load ΓLopt corresponding to ΓGopt and calculate the available gain. Then, plot the noise figure circles for F = 1.4, 1.5, 1.6 dB. For the 1.5-dB noise figure circle, determine the values of ΓG, ΓL that correspond to the maximum possible available gain. Design microstrip stub matching circuits for the computed values of ΓG, ΓL. 14.13. Problems 707 14.7 Computer Experiment. The Hewlett-Packard ATF-36163 pseudomorphic high electron mobility transistor (PHEMT) has the following S- and noise parameters at 6 GHz : S11 = 0.75∠−131o, S21 = 3.95∠55o, S12 = 0.13∠−12o, S22 = 0.27∠−116o Fmin = 0.66 dB, rn = 0.15, ΓGopt = 0.55∠88o Plot the F = 0.7, 0.8, 0.9 dB noise figure circles. On the 0.7-dB circle, determine the source reflection coefficient ΓG that corresponds to maximum available gain, and then determine the corresponding matched load coefficient ΓL. Design microstrip stub matching circuits for the computed values of ΓG, ΓL. 14.8 Computer Experiment. In this experiment, you will carry out two low-noise microwave amplifier designs, including the corresponding input and output matching networks. The first design fixes the noise figure and finds the maximum gain that can be used. The second design fixes the desired gain and finds the minimum noise figure that may be achieved. The Hewlett-Packard Agilent ATF-34143 PHEMT transistor is suitable for low-noise amplifiers in cellular/PCS base stations, low-earth-orbit and multipoint microwave distribution systems, and other low-noise applications. At 2 GHz, its S-parameters and noise-figure data are as follows, for biasing conditions of VDS = 4 V and IDS = 40 mA: S11 = 0.700∠−150o, S12 = 0.081∠19o S21 = 6.002∠73o, S22 = 0.210∠−150o Fmin = 0.22 dB, rn = 0.09, ΓGopt = 0.66∠67o a. At 2 GHz, the transistor is potentially unstable. Calculate the stability parameters K, μ, Δ,D1,D2. Calculate the MSG in dB. Draw a basic Smith chart and place on it the source and load stability circles (display only a small portion of each circle outside the Smith chart.) Then, determine the parts of the Smith chart that correspond to the source and load stability regions. b. For the given optimum reflection coefficient ΓGopt, calculate the corresponding load reflection coefficient ΓLopt assuming a matched load. Place the two points ΓGopt, ΓLopt on the above Smith chart and determine whether they lie in their respective stability regions. c. Calculate the available gain Ga,opt in dB that corresponds to ΓGopt. Add the corresponding available gain circle to the above Smith chart. (Note that the source stability circle and the available gain circles intersect the Smith chart at the same points.) d. Add to your Smith chart the noise figure circles corresponding to the noise figure values of F = 0.25, 0.30, 0.35 dB. For the case F = 0.35 dB, calculate and plot the available gain Ga in dB as ΓG traces the noise-figure circle. Determine the maximum value of Ga and the corresponding value of ΓG. Place on your Smith chart the available gain circle corresponding to this maximum Ga. Place also the corresponding point ΓG, which should be the point of tangency between the gain and noise figure circles. 708 14. S-Parameters Calculate and place on the Smith chart the corresponding load reflection coefficient ΓL = Γ∗ out. Verify that the two points ΓG, ΓL lie in their respective stability regions. In addition, for comparison purposes, place on your Smith chart the available gain circles corresponding to the values Ga = 15 and 16 dB. e. The points ΓG and ΓL determined in the previous question achieve the maximum gain for the given noise figure of F = 0.35 dB. Design input and output stub matching networks that match the amplifier to a 50-ohm generator and a 50-ohm load. Use “parallel/open” microstrip stubs having 50-ohm characteristic impedance and alumina substrate of relative permittivity of r = 9.8. Determine the stub lengths d, l in units of λ, the wavelength inside the microstrip lines. Choose always the solution with the shortest total length d + l. Determine the effective permittivity eff of the stubs, the stub wavelength λ in cm, and the width/height ratio, w/h. Then, determine the stub lengths d, l in cm. Finally, make a schematic of your final design that shows both the input and output matching networks (as in Fig.10.8.3.) f. The above design sets F = 0.35 dB and finds the maximum achievable gain. Carry out an alternative design as follows. Start with a desired available gain of Ga = 16 dB and draw the corresponding available gain circle on your Smith chart. As ΓG traces the portion of this circle that lies inside the Smith chart, compute the corresponding noise figure F. (Points on the circle can be parametrized by ΓG = c + rejφ, but you must keep only those that have |ΓG| < 1.) Find the minimum among these values of F in dB and calculate the corresponding value of ΓG. Calculate the corresponding matched ΓL. Add to your Smith chart the corresponding noise figure circle and place on it the points ΓG and ΓL. g. Design the appropriate stub matching networks as in part 14.8.