Applications

Stability

A very important task in microwave design (especially for amplifiers) is the question, whether the circuit tends to unwanted oscillations. A two-port oscillates if, despite of no signal being fed into it, AC power issues from at least one of its ports. This condition can be easily expressed in terms of RF quantities, so a circuit is stable if:

$$\vert\underline{r}_1\vert < 1 \qquad \vert\underline{r}_2\vert < 1$$ (1.39)

with $\underline{r}_1$ being reflexion coefficient of port 1 and $\underline{r}_2$ the one of port 2.

A further question can be asked: What conditions must be fulfilled to have a two-port be stable for all combinations of passive impedance terminations at port 1 and port 2? Such a circuit is called unconditionally stable. [3] is one of the best discussions dealing with this subject.

A circuit is unconditionally stable if the following two relations hold:

$$K = \frac{1-\vert S_{11}\vert^2-\vert S_{22}\vert^2+\vert\Delta\vert^2}{2\cdot \vert S_{12}\cdot S_{21}\vert} > 1$$ (1.40)

$$\vert\Delta\vert = \vert S_{11}\cdot S_{22} - S_{12}\cdot S_{21}\vert < 1$$ (1.41)

with $\Delta$ being the determinant of the S parameter matrix of the two port. $K$ is called Rollet stability factor. Two relations must be fulfilled to have a necessary and sufficient criterion.

A more practical criterion (necessary and sufficient) for unconditional stability is obtained with the $\mu$-factor:

$$\mu = \frac{1-\vert S_{11}\vert^2}{\vert S_{22}-S_{11}^*\cdot \Delta\vert + \vert S_{12}\cdot S_{21}\vert} > 1$$ (1.42)

Because of symmetry reasons, a second stability factor must exist that also gives a necessary and sufficient criterion for unconditional stability:

$$\mu' = \frac{1-\vert S_{22}\vert^2}{\vert S_{11}-S_{22}^*\cdot \Delta\vert + \vert S_{12}\cdot S_{21}\vert} > 1$$ (1.43)

For conditional stable two-ports it is interesting which which load and which source impedance may cause instability. This can be seen using stability circles [4]. A disadvantage of this method is that the radius of the below-mentioned circles can become infinity. (A circle with infinite radius is a line.)

Within the reflexion coefficient plane of the load ($r_L$-plane), the stability circle is:

$$\underline{r}_{center} = \frac{S_{22}^* - S_{11}\cdot \Delta^*}{\vert S_{22}\vert^2 - \vert\Delta\vert^2}$$ (1.44)

$$= \frac{\vert S_{12}\vert\cdot \vert S_{21}\vert}{\vert S_{22}\vert^2 - \vert\Delta\vert^2}$$ (1.45)

If the center of the $r_L$-plane lies within this circle and $\vert S_{11}\vert \le 1$ then the circuit is stable for all reflexion coefficients inside the circle. If the center of the $r_L$-plane lies outside the circle and $\vert S_{11}\vert \le 1$ then the circuit is stable for all reflexion coefficients outside the circle.

Very similar is the situation for reflexion coefficients in the source plane ($r_S$-plane). The stability circle is:

$$\underline{r}_{center} = \frac{S_{11}^* - S_{22}\cdot \Delta^*}{\vert S_{11}\vert^2 - \vert\Delta\vert^2}$$ (1.46)

$$= \frac{\vert S_{12}\vert\cdot \vert S_{21}\vert}{\vert S_{11}\vert^2 - \vert\Delta\vert^2}$$ (1.47)

If the center of the $r_S$-plane lies within this circle and $\vert S_{22}\vert \le 1$ then the circuit is stable for all reflexion coefficients inside the circle. If the center of the $r_S$-plane lies outside the circle and $\vert S_{22}\vert \le 1$ then the circuit is stable for all reflexion coefficients outside the circle.

Gain

Maximum available and stable power gain (only for unconditional stable 2-ports) [4]:

$$G_{max} = \left\vert \frac{S_{21}}{S_{12}} \right\vert \cdot \left( K - \sqrt{K^2-1} \right)$$ (1.48)

where $K$ is Rollet stability factor.

The (bilateral) transmission power gain of a two-port can be split into three parts [4]:

$$G = G_S \cdot G_0 \cdot G_L$$ (1.49)

with

$$G_S = \frac{(1 - \vert r_S\vert^2) \cdot (1 - \vert r_1\vert^2)}{\vert 1 - r_S\cdot r_1\vert^2}$$ (1.50)

$$G_0 = \vert S_{21}\vert^2$$ (1.51)

$$G_L = \frac{1 - \vert r_L\vert^2}{\vert 1 - r_L\cdot S_{22}\vert^2 \cdot (1 - \vert r_1\vert^2)}$$ (1.52)

where $r_1$ is reflexion coefficient of the two-port input.

The curves of constant gain are circles in the reflexion coefficient plane. The circle for the load-mismatched two-port with gain $G_L$ is

$$\underline{r}_{center} = \frac{(S_{22}^* - S_{11}\cdot \Delta^*) \cdot G_L}{G_L\cdot (\vert S_{22}\vert^2 - \vert\Delta\vert^2) + 1}$$ (1.53)

$$= \frac{\sqrt{1 - G_L\cdot (1-\vert S_{11}\vert^2-\vert S_{22}\ve... ...\cdot S_{21}\vert^2}} {G_L\cdot (\vert S_{22}\vert^2 - \vert\Delta\vert^2) + 1}$$ (1.54)

The circle for the source-mismatched two-port with gain $G_S$ is

$$\underline{r}_{center} = \frac{G_S\cdot r_1^*}{1 - \vert r_1\vert^2\cdot (1-G_S)}$$ (1.55)

$$= \frac{\sqrt{1 - G_S}\cdot (1-\vert r_1\vert^2)}{1 - \vert r_1\vert^2\cdot (1-G_S)}$$ (1.56)

with

$$r_1 = S_{11} + \frac{S_{12}\cdot S_{21}\cdot r_L}{1 - r_L\cdot S_{22}}$$ (1.57)

The available power gain $G_A$ of a two-port is reached when the load is conjugately matched to the output port. It is:

$$G_A = \frac{\vert S_{21}\vert^2\cdot (1-\vert r_S\vert^2)}{\vert 1-S_{11}\cdot r_S\vert^2 - \vert S_{22}-\Delta\cdot r_S\vert^2}$$ (1.58)

with $\Delta = S_{11}S_{22} - S_{12}S_{21}$. The curves with constant gain $G_A$ are circles in the source reflexion coefficient plane ($r_S$-plane). The center $r_{S,c}$ and the radius $R_S$ are:

$$r_{S,c} = \frac{g_A\cdot C_1^*}{1 + g_A\cdot(\vert S_{11}\vert^2 - \vert\Delta\vert^2)}$$ (1.59)

$$R_S = \frac{\sqrt{1 - 2\cdot K\cdot g_A\cdot\vert S_{12}S_{21}\vert +... ...1}\vert^2}} {\vert 1 + g_A\cdot(\vert S_{11}\vert^2 - \vert\Delta\vert^2)\vert}$$ (1.60)

with $C_1 = S_{11} - S_{22}^*\cdot\Delta$, $g_A = G_A / \vert S_{21}\vert^2$ and $K$ Rollet stability factor.

The operating power gain $G_P$ of a two-port is the power delivered to the load divided by the input power of the amplifier. It is:

$$G_P = \frac{\vert S_{21}\vert^2\cdot (1-\vert r_L\vert^2)}{\vert 1-S_{22}\cdot r_L\vert^2 - \vert S_{11}-\Delta\cdot r_L\vert^2}$$ (1.61)

with $\Delta = S_{11}S_{22} - S_{12}S_{21}$. The curves with constant gain $G_P$ are circles in the load reflexion coefficient plane ($r_L$-plane). The center $r_{L,c}$ and the radius $R_L$ are:

$$r_{L,c} = \frac{g_P\cdot C_2^*}{1 + g_P\cdot(\vert S_{22}\vert^2 - \vert\Delta\vert^2)}$$ (1.62)

$$R_L = \frac{\sqrt{1 - 2\cdot K\cdot g_P\cdot\vert S_{12}S_{21}\vert +... ...1}\vert^2}} {\vert 1 + g_P\cdot(\vert S_{22}\vert^2 - \vert\Delta\vert^2)\vert}$$ (1.63)

with $C_2 = S_{22} - S_{11}^*\cdot\Delta$, $g_P = G_P / \vert S_{21}\vert^2$ and $K$ Rollet stability factor.

Two-Port Matching

Obtaining concurrent power matching of input and output in a bilateral circuit is not such simple, due to the backward transmission $S_{12}$. However, in linear circuits, this task can be easily solved by the following equations:

$$\Delta = S_{11}\cdot S_{22} - S_{12}\cdot S_{21}$$ (1.64)

$$B = 1 + \vert S_{11}\vert^2 - \vert S_{22}\vert^2 - \vert\Delta\vert^2$$ (1.65)

$$C = S_{11} - S_{22}^* \cdot \Delta$$ (1.66)

$$r_S = \frac{1}{2\cdot C} \cdot \left( B - \sqrt{B^2 - \vert 2\cdot C\vert^2 } \right)$$ (1.67)

Here $r_S$ is the reflexion coefficient that the circuit needs to see at the input port in order to reach concurrently matched in- and output. For the reflexion coefficient at the output $r_L$ the same equations hold by simply changing the indices (exchange 1 by 2 and vice versa).