Microstrip cross

The most useful model of a microstrip cross have been published in [39,40]. Fig. 11.10 shows the equivalent circuit (right-hand side) and the scheme with dimensions (left-hand side). The hatched area in the scheme marks the area modeled by the equivalent circuit. As can be seen the model require the microstrip width of line 1 and 3, as well as the one of line 2 and 4 to equal each other. Furthermore the permittivity of the substrat must be $\epsilon_r = 9.9$. The component values are calculated as follows:

$$X = \log_{10}\left(\dfrac{W_1}{h}\right)\cdot \left( 86.6\cdot\df... ...} + 367 \right) + \left( \dfrac{W_2}{h} \right)^3 + 74\cdot\dfrac{W_2}{h} + 130$$ (11.226)

$$\begin{split}C_1 &= C_2 = C_3 = C_4 \\ &= 10^{-12}\cdot W_1\c... ..._1}{h}\cdot \left( 1-\dfrac{W_2}{h} \right) \right) \end{split}$$ (11.227)

$$Y = 165.6\cdot\dfrac{W_2}{h} + 31.2\sqrt{\dfrac{W_2}{h}} -11.8\cdot\left( \dfrac{W_2}{h} \right)^2$$ (11.228)

$$L_1 = 10^{-9}\cdot h\cdot \left( Y\cdot\dfrac{W_1}{h} - 32\cdot\dfrac{W_2}{h} + 3 \right) \cdot \left( \dfrac{h}{W_1} \right)^{1.5}$$ (11.229)

$$L_3 = 10^{-9}\cdot h\cdot \left( 5\cdot\dfrac{W_2}{h}\cdot \cos{\... ...t)} - \left( 1+\dfrac{7\cdot h}{W_1}\right)\cdot \dfrac{h}{W_2} - 337.5 \right)$$ (11.230)

The equation of $L_2$ is obtained from the one of $L_1$ by exchanging the indices ($W_1$ and $W_2$). Note that $L_3$ is negative, so the model is unphysical without external microstrip lines. The above-mentioned equations are accurate to within 5% for $0.3\le W_1/h \le 3$ and $0.1\le W_2/h \le 3$ (value of $C_1 \dots C_4$) or for $0.5\le W_{1,2}/h \le 2$ (value of $L_1 \dots L_3$), respectively.

Figure 11.10: single-symmetrical microstrip cross and its model

Some improvement should be added to the original model:

1. Comparisons with real life show that the value of $L_3$ is too large. Multiplying it by 0.8 leads to much better results.
2. The model can be expanded for substrates with $\epsilon_r \neq 9.9$ by modifying the values of the capacitances:

   $$C_x = C_x(\epsilon_r=9.9)\cdot \dfrac{Z_0(\epsilon_r=9.9, W=W_x)}... ...}(\epsilon_r=\epsilon_{r,sub}, W=W_x)} {\epsilon_{eff}(\epsilon_r=9.9, W=W_x)}}$$ (11.231)

   
   
   The equations of $Z_0$ and $\epsilon_{eff}$ are the ones from the microstrip lines.

A useful model for an unsymmetrical cross junction has never been published. Nonetheless, as long as the lines that lie opposite are not to different in width, the model described here can be used as a first order approximation. This is perfomred by replacing $W_1$ and $W_2$ by the arithmetic mean of the line widths that lie opposite. This is done:

• In equation (11.226) and (11.227) for $W_2$ only, whereas $W_1$ is replaced by the width of the line.
• In equation (11.228) and (11.229) for $W_2$ only, whereas $W_1$ is replaced by the width of the line.
• In equation (11.230) for $W_1$ and $W_2$.

Another closed-form expression describing the non-ideal behaviour of a microstrip cross junction was published by [41]. Additionally there have been published papers [42,43,44] giving analytic (but not closed-form) expressions or just simple equivalent circuits with only a few expressions for certain topologies and dielectric constants which are actually of no pratical use.