Microstrip open

A microstrip open end can be modeled by a longer effective microstrip line length $\Delta l$ as described by M. Kirschning, R.H. Jansen and N.H.L. Koster [35].

$$\frac{\Delta l}{h} = \frac{Q_1\cdot Q_3\cdot Q_5}{Q_4}$$ (11.185)

with

$$Q_1 = 0.434907\cdot \dfrac{\varepsilon_{r,eff}^{0.81}+0.26}{\varepsil... ...\dfrac{\left( W/h \right)^{0.8544} + 0.236}{\left( W/h \right)^{0.8544} + 0.87}$$ (11.186)

$$Q_2 = 1 + \dfrac{\left( W/h \right) ^{0.371}}{2.358\cdot \varepsilon_r + 1}$$ (11.187)

$$Q_3 = 1 + \dfrac{0.5274}{\varepsilon_{r,eff}^{0.9236}} \cdot \arctan\left( 0.084\cdot\left( W/h \right) ^{\tfrac{1.9413}{Q_2}} \right)$$ (11.188)

$$Q_4 = 1 + 0.0377\cdot \left( 6-5\cdot\exp{\left(0.036\cdot\left(1-\va... ...)\right)} \right)\cdot \arctan\left( 0.067\cdot\left(W/h\right)^{1.456} \right)$$ (11.189)

$$Q_5 = 1 - 0.218\cdot \exp{\left( -7.5\cdot W/h \right)}$$ (11.190)

The numerical error is less than $2.5$% for $0.01 \le W/h \le 100$ and $1\le\varepsilon_r\le 50$.

Another microstrip open end model was published by E. Hammerstad [36]:

$$\dfrac{\Delta l}{h} = 0.102\cdot \dfrac{W/h+0.106}{W/h+0.264} \cd... ..._r+1}{\varepsilon_r}\cdot \left(0.9+\ln{\left(W/h+2.475\right)} \right) \right)$$ (11.191)

Here the numerical error is less than $1.7$% for $W/h < 20$.

In order to simplify calculations, the equivalent additional line length $\Delta l$ can be transformed into an equivalent open end capacitance $C_{end}$:

$$C_{end} = C'\cdot \Delta l = \dfrac{\sqrt{\varepsilon_{r,eff}}}{c_0\cdot Z_L} \Delta l$$ (11.192)

With $C'$ being the capacitance per length and $c_0$ = 299 792 458 m/s being the vacuum light velocity.