## 2 Microwave Liquid Crystal Technology
### Orientation Mechanisms and Biasing Schemes
It differs for the various LC mixtures of Table 1, which are typically around 1 V. With \(V_{b}>V_{th}\), the electric torques are exceeding the elastic ones, and the orientation of most LC directors \(\overset{\rightarrow}{n}\) starts to align more and more toward the orientation of the field lines of \(\overset{\rightarrow}{E}_{b}\) (see the three pictures in the middle of Figure 9a), until nearly all LC directors \(\overset{\rightarrow}{n}\) are oriented along the field lines for \(V_{b}=V_{sat}\). Beyond this saturation state \(V_{b}>V_{sat}\), no change in permittivity can be observed anymore. When a RF field \(\overset{\rightarrow}{E}_{RF}\) is applied at the parallel plate capacitor, it will experience different relative permittivities \(\varepsilon\), and of course, different dielectric losses \(\tan\delta\), depending on the changing orientation of the LC directors \(\overset{\rightarrow}{n}\) with respect to the fixed orientation of the RF field vector \(\overset{\rightarrow}{E}_{RF}\), which is vertical polarized as the tuning field \(\overset{\rightarrow}{E}_{b}\) in Figure 9a, since both are not decoupled. Initially, when no bias voltage is applied between both plates, the LC directors \(\overset{\rightarrow}{n}\) are oriented in parallel to the surfaces, i.e., by default, they are perpendicularly aligned to the RF field \(\overset{\rightarrow}{E}_{RF}\). Hence, the RF field experiences a relative permittivity of \(\varepsilon_{r,\perp}\) and loss tangent \(\tan\delta_{\perp}\). With increasing biasing voltage, LC directors \(\overset{\rightarrow}{n}\) starts to align more and more toward the orientation of the field lines of \(\overset{\rightarrow}{E}_{b}\), i.e., in parallel to \(\overset{\rightarrow}{E}_{RF}\). Hence, \(\overset{\rightarrow}{E}_{RF}\) experiences a material's effective relative permittivity, which varies from \(\varepsilon_{r,\perp}\) continuously toward \(\varepsilon_{r,\parallel}\), which is accompanied by a change in the loss tangent from \(\tan\delta_{\perp}\); toward \(\tan\delta_{\parallel}\) (see also Figure 16 for a tunable inverted microstrip line). Thus, varying \(\varepsilon_{r}\) by changing the orientation of LC directors \(\overset{\rightarrow}{n}\) due to the bias field \(\overset{\rightarrow}{E}_{b}\), the capacitance of the parallel plate capacitor can be tuned, building up a tunable LC varactor. The time it takes to change from \(\varepsilon_{r,\perp}\) to \(\varepsilon_{r,\parallel}\) by applying a fast voltage source from \(V_{b}=0\) V to \(V_{b}>V_{sat}\) is the rise or switch-on response time. \[\tau_{\text{on}}\propto\frac{\gamma_{\text{rot}}}{\varepsilon_{0}\Delta \varepsilon_{r,1\text{kHz}}\left(E_{b}^{2}-E_{th}^{2}\right)}=\frac{\gamma_{ \text{rot}}h_{LC}^{2}}{K_{11}\pi^{2}\left(\frac{V_{1}^{2}}{V_{m}^{2}}-1 \right)}=\frac{\tau_{\text{off}}}{\left(\frac{V_{1}^{2}}{V_{m}^{2}}-1\right)} \to T_{90}^{10} \tag{5}\]  
When the bias voltage is released, the LC directors \(\overset{\rightarrow}{n}\) (in parallel to \(\overset{\rightarrow}{E}_{RF}\)) are going back to the initial state in parallel to the alignment layers (perpendicular to \(\overset{\rightarrow}{E}_{RF}\)) by means of the surface anchoring forces. Hence, the time it takes to change from \(\varepsilon_{r,\parallel}\) to \(\varepsilon_{r,\perp}\) is the decay or switch-off response time. \[\tau_{\text{off}}\propto\frac{\gamma_{\text{rot}}}{\varepsilon_{0}\Delta \varepsilon_{r,1\text{kHz}}E_{th}^{2}}=\frac{\gamma_{\text{rot}}h_{LC}^{2}}{ \varepsilon_{0}\Delta\varepsilon_{r,1\text{kHz}}.V_{th}^{2}}=\frac{\gamma_{ \text{rot}}h_{LC}^{2}}{K_{11}\pi^{2}}\to T_{10}^{90}. \tag{6}\]  
The switch-on response time \(\tau_{\text{on}}\) according to Equation (5) is always much faster by \(1/(\left[V_{b}/V_{th}\right]^{2}-1)\) than the switch-off response time \(\tau_{\text{off}}\) according to Equation (6), since it is driven by the voltage \(V_{b}<V_{sat}\), which is much stronger than the low anchoring forces to reorient the LC molecules back into the initial state after releasing the voltage.