# Energy-Efficient Amplifiers Based on Quasi-Floating Gate Techniques
## 2 Quasi-Floating Gate Transistors
### Wideband Capacitive Coupling Using Quasi-Floating Gate Transistors  
Capacitive (or AC) coupling is widely used to isolate the DC bias setting circuit of active devices from the driving signal source. A typical example is shown in Figure 1a, where a resistive divider sets the quiescent gate voltage of the transistor. Coupling capacitor C blocks the input DC voltage and allows the input signal to pass through to the gate. A more general biasing scheme is shown in Figure 1b, where DC bias voltage \(V_{B}\) is set using resistor R\({}_{B}\). Note that the circuit of Figure 1a can be regarded as a particular case of Figure 1b, using the [PERSON]'s theorem, where \(V_{B}=(V_{DD}-V_{SS})\cdot R_{2}/(R_{1}+R_{2})\) and \(R_{B}=R_{1}\mid 1R_{2}\). The circuit of Figure 1b provides a first-order, low-pass filtering from input \(V_{B}\) to the gate and a first-order, high-pass filtering from input \(V_{sig}\) to the gate, both with time constant \(R_{B}\).C. Due to the limited practical values of on-chip passive resistors and capacitors, this time constant cannot be too large. Hence, despite the name \"AC coupling,\" this technique is rather employed in ICs for RF narrowband signals with frequencies above the relatively large cutoff frequency of the high-pass input filter. However, several relevant applications require processing of very low frequency signals (such as biomedical systems, structural health monitoring, geonevironment monitoring, etc.) or wideband signals with significant content at low frequencies (for instance, baseband processing circuits in a direct conversion receiver). In these cases, DC-blocking circuits able to allow near-DC frequency components pass through are required. An efficient approach to achieve the large time constant required is replacing resistor \(R_{B}\) in Figure 1b by a pseudo-resistor (also known as quasi-infinite resistor, QIR) [6,7]. The resulting circuit is shown in Figure 1c, which is known as a QFG transistor [5]. Pseudo-resistors are small-area integrated devices that can provide extremely large incremental resistances (in the order of G\(\Omega\)-T\(\Omega\)) when they are properly biased. Therefore, very large RC time constants can be obtained using small capacitors that can be implemented on-chip. From Figure 1c, the transfer functions from inputs \(V_{B}\) and \(V_{sig}\) to \(V_{g}\) are:  
\[\frac{V_{g}(s)}{V_{B}(s)}=\frac{1}{1+s\tau} \tag{1}\]  
\[\frac{V_{g}(s)}{V_{sig}(s)}=\alpha\frac{s\tau}{1+s\tau} \tag{2}\]  
with \(\tau=R_{B}\cdot(C+C_{g})\) as the time constant, \(C_{g}\) the parasitic capacitance at the gate terminal and \(\alpha=C/(C+C_{g})\). Note that input \(V_{B}\) is low pass filtered with an extremely large time constant \(\tau\), so that the DC bias voltage \(V_{BIAS}\) is set to the gate, and any AC noise or interference coming from this input is rejected. The signal at input \(V_{sig}\) is high pass filtered with the same time constant, so, in practice, only the DC level at this input is blocked. This input signal is also attenuated by a factor \(\alpha\), due to the nonzero parasitic gate capacitance \(C_{g}\). The value of \(C_{g}\) sets the minimum required value for \(C\) to avoid excessive attenuation. Multiple-input QFG transistors can also be used [5], allowing weighted averaging of signals in a simple and compact way. Figure 1: (**a**) Conventional capacitive coupling. (**b**) General scheme. (**c**) Quasi-floating gate transistor. Note the different behavior of the circuit of Figure 1c for DC and AC small-signal and large-signal dynamic (or transient) operation. Capacitor C is an open circuit in DC, so the quiescent gate voltage is \(V_{g}{}^{Q}=V_{BIAS}\), since there is not current flowing through the pseudo-resistor. Hence, the bias point can be accurately set. For AC small-signal operation, the capacitor acts as a short circuit (more precisely, as an attenuator, due to the input capacitive divider), so that the AC gate voltage is \(v_{g}=\alpha v_{ac}\), with \(v_{ac}\) the AC voltage at input \(V_{sig}\).