5.7: Negative Transconductance Differential Oscillator

In RFICs it is common to use an oscillator with a tank circuit across a pair of matched transistors in a differential configuration. Such an oscillator is shown in Figure \(\PageIndex{1}\)(a). So while this circuit is in a differential configuration, it is analyzed and designed as a reflection oscillator at RF.

The cross-connected differential common source pair creates a negative resistance while the fixed inductors (the \(L\)s) and the voltage-tunable capacitors, the \(C\)s, form the variable LC tank circuit. The tunable capacitors are typically implemented using semiconductor varactor diodes whose

Figure \(\PageIndex{1}\): Negative-gm differential FET VCO: (a) schematic; and (b) small-signal model used in analyzing the oscillator; (c) small-signal model with \(C_{gd}\) incorporated in the tank circuit; (d) negative resistance network of the VCO; and (e) small-signal model used in deriving the input impedance of a negative resistance network. This is a modified form of a Colpitts oscillator. The \(L-C-C_{gd}\) resonant circuit operates below resonance and presents an effective inductance (a positive reactance) but with an admittance derivative with respect to frequency that is less than that of an actual inductor. This is essential for stability. The effective inductor, \(L_{3}\) in Figure 5.2.7(b), connects the output of each of the transistors to its respective input. For each transistor \(C_{gs}\) is \(C_{1}\), and \(C_{ds}\) is \(C_{2}\), in 5.2.7(b).

Figure \(\PageIndex{2}\): Reduced model of the differential FET VCO of Figure \(\PageIndex{1}\):(a) small-signal model with negative resistance FET network replaced by the equivalent resistance and capacitance; and (b) simplest parallel small-signal model combining the tank and the negative resistance network model.

capacitance can be adjusted by the tuning voltage \(V_{\text{tune}}\). The bias transistor, the bottom FET, sets the current in the differential transistors and this current directly impacts the power consumption of the oscillator and the phase noise. The circuit is symmetrical so that the node between the two variable capacitors, the \(V_{\text{tune}}\) terminal, looks like an RF short as does the common source node of the differential pair, the node labeled \(\mathsf{X}\). This is key to developing the small-signal model shown in Figure \(\PageIndex{1}\)(b), where the dominant parasitic capacitances of the transistors, the drain-source capacitance (\(C_{ds}\)), the gate-source capacitance (\(C_{gs}\)), and the gate-drain capacitance (\(C_{gd}\)) are seen. \(C_{gd}\) becomes part of the tank circuit. This leads to the simpler small-signal model shown in Figure \(\PageIndex{1}\)(c). Removing the tank circuit leads to the small-signal active device models shown in Figure \(\PageIndex{1}\)(d and e) which present a negative resistance to the tank circuit and load.

The input admittance of the negative resistance network (Figure \(\PageIndex{1}\)(e)) can now be determined. Analysis begins by summing currents at the A and B nodes, respectively:

\[\begin{align}\label{eq:1}i_{+}&=(\jmath\omega C_{gs})v_{+}+(\jmath\omega C_{ds})v_{+}+g_{m}v_{-}\\ \label{eq:2}i_{-}&=(\jmath\omega C_{gs})v_{-}+(\jmath\omega C_{ds})v_{-}+g_{m}v_{+}\end{align} \]

The differential input admittance is then

\[\label{eq:3}Y_{\text{in}}=\frac{i_{+}-i_{-}}{v_{+}-v_{-}}=-g_{m}+\jmath\omega (C_{gs}+C_{ds}) \]

Thus the negative resistance network is modeled as a negative resistance of value \(R_{\text{in}} = (−1/g_{m})\) in parallel with a capacitance \(C_{\text{in}} = (C_{gs} + C_{ds})\). The dependence of \(R_{\text{in}}\) on \(g_{m}\) gives this oscillator its name “negative transconductance oscillator” or “negative-gm oscillator.” The gate-drain capacitance \(C_{gd}\) is in parallel with the tank capacitance \(C\) and so a new equivalent capacitance \(C_{p} = C + C_{gd}\) can be defined. Loss in the resonator circuit is modeled by a resistor \(R_{p}\) in parallel with \(C_{p}\). The small-signal model of the oscillator is now as shown in Figure \(\PageIndex{2}\)(a). This further reduces to the model shown in Figure \(\PageIndex{2}\)(b). Oscillations will initiate if \(|1/R_{\text{in}}| = |g_{m}| > 1/R_{p}\). Also the oscillation frequency, \(f_{0}\), is the frequency at which the shunt reactance is zero, that is,

\[\label{eq:4}f_{0}=\frac{1}{2\pi}\frac{1}{\sqrt{L(C_{p}+C_{\text{in}})}}=\frac{1}{2\pi}\frac{1}{\sqrt{L(C+C_{gd}+C_{gs}+C_{ds})}} \]

As oscillation builds up, \(|g_{m}|\) reduces to the value of \(1/R_{p}\) and stable oscillation is obtained. The negative-gm oscillator has the ideal characteristic if the negative conductance is the only element dependent on amplitude. Unfortunately the values of \(C_{gd},\) \(C_{gs},\) and \(C_{ds}\) also vary as the amplitude of the signal increases. This complicates design at microwave frequencies as these variations could lead to multiple simultaneous oscillations.

Example \(\PageIndex{1}\): Oscillator Analysis

Determine the frequency of oscillation of a Colpitts common emitter BJT oscillator.

Solution

Figure 5.8.1 shows two different implementations of a common emitter Colpitts BJT oscillator. The form in Figure 5.8.1(a) is the most direct implementation, with a clearly defined insertion of the Colpitts network in the collector-to-base feedback path. In Figure 5.8.1(a), the resistors \(R_{1}\) and \(R_{2}\) provide base biasing, and \(L_{C}\) is an RF choke. The oscillation frequency of this oscillator can be derived from the small-signal model of the oscillator. Since \(R_{1}\) and \(R_{2}\) will be relatively large resistances, and since \(L_{C}\) is an RF choke (it will look like an RF open circuit), the small-signal model of the oscillator is as shown below.

Figure \(\PageIndex{3}\)

In this small-signal model, \(r_{\pi}\) is the base input resistance and \(r_{o}\) is the output resistance— both of these will be relatively large. The transconductance of the transistor is \(g_{m}\). The network equations are obtained by summing the currents leaving the base node, with \(Y_{1},\: Y_{2},\) and \(Y_{3}\) being the admittances of \(C_{1},\: C_{2},\) and \(L_{3}\) respectively:

\[\begin{align}\label{eq:5}Y_{2}V_{B}+G_{\pi}V_{B}+Y_{3}(V_{B}-V_{\text{OUT}})&=0 \\ \label{eq:6}Y_{1}V_{\text{OUT}}+g_{m}V_{B}+Y_{3}(V_{\text{OUT}}-V_{B})+G_{o}V_{\text{OUT}}&=0\end{align} \]

and \(G_{\pi} = 1/r_{\pi}\), \(G_{o} = 1/r_{o}\). In matrix form

\[\label{eq:7}\left[\begin{array}{cc}{(Y_{2}+Y_{3}+G_{\pi})}&{(-Y_{3})} \\ {(g_{m}-Y_{3})}&{(Y_{1}+Y_{3}+G_{o})}\end{array}\right]\left[\begin{array}{c}{V_{B}}\\{V_{\text{OUT}}}\end{array}\right]=0 \]

This can be simplified by noting that \(r_{\pi}\) and \(r_{o}\) will have admittances smaller than \(Y_{1},\: Y_{2},\) and \(Y_{3}\). Thus Equation \(\eqref{eq:7}\) becomes

\[\label{eq:8}\left[\begin{array}{cc}{(Y_{2}+Y_{3})}&{(-Y_{3})}\\{(-Y_{3})}&{(Y_{1}+Y_{3})}\end{array}\right]\left[\begin{array}{c}{V_{B}}\\{V_{\text{OUT}}}\end{array}\right]=0 \]

Equation \(\eqref{eq:8}\) has a solution only if the determinant of the matrix is zero. That is,

\[\begin{align}(Y_{2} + Y_{3})(Y_{1} + Y_{3}) − Y_{3}Y_{3} &= Y_{1}Y_{2} + Y_{2}Y_{3} + Y_{1}Y_{3} + Y_{3}^{2} − Y_{3}^{2}\nonumber \\ \label{eq:9}&=Y_{1}Y_{2} + Y_{2}Y_{3} + Y_{1}Y_{3} = 0\end{align} \]

Now \(Y_{1} = \jmath\omega C_{1}\), etc., where \(\omega = 2\pi f\) is the radian oscillation frequency. Thus Equation \(\eqref{eq:9}\) becomes

\[\label{eq:10}-\omega^{2}C_{1}C_{2}+\frac{C_{1}}{L_{3}}+\frac{C_{2}}{L_{3}}=-\omega^{2}C_{1}C_{2}+\frac{C_{1}+C_{2}}{L_{3}}=0 \]

Rearranging, the frequency of oscillation is

\[\label{eq:11}f=\frac{1}{2\pi}\sqrt{\frac{1}{L_{3}}\frac{(C_{1}+C_{2})}{C_{1}C_{2}}} \]

The same result is obtained for the alternative form of the Colpitts oscillator shown in Figure 5.8.1(b).