8.11. Quadrature Oscillators

In our study of transceiver architectures in Chapter 4, we observed the need for quadrature LO phases in downconversion and upconversion operations. We also noted that flip-flop-based divide-by-two circuits generate quadrature phases, but they restrict the maximum LO frequency. In applications where dividers do not offer sufficient speed, we may employ polyphase filters or quadrature oscillators instead. In this section, we study the latter.

8.11.1. Basic Concepts

Two identical oscillators can be “coupled” such that they operate in-quadrature. We therefore begin our study with the concept of coupling (or injecting) a signal to an oscillator. Figure 8.96 depicts an example, where the input voltage is converted to current and injected into the oscillator. The differential pair is a natural means of coupling because the cross-coupled pair can also be viewed as a circuit that steers and injects current into the tanks. If the two pairs completely steer their respective tail currents, then the “coupling factor” is equal to I₁/I_SS.^[10] This topology also exemplifies “unilateral” coupling because very little of the oscillator signal couples back to the input. By contrast, if implemented by, say, two capacitors tied between V_in and the oscillator nodes, the coupling is bilateral.

^[10] In this circuit, we typically scale the transistor widths in proportion to their bias currents; thus, g_m3,4/g_m1,2 = I₁/I_SS. For small-signal analysis, the coupling factor is equal to g_m3,4/g_m1,2.