negative feedback branch has a positive phase shift. The phase shift goes through zero at the resonant frequency. As the lamp heats up, it decreases the positive feedback. The Q of the crystal in Meacham's circuit is given as 104,000. At any frequency different from the resonant frequency by more than a small multiple of the bandwidth of the crystal, the negative feedback branch dominates the loop gain and there can be no self-sustaining oscillation except within the narrow bandwidth of the crystal. In 1944 (after Hewlett's design), J. K. Clapp modified Meacham's circuit to use a vacuum tube phase inverter instead of a transformer to drive the bridge. A modified Meacham oscillator uses Clapp's phase inverter but substitutes a diode limiter for the tungsten lamp. Hewlett's oscillator William R. Hewlett's Wien bridge oscillator can be considered as a combination of a differential amplifier and a Wien bridge, connected in a positive feedback loop between the amplifier output and differential inputs. At the oscillating frequency, the bridge is almost balanced and has very small transfer ratio. The loop gain is a product of the very high amplifier gain and the very low bridge ratio. In Hewlett's circuit, the amplifier is implemented by two vacuum tubes. The amplifier's inverting input is the cathode of tube V1 and the non-inverting input is the control grid of tube V2. To simplify analysis, all the components other than R1, R2, C1 and C2 can be modeled as a non-inverting amplifier with a gain of 1+Rf/Rb and with a high input impedance. R1, R2, C1 and C2 form a bandpass filter which is connected to provide positive feedback at the frequency of oscillation. Rb self heats and increases the negative feedback which reduces the amplifier gain until the point is reached that there is just enough gain to sustain sinusoidal oscillation without over driving the amplifier. If R1 = R2 and C1 = C2 then at equilibrium Rf/Rb = 2 and the amplifier gain is 3. When the circuit is first energized, the lamp is cold and the gain of the circuit is greater than 3 which ensures start up. The dc bias current of vacuum tube V1 also flows through the lamp. This does not change the principles of the circuit's operation, but it does reduce the amplitude of the output at equilibrium because the bias current provides part of the heating of the