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Wein Bridge Oscillators Additional Notes MathCAD Application File Wienbridge Oscillator - Transfer Function - Frequency Analysis - P. F. Ribeiro i 1 20 j 1 20 x 450 i 5 w 5000 500 j R1 10000 C1 0.01 10 R2 10000 C2 0.01 10 Ra x Ra i i j 6 6 i Rb 250 10 500 wo jj 1 1 C1 R1 4 wo 1 10 Considering the transfer function of the circuit: f i j Ra 1 i R2 C1 jj w j Rb jj wj2R1R2C1C2 jj wj(R1C1 R2C1 R2C2) 1 Sensitivity of the Loop Gain versus frequency for different Amplifier Gains f1 j f 10 j wj f 20 j wj wj s 1 jj 999 Given A 3 s 2R12C12 s 3R1C1 1 R2 C2 s A 5 ss Find ( s ) ss 3.222 10 4 i 10 t 0 0.0001 0.01 y ( t) A e Re( ss) t sin ( Im( ss ) t) [Try R1=11395, R2=10005] y ( t) Frequency of Oscilation t 1 R1 R2 C1 C2 Gain Ra i Av 1 i Rb 4 1 10 Gain Ra i Av 1 i Rb Avi i Sensitivity of the Loop Gain versus Amplifier Gain for different frequencies 3-D of the Magnitude of the Transfer Function fi 1 f i 10 Rai Rai f MathCAD Application File An Investigation of the Wien-Bridge Oscillator Troy Cok and P.F. Ribeiro The Wien-bridge oscillator, shown below in Figure 1, is a circuit that provides a sinusiodal output voltage using no voltage source. The RC circuit uses the initial charge on one of the capacitors to provide voltage to the rest of the circuit. Figure 1: Wien-Bridge Oscillator Circuit The gain of this circuit can be examined in terms of the individual component values. The noninverting amplifier gain is determined by the res istors R1 and R2, according to: G 1 R2 R1 The loop gain (or transfer function) of the Wien-bridge oscillator is determined by the noninverting gain and the remaining circuit element s. T j R C G j 1 2R2C2 3 jRC T j R C G j 1 2R2C2 3 jRC For stability, the phase shift is preferred to be zero. In order to accomplish this, the real part of the denominator of the transfer function must be zero. The real part of the denominator will be zero if the operating frequency is at resonance. The resonant frequency is: o 1 R C T j At resonance, the transfer function reduces to G 3 So, if the noniverting gain is 3, the loop gain will be 1. To investigate the circuit in more detail, we can use a PSPICE simulation. To begin, we will try to get a unity gain. The individual component values are determined according to the transfer function. Using standard resistor values, R1 will be set to 10 k. G 1 R2 R1 R1 10k solve R2 ( G 1) R1 G 3 R2 ( G 1) R1 4 R2 2 10 For a resonance frequency of 1 kHz, the resistor and capacitor values can be: o 1kHz R 10k o 1 R C C 1 R o C 0.1 F Varying the frequency of the transfer function can be examine for the calculated component values. Both the theoretical and computer simulated data are plotted using radians. T j i R C G j 1 2R2C2 3 jRC B 20 log T 6 spice unity 2 10 20 100000 7 Tspice unity Gain (dB) Frequency Response B( ) Tspice 0 20 40 10 100 1 10 spice Frequency (rad) 3 1 10 4 1 10 The peak gain of the frequency analysis occurs at the resonance frequency for each circuit model. The two traces exhibit nearly identical Bode plots. 5 A better design would cause the circuit to exhibit a cons tant (not decaying) oscillation. We can attempt to update the circuit using a form of amplitude stabilization. There are a couple of available design methods, but one of the better schemes involves the introduction diodes into the circuit. Along with the diodes, two additional resistors are added to form an amplitude control network. The schematic for this circuit is shown below. The new resistors are determined according to the following equation. This ensures that the noninverting gain of the circuit will be slightly more than 3 when the diodes are off and slightly less that 3 when one is active. R2 R3 R1 2 So, if R1 is now 15 k, R2 and R3 can be 15 k and 16 k respectively. R1 15k R2 R3 R1 R2 R2 15k R3 18k 2.2 R3 R4 R3 R4 R1 2 Here, the parallel combination of R3 and R4 must be slightly less than R2. Since R3 is a bit greater than R2, a mid-range resistor value of R4 will suffice. R4 33k R2 R3 R4 R3 R4 R1 1.776 R2 R3 R4 R3 R4 R1 1.776 The updated circuit can again be examined using PSPICE. The resulting transient waveforms are shown below. With the modification, the circuit appears to operate with a steady oscillation as time passes. 0 tspice unity 1 Uspice unity Wien-bridge with Amplitude Stabilization Steady Oscillation over Time 10 Voltage (V) Voltage (V) 10 Uspice 0 10 0 0.02 tspice Time (s) 0.04 Uspice 0 10 0 0.5 1 tspice Time (s) 1.5 2 Wien-Bridge Oscillator Design Application Notes 1 Wien-Bridge Oscillator Design Application Notes 2