Learn about the Sulzer network, an alternative to classic R-C networks that is easier to tune over wide frequency ranges.

In my previous article, I explained the classical oscillator networks you're probably familiar with: the Wien and Bridged-T networks. At the end, I explained that they are undesirable in some applications because tuning them is difficult over a wide frequency range. 

In this article, I'll lay out an alternative network you may not have heard of: the Sulzer network.

 

Who Is Peter Sulzer?

It may surprise you to learn that back in 1952 Peter Sulzer of the National Bureau of Standards (now NIST) published in Electronics magazine an oscillator built around an innovative R-C network that can be tuned over a wide frequency range conveniently with a single variable resistor, and with a constant attenuation over frequency. This circuit can be a viable alternative to the Bridged-T oscillator, which Sulzer in fact developed a little more than a year before inventing the network that is the subject of this article.

Outside of two hobby construction articles that appeared in Popular Electronics in 1956 and The Radio Constructor in 1958, I’ve never seen this network in any textbook, despite the network being a clever circuit that eliminates the tracking problem between dual variable elements.

The Sulzer Network Topology

First, let’s look at the network topology Sulzer published—more elaborate than the traditional R-C network—with its single tuning resistor. Later, we'll compare its simulation results with that of the classical networks.

An example network with some typical component values is shown in Figure 1.

 

Figure 1. Sulzer’s phase shift network for R-C oscillators

 

As with the classical oscillators, the feedback loop output is connected to V_IN of this network, and V_OUT is fed back to the feedback gain block input. We’ll assume the gain block input draws negligible current from the V_OUT node.

The current into the V_IN terminal splits into two paths of two voltage dividers: 

1. A divider between R₁ and C₁
2. A divider between R₂ and R₃

_​The two current paths join again with the total current dropped across R₄ to ground.

The output voltage is formed at the common connection of the variable tuning resistor R_TUNE and capacitor C₂. The values of C₁ and C₂ determine the frequency band that the oscillator can be tuned over.

What Sulzer Modified: The All-Pass Network

Now, let’s trace through Sulzer’s original conceptual development of his network to see how the phase shift from V_IN to V_OUT will pass through zero at some resonant frequency.

Start with the simple ideal phase shift network of Figure 2. 

 

Figure 2. Ideal circuit for an all-pass network

 

This network is fed by two sinusoidal input voltage sources 180 degrees apart in phase, V_IN and -V_IN. The output voltage is taken from the common connection of a resistor and capacitor as shown.

This is a well-known “all-pass” network, with an output magnitude constant with frequency and equal to the input magnitude, and with the output phase varying continuously from 180 down to 0 degrees as the frequency is raised.

The phasor vector diagram in Figure 3 shows graphically how this all-pass network works.

Figure 3. Phasor vector diagram for the ideal all-pass network

 

The two phasors for the two input sources extend to the right and left on the real axis with a magnitude of V_IN. Since the capacitor voltage V_C and the resistor voltage V_R must sum up to the total voltage between the two input sources in the circuit network, the vector sum of the phasors representing VC and VR must graphically span between the two input vectors. Two phasors representing the output voltage V_OUT at the intermediate point between V_C and V_R are shown at two different frequencies in the figure above.

We know the vector relationship between V_C and V_R. Since $$\frac {V_R}{R} = I_C = jωCV_C$$  then $$V_R = (jωRC)V_C$$ . Due to the factor of j in this relation, the phasors of VC and VR must graphically be at right angles, representing a 90 degree phase shift. As the frequency increases, the length of VR will increase and that of VC will decrease. The only way to satisfy these geometric constraints is for the phasor of V_OUT to trace out a semicircle as shown as the frequency is increased. This implies a constant output magnitude equal to the input magnitude and a phase varying from 180 to 0 degrees.

Note that this phasor diagram does not have a path for the output phase to pass in and out of zero degrees, as do the classical networks, so a single all-pass network of this type will not be useful in an oscillator. Sulzer modified this network in search of a useful circuit that can be tuned with a single variable resistor.

We trace his design process by replacing the -V_IN source in Figure 2, a duplicate of the input voltage phase shifted by 180 degrees, with a source that is the input voltage instead shifted in phase by -90 degrees.

Mathematically, this is represented by the phasor -jV_IN., as shown in the schematic of Figure 4 and the phasor diagram of Figure 5.

Figure 4. Ideal circuit for the modified all-pass network

 

The phasors for V_C and V_R will still be subject to the same right angle relationship as in the previous network, but the terminal point for the sum of V_C and V_R will be rotated counterclockwise to the -90 degree direction as seen in Figure 5.

Figure 5. Phasor vector diagram for modified all-pass network

 

This will stretch counterclockwise the circular dotted line path that the phasor of V_OUT traces as the frequency is varied.

 

From Phasor Diagram to Practical Circuit Network

So we are close to having a network usable in an oscillator, as now the path passes through a point of zero phase shift. However, the geometries work out so that this point will be at the origin of this phasor diagram, which means that as the phase shift passes through zero, the output magnitude will also be zero. This would require an infinite gain to make up by the active feedback loop, so it is clearly an impractical design.

However, Sulzer’s design idea at this point was to add a fraction of V_IN with no phase shift to V_OUT. Graphically, this would essentially slide the entire dotted path in Figure 5 to the right, so that now the path would cross the real axis to the right of the origin. This means there will be a frequency where the phase through the network is zero at a practical output magnitude for a feedback loop to build up sustained oscillations. The frequency along the path where it crosses the real axis can be tuned by adjusting the single resistance R.

So how did Sulzer translate this abstract phasor diagram into a practical circuit network?

 

Modern Simulation Tools to Solve Old Challenges

We can now see all the components of the ideal network of Figure 4 in approximate form in the circuit schematic of Figure 1. The voltage divider of R₁ and C₁ forms at V_A an approximation to the -90-degree phase-shifted version of V_IN, assuming we are at frequencies greater than the low pass rolloff frequency of this divider, and at an attenuated magnitude. The resistive divider of R₂ and R₃ forms at V_B a version of V_IN with no phase shift, and with an attenuation to approximately match that of the R₁-C₁ divider. The R_TUNE-C₂ series connection is simply the R and C of the ideal phase shift network. Then the common resistor to ground, R₄, will add to V_OUT an approximate fraction of V_IN by dropping a voltage from VC to ground.

Now that’s a lot of approximation in developing this circuit topology, and Sulzer had to leave his analysis at this fairly rough point in his 1952 article and carry on past this point experimentally. (His graphical phasor analysis actually contains a few errors that I’ve corrected here.)

But by using modern simulation tools and some numerical optimization, we can go beyond his analysis and find the optimum value of the resistor R₄ that, due to a fortunate way all the math numerically works out, gives us a network remarkably useful for an oscillator.

For example, with the component values shown in Figure 1, the value of R₄ that optimizes transmission flatness with frequency turns out to be 585 ohms. By running many automated SPICE simulations in a programmed script varying R_TUNE and extracting the resonant frequency where the network phase shift passes through zero in each simulation, we end up with the plot shown in Figure 6.

Figure 6. Simulated resonance frequency and transmission of the optimized Sulzer network over R_TUNE values with R₁=2kΩ C₁=100nF R₂=2kΩ R₃=510Ω C₂=5nF R₄=585Ω

 

This plot shows a resonant frequency tuned over two orders of magnitude by varying the single resistor R_TUNE through four orders of magnitude, while the transmission amplitude varies only a tiny fraction of a dB! This excellent amplitude flatness over frequency would be very hard to achieve with the mechanically tracking dual tuning components required by classical R-C networks. For most of this characteristic, the oscillation frequency follows the square root of the tuning resistance quite linearly.

You may wonder at this point how critical the value of R₄ has to be to achieve this performance.

In another fortunate way the math all works out, the optimum for R₄ turns out to be fairly broad. The next plot, Figure 7, shows the amplitude flatness with five different values for R₄, one the optimum value of 585 ohms and four other values representing large resistor variations. 

Figure 7. Simulated transmission through the Sulzer network over R_TUNE values for five different R₄ values with R₁=2kΩ C₁=100nF R₂=2kΩ R₃=510Ω C₂=5nF

Even far off from the optimum, the transmission amplitude is still flat to within less than a half a dB, a small enough variation so that whatever amplitude stabilization mechanism used in the overall oscillator, such as the incandescent bulb technique, should easily reduce the output amplitude variation to a very low level.

At a center value of R_TUNE, for example 5kΩ, we can take one of the simulation results and look in Figure 8 at the same sort of gain-phase plots as we saw before for the classical networks.

 

Figure 8. Simulation results for the optimized Sulzer network with R₁=2kΩ C₁=100nF R₂=2kΩ R₃=510Ω C₂=5nF R₄=585Ω  R_TUNE=5kΩ

 

At the oscillation frequency where the phase shift passes through zero, the feedback loop gain must make up for a loss of about 9 dB, comparable to that for a Wien oscillator. The overall phase variation shown here is also comparable to that of the Bridged-T network for the typical value of α=4. The magnitude characteristic for the Sulzer network is roughly that of a band-reject filter, so in building a practical oscillator we can use the feedback polarity for the Bridged-T network.

Design Variations

Other choices of the component values in the versatile network of Figure 1 are possible to trade off network attenuation for total tuning range.

For example, raising the values of R₁ and R₂ from 2kΩ to 4.7kΩ reduces the amplitudes of V_A and V_B and widens the frequency range over which the -90 degree phase shift approximation is valid.

Running a simulation over a sweep of tuning resistance values now, with a new optimum value of 515 ohms for R₄, gives the very wide tuning range seen in Figure 9.

 

Figure 9. Simulated resonance frequency and transmission of the optimized wide-range Sulzer network over R_TUNE values with R₁=4.7kΩ C₁=100nF R₂=4.7kΩ R₃=510Ω C₂=5nF R₄=515Ω

 

The values of C₁=100nF and C₂=5nF allow tuning from 150Hz to over 100kHz. Larger capacitor values would allow covering virtually the entire audio range quite linearly in one band, and indeed this sort of oscillator is what Sulzer targeted in his original article. The tradeoff for this wide a frequency range, about three orders of magnitude, is that the attenuation through the Sulzer network has increased to about 15 dB, and requires a higher gain in the active feedback amplifier.

Taking a simulation at the center value of R_TUNE=5kΩ shows the deeper attenuation that must be overcome by the gain block in Figure 10. 

 

Figure 10. Simulation results for the optimized wide-range Sulzer network with R₁=4.7kΩ C₁=100nF R₂=4.7kΩ R₃=510Ω C₂=5nF R₄=515Ω  R_TUNE=5kΩ

 

Potentially this could mean slightly higher distortion, as this greater attenuation takes away from the loop gain that also suppresses any gain nonlinearity.

An Alternative Design

I hope you find Sulzer’s network of enough interest to try it out on a breadboard yourself. The textbook oscillator networks we have used for decades are attractive because they are simple enough to design with analytic derivations and have good phase and attenuation characteristics. Even though Sulzer’s network requires numerical simulation to design effectively, this invention seems to be very valuable for solving the practical tuning problems of the classical R-C oscillator networks.