Related to the twin-tee, the Hall network is a notch filter that’s also tunable over a wide frequency range using a single potentiometer.^{1} In its passive form, the notch bandwidth is broad. However, active electronics can narrow the response and extend the network to other applications.
The Hall network is a third-order RC circuit (Fig. 1). R1 and R2 represent the split portions of the tuning potentiometer, which typically includes series end resistors to prevent adjustment all the way to zero resistance (where the network ceases to function). Signals at frequencies well below the notch pass through R3 and signals at frequencies well above the notch pass through the three capacitors.
At the notch frequency, the separate paths have equal transmission magnitudes but opposite phase. Therefore, they sum to zero. In one example, frequency response for a 1-kHz notch frequency shows that phase response is approximately zero degrees at the notch (Fig. 2).
Because the network is third order and has bridged nodes, the math can be rather arduous.^{2} Patient application of standard circuit methods eventually produces three important equations. Equation 1 gives the general transfer function of the network:
Equation 2, which gives the notch frequency in hertz, is correct only if R3 is the required value per Equation 3:
To achieve infinite null for the general case of C3 = C1 and C1 = K ×?C2, the required value of R3 is given by Equation 3:
For applications that require the notch depth be as deep as possible, use a small series potentiometer to fine tweak R3 to compensate for component tolerances. Once R3 is fine-tweaked, it doesn’t need further adjustment for any tuning setting. Note that R3 affects tuning, but it should not be used for tuning.
Historical Context
Back in 1955, Mr. Hall indicated that C3 equals C1 (the math confirms that ideally those would be matched for the circuit to work properly over its tuning range), but said nothing about the relationship of C2. In Rufus P. Turner’s 1968 book Waveform Measurements,^{3} an active bandpass filter used the Hall network with C1 = C3 = 10C2. Such a ratio has been used in a few other places featuring the Hall network.
In 1975, an article appearing in Ham Radio magazine detailed two applications of the Hall network.^{4} The author, Courtney Hall (no relation to the designer), used three capacitors all of the same value, and noted that they should all be matched for optimum notch depth.
The only known commercial use of the Hall network is as a tunable bandpass filter for the General Radio Type 1232-A tuned amplifier and null detector, which debuted in 1961.5,6 An examination of the instrument’s manual reveals that all capacitors have the same value. This product is still sold today by IET Labs Inc. as part of the GenRad 1620 capacitance bridge (www.ietlabs.com/). Minimum notch bandwidth (i.e., highest Q) occurs when all capacitors maintain the same value (K = 1). In fact, this article uses that ratio for all examples.
Tuning And Q Characteristics
The tuning characteristic (Equation 4) is very shallow near the middle of the potentiometer and very steep near the ends (Fig. 3). The curve was derived by a modified version of Equation 2 where R1 was replaced with a tuning factor (TF) that varies from 0 to 1 as the pot rotates from full counter-clockwise to full clockwise and R2 is replaced with 1 – TF:
The 0.5 factor normalizes the minimum frequency at the center of pot rotation to 1.0.
The Q (resonant frequency divided by the –3-dB bandwidth) of any passive RC network is low. With three identical capacitors, the Hall network’s Q is a maximum of 0.177 with a 0.5 tuning factor and decreases to zero when the tuning factor is 0.0 or 1.0:
Active electronics can increase Q up to a practical maximum of several tens. However, the best results for applications requiring high Q come when the tuning factor falls roughly into the 0.1 to 0.9 range.
Design For Tuning Range
Since the tuning curve is symmetric at about the midpoint of R1 + R2 (Fig. 3, again), it makes sense to add resistances to either side of the potentiometer (Fig. 4) to limit the adjustment range to the region between 0.5 and 1.0. In terms of symmetry, the 0.5 to 0.0 region also could have been used, but it makes more intuitive sense to the author to pick the right side of the plot.
Depending on the application, the tuning ratio (the maximum divided by the minimum desired frequencies over the full range of the potentiometer) might range from 1.1 or less up to a practical maximum of about 4 with a linear potentiometer. High tuning ratios, which must utilize the steep portion of the tuning curve (Fig. 3, again), become impractical because tuning is very compressed at the high end. Special taper potentiometers can extend the practical tuning ratio to over 10 (see references 5 and 6).
The choice of starting point on the tuning curve affects tuning linearity. Starting at 0.5 maximizes the tuning nonlinearity for any tuning ratio. Using higher starting points reduces tuning nonlinearity, but a point of diminishing returns occurs at around 0.7. With a starting tuning factor of 0.7, tuning ratios less than about 1.2 results in fairly linear tuning, and tuning ratios up to about 3 aren’t excessively compressed at the high end.
When determining the end resistors, one must first establish the tuning factor range. For good results, the starting or low tuning factor (TF_{LOW}) should be around 0.7 when the potentiometer wiper is at the full counter-clockwise position. Then Equation 6 is used to calculate the required high tuning factor (TF_{HIGH}) to achieve the desired tuning ratio when the wiper is at the full clockwise position:
Equation 6 was derived from Equation 4 by first normalizing the result to unity for TF_{LOW} and then solving the resulting quadratic equation for TF_{HIGH} with Equation 4 set to the desired “tuning_ratio.”
With the extreme tuning factors known, the next step is to determine a good resistance for the potentiometer (R_{P}). Equation 7 was empirically derived to provide a geometric center estimate for a potentiometer’s practical resistance:
The equation is based on experience that for a given frequency, there’s a range of R and C values that provides good results. Frequency is nominally the geometric center of the tuning range in hertz, but it can be any frequency within the tuning range due to the broad flexibility in using the result.
When choosing a convenient value for R_{P}, it should be between roughly one-fourth to about four times the geometric center value from Equation 7. The end resistors, R_{A} and R_{B}, are then calculated as:
The examples in this article employ a Hall network designed to tune from 750 to 1500 Hz, which is a tuning ratio of 2. TF_{LOW} was chosen to be 0.7 and TF_{HIGH} computed to be 0.944. Since the geometric center for R_{P} computed to be about 9.5-kΩ, a 10-kΩ potentiometer was chosen.
R_{TOTAL} computed to be 40,917 Ω, R_{A} computed to be 28,642 Ω, and R_{B} computed to be 2291 Ω. R_{A} was rounded to a standard value of 27-kΩ and R_{B} was rounded to a standard value of 2.2-kΩ. With these values, the actual tuning ratio computed to 2.01:
If the rounded resistor values produce a tuning ratio too different from desired then adjust R_{B} up or down a standard value as needed. Figure 5 shows the tuning curve.
Design For Capacitors
For three identical capacitors, Equations 2 and 3 for the notch frequency and R3 simplify to Equations 12 and 13:
The required capacitance for each of the three capacitors is calculated by inverting Equation 12 and noting that at the minimum notch frequency (full counter-clockwise of the potentiometer), R1 = R_{A} and R2 = R_{P} + R_{B}:
The capacitance at the minimum frequency of 750 Hz computed to be 6.75 nF, which was rounded to the standard value of 6.8 nF.
From Equation 13, R3’s value for theoretical infinite notch computed to be 6 × (27,000 + 10,000 + 2200) = 235,000 Ω. It can be rounded to the standard value of 240 kΩ for applications that don’t require optimum notch depth. For the circuits in this article, R3 was 220 kΩ in series with a 50-kΩ series potentiometer and adjusted for optimum notch depth. The completed design is now ready to be employed in three useful applications (Fig. 6).
Active Tunable Notch Filter
The ground terminal of an active tunable notch filter’s Hall network can be bootstrapped with adjustable positive feedback from the output (Fig. 7). The positive feedback factor (PFF) varies from 0 when R6 is at full counter-clockwise (CCW) to 1.0 or at full clockwise (CW). Increasing the PFF boosts the Q of the circuit, narrowing the notch bandwidth. The resulting Q is the basic Hall network Q at the given tuning factor (Equation 5) multiplied by the Q boost factor:
The practical maximum for Q is around 30, since adjustment for higher values becomes very touchy.
Unfortunately, notch depth in a physical circuit gets shallower with increasing Q, even with the best setting of the optional “Notch Depth Optimize” adjustment. The mathematical model doesn’t show this effect. Although not proven, it’s suspected that small phase lags in U2 and U3 are a contributing factor. Therefore, some kind of phase compensation would improve operation. For the prototype circuit, the measured null depth at 1 kHz with PFF = 0 was –54 dB, and with PFF = 0.9 it was –45 dB.
TL084 op amps were used, although most any op amp would work fine. When the PFF is zero (R6 at full CCW), the circuit is simply the standard Hall network with unity-gain buffers. As the potentiometer turns clockwise, the notch width narrows and depth decreases until the circuit has no filtering at unity positive feedback and may oscillate. Except for the notch region, this circuit has unity gain for any setting of the tuning control or PFF.
Active Tunable Bandpass Filter
A bandpass filter is formed by subtracting the output of the notch filter from the input (Fig. 8). For best results, resistors R7, R8, R9, and R10 should be matched.
A useful characteristic of this bandpass filter is that the peak response at resonance is 1.0, independent of Q as set by the PFF. This means that as Q is adjusted, signals at the resonant frequency remain unchanged in amplitude while signals at frequencies away from resonance vary in amplitude. The notch output is simultaneously available. The Q of the bandpass is identical to that of the notch.
One-Pot Tunable Sinewave Oscillator
A low-distortion one-pot tunable sinewave oscillator (Fig. 9) is constructed by combining the Hall network with the light bulb from Bill Hewlett’s famous oscillator.^{7,8} The resistance of the lamp filament, which increases with the thermally averaged rms voltage across it, is used to control the positive feedback gain to the correct level (Fig. 10). The thermal time constant of the filament is long compared to a half-cycle of oscillation. Thus, its resistance remains constant throughout the oscillation waveform—an important characteristic for low distortion.
There’s no voltage across the lamp filament before oscillation commences, so it has relatively low cold resistance. Thus, the positive feedback path voltage-divider gain to the U1 non-inverting input is relatively high, and the negative feedback path gain at the notch frequency is relatively low.
Because the positive feedback path gain exceeds the negative feedback path gain, the system poles are in the right-half s-plane and oscillation grows exponentially. The lamp filament then heats with the increasing oscillation voltage across it. Heating causes its resistance to increase, reducing the positive feedback path gain and the system poles’ shift to the left.
This thermal loop produces two effects. First, oscillation stabilizes at the amplitude where the positive and negative feedback path gains are equal and the system poles are exactly on the jw axis. Second, the filament resistance-voltage characteristic regulates the amplitude since there’s a unique voltage that results in equal feedback gains.
Hewlett determined that a good operating point on the lamp resistance-voltage curve resides in the lower power region. Here, the change in resistance with applied voltage is high resulting in high stabilization gain.
Another advantage of this operating level is that because the filament’s thermal time constant is long compared to higher operating voltages, distortion is lower. Hewlett’s oscillator used a 120-V, 3-W lamp operating at about 7 V rms. Correspondingly, the lamp (Chicago Miniature 8-3995, 130 V, 20 mA in a common T-3 1/4 miniature bayonet base) in this application was chosen to operate at about 7 V rms. The lamp produces no visible light at this low power level.
Texas Instruments’ TLE2071 was chosen for U1 because it’s capable of higher than typical output current to drive the lamp. Moreover, it features wider than typical bandwidth. Amplifier gain at the oscillation frequency runs into the many hundreds. Therefore, gain-bandwidth product of the amplifier must be sufficiently high—typically over 1000 times the oscillation frequency.
Due to the low resistances in the positive feedback path, it’s important that the ground impedance in the circuit be very low. Otherwise the high gain at the oscillation frequency will continually amplify small voltage drops across the ground, to the point that it may be difficult to set the oscillation amplitude.
Depending on the particular ground impedance and connections, the circuit might oscillate to the full rail-to-rail output voltage of U1 independent of the setting of R6. Or, it might have a hysteresis effect, in which case the circuit may or may not oscillate depending on R6’s setting, yet no setting will stabilize the amplitude at a desired level.
If these problems occur, one quick, albeit not permanent, remedy (short of improving the ground system) is to tweak R4 off of the theoretical ideal value. This will make the notch shallower, requiring less gain for oscillation.
Each of the aforementioned problems was experienced with various prototype versions of the oscillator. However, the circuit works very well if constructed properly.
That said, the initial setting of R6 should be full clockwise. This ensures that oscillation starts, although the waveform will be clipped at the op-amp rails (use a higher resistance for R5 if oscillation doesn’t start). Then adjust R6 counter-clockwise until the output amplitude becomes approximately a 20-V p-p sinewave.
The filament’s thermal servo system is under-damped, so the amplitude will fluctuate for a number of seconds before stabilizing. The ultra-low-distortion sinewave (less than 0.003%) is worth the wait.
For different designs, it’s best to choose a lamp with a rated voltage around 20 times the target rms output voltage. A lamp with the lowest rated current for a given voltage eases the op amp’s current drive requirement.
The next step is to determine the resistance of the filament at the target operating voltage. R5 should be roughly in the range of 1/300 to 1/1000 of that filament resistance. R6 fine-adjusts the operating point. The op amp must have sufficient output current to drive the lamp peak current, as well as a large signal gain-bandwidth product that’s at least 1000 times higher than the frequency of oscillation.
One undesirable characteristic of this circuit is that the sinewave amplitude varies with tuning. It’s caused by variation of transfer gain of the Hall network at the notch frequency with tuning. For stable oscillation, the positive feedback factor must also vary by the same amount. This means that as the Hall network transfer gain decreases, the oscillation amplitude must increase to cause the positive feedback factor to decrease by the same amount (i.e., the filament’s resistance must increase).
Depending on the tuning range and specific values of the components, the amplitude variation with tuning could range from “ceasing to oscillate” to “clipping at the op-amp power rails.” R6 can always be adjusted to compensate, but that’s inconvenient. The lamp-stabilized oscillator is attractive for its simplicity if there’s a narrow tuning range or it becomes permanently set at a specific frequency.
Electronic Servo System
The only way to obtain stable oscillation amplitude over a wide frequency range is to use an electronic servo system instead of the lamp to control the gain. This approach nearly achieves the low distortion of the lamp circuit while rapidly settling to the amplitude set point.
Because of the much higher impedances in the positive feedback path, this circuit is practically immune to the ground impedance issues plaguing the lamp version. A sample circuit was adapted from the amplitude control for the Hewlett-Packard model 239A oscillator (Fig. 11).^{9}
Components D1, R11, and C6 detect the peak of the sinewave. R11 limits peak capacitor charging current from the op amp to prevent distortion. The output of U2A is the control error voltage, which is the detected peak voltage minus twice the reference voltage (VREF = 5 V for this example) at the non-inverting input. Thus, at zero error, the peak sinewave output voltage equals twice the reference voltage plus the diode drop, or about 10.6 V, for this example.
U2B is a proportional plus integral controller that applies a variable gate bias to JFET Q1, which operates in its ohmic region as a voltage-controlled resistor. The integral gain equals 1/(R8 × C5) and the proportional gain equals R7/R8. R7, which is used to adjust the damping, becomes a compromise between not too under-damped and low distortion.
Ideally, the channel resistance of Q1 would be constant throughout a cycle of oscillation, but the resistance is modulated by the ripple voltage on C6 amplified by the proportional gain. The lowest distortion occurs when R7 is as small as can be tolerated (i.e., lowest proportional gain and lowest damping). Therefore, the settling time isn’t too oscillatory when changing frequencies. Zener diode D2 limits the voltage swing of U2B to the gate control range of Q1. D2’s voltage may need to be adjusted, depending on the characteristics of a particular JFET.
This example uses a type J310 JFET, but most any JFET should work well with appropriate adjustments to R4 and perhaps D2. The channel resistance is high when the gate-source voltage is near pinch-off and low when the gate-source voltage falls to near zero.
The required attenuation factor in the positive feedback path of U1, which is many hundreds, divides into two parts: the voltage division of potentiometer R3; and the voltage division between R4 and the resistance of the JFET channel. By breaking the attenuation into two parts, a single, relatively low-resistance potentiometer value can be used to scale R4 up to any feedback resistance.
If the oscillation amplitude falls below the set point, then the error output voltage of U2A is positive. This drives the output of U2B negative, creating high channel resistance for Q1. In turn, the positive feedback factor becomes relatively high, causing the oscillation amplitude to rise until the error voltage is zero.
In the ohmic region of operation, the JFET channel’s resistance is fairly linear for small voltages across the channel (less than a few tenths of a volt positive or negative). C4, R5, and R6 form an ac voltage divider that couples half of the small signal at the JFET’s drain to its gate, which significantly improves linearity for low distortion.^{10 }
R3’s adjustment range divides into three regions:
- Oscillation won’t occur if R3 is too counter-clockwise. U2B will drive the JFET gate bias toward pinch-off in an attempt to maximize the gain.
- Oscillation amplitude won’t be controlled, and the peaks may go into clipping if R3 is too clockwise. U2B will drive the JFET gate bias positive in an attempt to minimize the gain.
- Between the excessive conditions is a zone that controls the oscillation amplitude and sinewave amplitude is independent of the R3 setting. The optimum setting for lowest distortion is at the onset of where U2B’s output starts to quickly go negative. This should be checked over the entire tuning range. Furthermore, R3 should be adjusted for the best compromise so U2B’s output is always near 0 V (preferably not positive more than around a tenth of a volt). Selection of R4’s resistance must be based on the characteristics of the particular JFET. The useful adjustment range will be very narrow if R4 is too small—the circuit won’t oscillate if R4 is too large. Although there’s broad leeway, a good value for R4 equates to the optimum adjustment of R3 being about one-third to one-half up from the ground end. This value has to be determined through experimentation.
An H-P model 3580A spectrum analyzer was used to measure the distortion (at 1 kHz) of both circuits. The amplitude of the fundamental was adjusted to be at the 0-dB reference level. The sweep range was 0 to 10 kHz, and the resolution bandwidth was 30 Hz.
With the lamp version, no harmonic was visible above the instrument noise floor 90 dB below the reference level. Thus, distortion is less than 0.003%. The JFET version showed a slight indication at the second harmonic.
Some intriguing history exists with these two oscillator circuits. Hewlett used the lamp filament to increase negative feedback to stabilize his Wien bridge oscillator while in the Hall circuit the lamp filament is used to decreases positive feedback to stabilize the oscillation. Another interesting tidbit is that the lamp control comes from H-P’s first oscillator and the JFET control comes from the company’s last purely analog oscillator (before digital synthesis took over in the 1980s).
One rather bold question arises when it comes to notch-type oscillators: “How do they work at all?” Considering the transfer gain at the notch frequency is theoretically zero, even an infinite gain op-amp would be insufficient.
The answer? The net zero phase operating frequency, including the small time delay through the op amp, is not precisely the notch frequency. Thus, a finite, albeit small, transmission exists through the network. The op amp then amplifies that small voltage by many hundreds to produce the sinewave output.
Acknowledgement
I would like to give a big personal thanks to the designer of the Hall network, Henry P. Hall, now retired from a distinguished career at General Radio and recipient of the IEEE Joseph F. Keithley Award in Instrumentation and Measurement in 2004. He graciously provided me with helpful information, inspiration, and encouragement.
References
- Hall, Henry P., “RC Networks with Single-Component Frequency Control,” IRE Transactions, Sept. 1955, pages 283 and 284.
- Kuhn, Kenneth A., “Scan of hand derivation of symbolic equations for Hall network as done during a number of lunch hours in 1979,” www.kennethkuhn.com/electronics/hall_network.jpg.
- Turner, Rufus P., Waveform Measurements, Hayden Book Co., New York, 1968, pp. 34-35.
- Hall, Courtney, “Tunable Notch Filter,” Ham Radio Magazine, Communications Technology Inc., Greenville, N.H., September, 1975, pp. 16-20.
- Instruction Manual, Type 1232-A Tuned Amplifier and Null Detector, General Radio Company, West Concord, Mass., 1961, page 23 (schematic diagram).
- The General Radio Experimenter, General Radio Co., West Concord, Mass., July, 1961, pp. 3-7. A pdf can be downloaded from www.ietlabs.com/genrad/experimenters/ .
- Hewlett, William R., “A New Type Resistance-Capacity Oscillator,” thesis submitted to Leland Stanford University, Stanford, Calif., June 1939.
- Operating and Service Manual, Model 200B Audio Oscillator, Hewlett-Packard Co., Palo Alto, Calif., circa 1955, page 8 (schematic diagram).
- Model 239A Oscillator Operating and Service Manual, Hewlett-Packard Co., Loveland, Colo., 1978, pp. 8-9 and 8-10.
- Evans, Arthur D., Designing with Field-Effect Transistors, McGraw-Hill Book Co., N.Y., 1981, pp. 244-245.