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Circuit Surgery Regular clinic by Ian Bell Gyrators Part 2 – Equaliser Circuits Fig.1. Graphic equaliser. L ast month, we started to look at gyrator circuits following a mention of their use in a response to a post on the EEWeb forum. Gyrators are part of a range of fascinating circuits, often with niche applications, that can generally be described as ‘impedance converters’. In terms of its formal definition, a gyrator is like a variant of the transformer which converts input voltage to output current, and vice versa, with a scaling factor called gyration resistance. In more practical terms, a key property of gyrators is that they can be used to create circuits that behave ‘like inductors’ without actually using any inductors. They effectively ‘impedance convert’ a capacitor to an inductor. Where large inductors are required, the total weight, size and cost of a gyratorbased implementation can be much lower. Furthermore, real inductors tend to be less ideal and suffer from more problems than real capacitors, so converting a good quality capacitor to an inductor may give good performance despite the additional circuity. A common use of gyrators is to implement resistor-inductor-capacitor (RLC) resonant circuits for use in bandpass or bandstop (notch) filters. The fact that the effective inductance of a gyrator is controlled by a resistor means that it is relatively straightforward to make such filters tuneable – a potentiometer can be used, so a variable capacitor or inductor is not required. Perhaps the best-known use of gyratorbased filters is in analogue audio equalisers, which require multiple bandstop/bandpass filters, so this month we’ll look at the circuitry involved. 62 Equalisers of the impact of settings on the sound and other equipment compared to simpler graphic equalisers. The fact that gyrators’ behaviour can be modified via variable resistors makes them suitable for parametric equalisers as well as inductor-less graphic equalisers. Of course, modern sound processing is often done digitally and equalisation is no exception; however, analogue systems are still used and the circuits are interesting to investigate. Equalisers provide a means of adjusting the volume of an audio signal within a set Gyrator circuit recap of different frequency bands. This allows One of the simplest and most popular much more specific control of the sound gyrator circuits is shown in Fig.2. We than simple bass and treble controls. discussed this in detail last month and Equalisers are used in a range of situations showed that the equivalent inductor is not and applications, including recording perfect, even if the circuit is built with ideal studios, sound systems in live venues components. There is some series and Gyrators Part 2 – Equaliser and guitar pedals. Equalisers require aCircuits parallel resistance (see Fig.3). The values set of filters which can provide either cut in Fig.2 and Fig.3 are related as follows: or boost to the signal within a specified 𝐿𝐿 = (𝑅𝑅! − 𝑅𝑅" )𝑅𝑅" 𝐶𝐶 range – that is, a set of filters which can be adjusted to provide bandpass or bandstop 𝑅𝑅# = 𝑅𝑅" behaviour at the required frequencies. Graphic equalisers (see Fig.1) typically 𝑅𝑅$ = 𝑅𝑅! − 𝑅𝑅" have a number of sliders to adjust signal cut and boost in a set of fixed frequency bands. Some equalisers (for example, in recording A real op amp will introduce further studios) have 31 bands, but others, such as imperfections, 𝑅𝑅( particularly at high 𝐴𝐴%&' = guitar pedals, have fewer. Typically, each frequencies where it bandwidth limits 𝑅𝑅)* + 𝑅𝑅( band provides boost or cut of up to ±12 the circuit (discussed last month). The op to ±15dB. The centre frequency of each amp slew rate will also limit the response band varies logarithmically – each band to fast changing inputs. is at a fixed multiple of the previous band, 𝑅𝑅)* 𝑅𝑅)* + 𝑅𝑅( 1 + boost = typically ×2 (octave), ×√2 = ×1.414 (half 𝐴𝐴+,,-' Cut=and 𝑅𝑅( 𝑅𝑅( 3 octave), or × √2 = ×1.260 (third octave). To make an equaliser we need filters The specific frequencies are standard in which can vary between bandstop or the industry. 1 Parametric equalisers provide R2 𝑓𝑓. = more control over each band 2𝜋𝜋√𝐿𝐿𝐿𝐿 iin th a n g r a phic equalis er s , Zin – Zin with adjustable frequency C and possibly bandwidth. + 2𝜋𝜋𝜋𝜋. 𝐿𝐿 They may also be switchable 𝑄𝑄 = 𝑅𝑅 between peaked (bandpass/ L R1 bandstop) and high-pass/lowpass response (referred to as a ‘shelving’ response in audio jargon). They are more complex, and to be used effectively they Fig.2. Gyrator circuit to produce behaviour equivalent require a deeper understanding to an inductor. Practical Electronics | October | 2023 Zin RIF iin RS Vin – RP RIF RIF Vout RIF Vin + Vout + RG RG RP – RP L Fig.3. Equivalent circuit for the gyrator shown in Fig.2. Fig.5. Cut/boost circuit – equivalent circuit in full cut mode. Fig.6. Cut/boost circuit – equivalent circuit in full boost mode. the ends of the potentiometer range as shown. The input resistor and feedback resistors must have the same value (RIF). The grounded resistor (RG) is replaced by a frequency-dependent circuit to implement a filter. As is commonly done in the analysis of linear op amp circuits with negative feedback, we assume that the op amp controls its output to achieve zero volts across its inputs. This means that the voltage across RP is always zero, however, this does not mean the current in RP is always zero because current flowing through the wiper can produce equal and opposite voltages across the two parts of the potentiometer track. We also make use of the common assumption in op amp circuit analysis that zero current flows into the op amp’s input. We can see that the cut and boost gains are reciprocals (A Cut = 1/A Boost). This means in decibels ACut dB = −ABoost dB, so the circuit has symmetrical maximum cut and boost. Fig.7 shows the general case where the wiper is not at the extreme ends of the potentiometer. The potentiometer is represented as two resistors (R P1, R P2) – the track resistances from the wiper to the two ends. Consider what happens when the wiper is exactly in the middle of the track, so R P1 = RP2. As previously noted, the voltage across the potentiometer is zero, which means that V P1 = VP2. With the wiper at the halfway position the two parts of the potentiometer track resistance (RP1 and RP2) are equal, therefore, for VP1 and VP2 to have equal magnitudes the currents IF (feedback current) and IIn (input current) must be equal too. The current in the feedback resistor is equal to the current in RP1 (as shown) because we assume that no current flows into the op amp input. Similarly, the current in RP2 is equal to the input current. The feedback and input resistors are at the same voltage at their op-amp-input ends due to the zero voltage difference between the op amp’s inputs. Given that the currents through them and their resistances are also equal, the voltages at their other ends must be equal too. Therefore, the input and output voltages must be equal. We conclude that when the potentiometer wiper is at the centre position the circuit has unity gain. bandpass at a given centre frequency; one that can be adjusted continuously between some maximum attenuation and maximum gain. The amount of attenuation or gain needs to be adjustable by a single control (potentiometer) which does not affect the centre frequency of the filter as it is changed. If the ‘amount’ is set to maximum boost (gain) the circuit will act as a relatively strong bandpass filter at the centre frequency. If the amount is set to a maximum cut (attenuation) the circuit will act as a relatively strong bandstop filter at the centre frequency. In the middle of this range will be a neutral point where there is no filtering – the circuit has unity gain at all frequencies. Increasing the gain from the neutral point will result in an increasing strong peak in the bandpass response, and similarly for Equivalent circuits bandstop in the attenuation direction. For Fig.5 shows the situation with the a graphic equaliser, each band will have potentiometer wiper fully at the cut end. a cut/boost (amount) control at a fixed The circuit is effectively a potential divider, frequency. For a parametric equaliser, formed by the input resistor (RIF) and RG, the centre frequency and possibly other connected to a unity-gain amplifier (the Gyrators Part 2 – Equaliser Circuits parameters can also be adjusted. feedback resistor provides 100% feedback). Fig.4 is a cut and boost circuit which The potentiometer resistance (shown faded provides the gain control function we out) has very little effect on the circuit 𝐿𝐿 = (𝑅𝑅! − 𝑅𝑅" )𝑅𝑅" 𝐶𝐶 require. A basic form of the circuit, which because, as just noted, it has zero voltage does not include any filtering is shown. across 𝑅𝑅 it and current in RG does not flow # = 𝑅𝑅" We will describe its operation in terms of through it. The circuit therefore acts as an gain and attenuation adjustment before attenuator the gain set by the potential 𝑅𝑅$ = 𝑅𝑅with ! − 𝑅𝑅" looking at how the filtering is included. divider effect. Using the potential divider Like the gyrator, this is not the easiest equation, we get: of circuits to intuitively understand, but Part full 2 – Equaliser 𝑅𝑅( we can look at theGyrators full cut and boost Circuits 𝐴𝐴%&' = operation reasonably straightforwardly to 𝑅𝑅)* + 𝑅𝑅( determine the gain/attenuation range. The = (𝑅𝑅! −the 𝑅𝑅" )𝑅𝑅situation " 𝐶𝐶 potentiometer RP is the cut/boost control, Fig.6𝐿𝐿shows with the potentiometer wiper fully at the boost with the maximum effects occurring at 𝑅𝑅𝑅𝑅)* 𝑅𝑅")* + 𝑅𝑅( # = 𝑅𝑅 end.=The 𝐴𝐴+,,-' 1 +circuit = is effectively a standard RIF 𝑅𝑅 ( ( non-inverting op 𝑅𝑅 amp amplifier, with 𝑅𝑅$ = 𝑅𝑅! − 𝑅𝑅" the gain set in the usual way by the Boost feedback (RIF) and grounded (RG) resistors. RP – Vout RIF Again, the potentiometer resistance Vin 1 + (shown 𝑓𝑓. =faded out) 𝑅𝑅 has very little effect Cut 2𝜋𝜋√𝐿𝐿𝐿𝐿 ( on the𝐴𝐴circuit %&' = for the same reasons as 𝑅𝑅)* + 𝑅𝑅( the full cut case. The gain is given by RG the non-inverting amplifier formula: Fig.4. Cut/boost circuit. Practical Electronics | October | 2023 2𝜋𝜋𝜋𝜋. 𝐿𝐿 𝑄𝑄 = 𝑅𝑅)* 𝑅𝑅)* + 𝑅𝑅( 𝐴𝐴+,,-' = 1 +𝑅𝑅 = 𝑅𝑅( 𝑅𝑅( 1 RIF IF Vin RIF RP1 VP1 RP2 VP2 – Vout + IIn RG IIn + IF Fig.7. Cut/boost circuit – wiper part way along track. 63 respectively, and that the gain for N = 0.5 (wiper halfway) is unity (0dB). The frequency response curves are flat because the circuit has no frequencydependent components and the range is below the bandwidth of the semi-idealised op amp model (LTspice UniversalOpamp2, as used last month). Effect of RG Fig.8. Cut/boost circuit LTspice schematic. Component values and simulation Selection of component values is fairly straightforward. The value of the potentiometer is not very critical – it does not affect the full cut and boost gains or the midpoint unity gain but will have some influence on intermediate values. Values in the range 5kΩ to 20kΩ are usually suitable, and 10kΩ is typical. Larger values are acceptable, but larger resistances will cause more noise in the circuit. The values of RG and RIF are found based on the required maximum cut/boost and will typically be in the range of a few kiloohms. For example, if we want a full boost gain of 12dB (= ×1012/20 = ×3.981) and we choose RIF = 5kΩ, then from the non-inverting amplifier gain formula RG = 5000/(3.981 − 1) = 1.677kΩ. The full cut gain will be −12dB. To simulate the cut/boost circuit in LTspice we need to model the potentiometer as we vary the wiper position. If we define the wiper position as a value N, where N = 0 with the wiper at one end and N = 1 with the wiper at the other, then RP2 = NRP and RP1 = (1 − N)RP. In LTspice we can use a node voltage or parameter value to control N and vary this in the simulation. We can implement the two resistance formulae using behavioural resistors but need to prevent the resistors from becoming zero or negative in value. This can be done with an LTspice limit function. We discussed modelling a potentiometer in this way in more depth in Part 5 of our series on Electronically Controlled Resistance (Circuit Surgery, January 2023). Fig.8 shows an LTspice schematic for simulating the cut/ boost circuit using the values just discussed and implementing the potentiometer as described in the previous paragraph. The wiper position is set using a parameter value (N) that is stepped over 11 values from 0 to 1 (step of 0.1). The results are shown in Fig.9 and confirm the full cut and boost are −12 and +12dB The amount of boost or cut (except at the unity gain central point) depends on the value of the grounded resistor. Increasing RG reduces the gain in the boost half of the wiper range (note that full boost (ABoost) is inversely proportional to RG). Increasing RG also reduces the amount of attenuation in the cut range (similarly, full cut (ACut) is proportional to RG – increasing ACut (gain) means a reduction in attenuation). So, depending on which half of the pot the wiper is on, changing RG has the same effect on either the amount of cut or the amount of boost. As RG tends towards infinity (open circuit) the gain becomes unity for all wiper positions – the equations for full boost and cut (ABoost and ACut) given above both simplify to ≈ RG/RG = 1 if RG is much larger than RIF so that (RG + RIF) ≈ RG. Fig.10 shows the equivalent circuit of the cut/boost circuit with infinite RG. We see that it is an inverting amplifier (100% feedback). The input resistor has no effect, assuming zero current flow into the op amp (zero voltage drop across it). RP has little effect as the wiper is effectively disconnected and there is zero volts across it, as previously discussed. We can demonstrate the effect of the value of RG by altering the circuit in Fig.5 to make the value of RG a parameter and step this through a range of resistor values (with a fixed wiper position). The results are shown in Fig.11 and Fig.12 and are for a wiper position halfway into the boost and cut ranges (N = 0.75 and N = 0.25 respectively). In each case, a given resistance RIF Vin RIF RP – Vout + Fig.10. Cut/boost circuit – equivalent circuit with infinite RG. Fig.9. Results from simulation of circuit in Fig.8. 64 Fig.11. Simulation results for various RG values in the cut/boost circuit with wiper position N = 0.75 (boost). Practical Electronics | October | 2023 Fig.12. Simulation results for various RG values in the cut/boost circuit with wiper position N = 0.25 (cut) Fig.13. (left) Passive RLC notch (bandstop) filter. value produces the same amount of cut or boost. For example, for RG = 3kΩ (red traces) the gains are 3dB (boost) and −3dB (cut). The largest resistor value (1MΩ) provides close to unity gain (0dB). RLC notch For an equaliser circuit we need to be able to apply cut and boost in specific frequency bands with the same characteristics for cut and boost. If we replace RG with a notch circuit – one that has very high er Circuits impedance at all frequencies apart from a narrow range, where the impedance is low – then the cut/boost circuit will (𝑅𝑅! unity 𝐿𝐿 = − 𝑅𝑅" )𝑅𝑅 " 𝐶𝐶 at most frequencies (near have gain infinite RG). However, around the notch 𝑅𝑅# = 𝑅𝑅" the low value of the circuit frequency impedance will result in either increased 𝑅𝑅$ = 𝑅𝑅! − 𝑅𝑅" gain or increased attenuation, depending on the wiper position, in accordance with the discussion in the previous paragraphs. Of course, 𝑅𝑅(with the wiper at the centre of 𝐴𝐴the %&' = potentiometer the gain will be unity – 𝑅𝑅)* + 𝑅𝑅( this will apply at all frequencies. The required notch behaviour can be obtained using an RLC series circuit. This well-known circuit 𝑅𝑅)* 𝑅𝑅)* + 𝑅𝑅( is show as an LTspice 𝐴𝐴+,,-' = 1 + = schematic in𝑅𝑅Fig.13, configured as a 𝑅𝑅( ( bandstop filter. The centre frequency (LC resonant frequency) is given by: 𝑓𝑓. = 1 2𝜋𝜋√𝐿𝐿𝐿𝐿 Practical Electronics | October | 2023 𝑄𝑄 = 2𝜋𝜋𝜋𝜋. 𝐿𝐿 Fig.14. Simulation results from circuit in Fig.13. Upper trace is gain from input to output, lower trace is resistance of RLC series circuit. If we want f0 = 250Hz and we choose C = 200nF then L is given by 1/(2πf0)2C = 2.03H. We set R = 1kΩ, somewhat arbitrarily. The simulation plots in Fig.14 show the gain and the resistance of the series combination of the resistor, capacitor and inductor close to f0. We see the sharp notch bandstop gain response, and the fact that the series resistance is minimum at f0 and is equal to R. Since we are using an ideal inductor and capacitor, their combined impedance is zero at f0 in this circuit. The maximum attenuation (notch depth) shown in this simulation depends on the number of data points – the more datapoints there are the closer they are likely to be to the exact value of f0 (ideal infinite attenuation). Cut/boost with RLC obtain a cut/boost filter with a centre frequency of 250Hz. These values are used in the LTspice circuit in Fig.16. This is configured to step through a set of potentiometer wiper positions in the same way as for the circuit in Fig.8. The results in Fig.17 shows the frequency response at the various potentiometer settings. We see that the response can be adjusted from bandstop, through flat unity gain to bandpass. The maximum gain and attenuation is ±12dB, as designed. The circuit has unity gain at frequencies well away from the centre frequency, irrespective of the potentiometer position. For a graphic equaliser we need to be able to control multiple frequency bands. Fortunately, this is easy to do with the circuit in Fig.12 – we just add more potentiometers and RLC circuits in parallel, as shown in Fig.18. The fact that there is zero volts across the potentiometers minimises their interaction, so that each potentiometer and its associated RLC circuit can When the RLC series circuit is used in place of RG in the cut/boost circuit, as shown in Fig.15, we are using the impedance variation of the whole circuit to control the gain – an output is not taken from the potential divider formed by R and LC as it is in the circuit Fig.13. At resonance (f0) the RLC series circuit has RIF its minimum impedance (equal to R), Boost which will result in the maximum cut RP – or boost for the circuit in Fig.15. Thus, Vout RIF Vin the maximum cut/boost of the circuit + in Fig.15 is set using RG in exactly Cut the way as for the circuit in Fig.4 and RG Fig.8. Therefore, if we use the same values for the resistors as in Fig.8 (as C described above) we will obtain a cut/ boost filter with a maximum gain or L attenuation of ±12dB. The centre frequency for cut/boost for the circuit in Fig.15 is found using the LC resonance frequency equation given above. If we use the same values as for the circuit in Fig.13 we will Fig.15. Cut/boost bandstop/bandpass filter. 65 there will be gaps between them where the gain cannot be controlled. If the responses are too flat there will be too much interaction between controls for adjacent bands and it will be difficult to independently adjust adjacent bands. In practice, even with suitable filters, the response will not be perfectly flat with all the gains set equally – there will be some ripple in the gain. Gyrators Part 2 – Equaliser Circuits Although the ideal case is no ripple it is not as important as" 𝐶𝐶it might seem because the 𝐿𝐿 = (𝑅𝑅! − 𝑅𝑅" )𝑅𝑅 point of graphic equalisers 𝑅𝑅# = 𝑅𝑅" is to adjust different bands differently, so a flat response 𝑅𝑅$ = 𝑅𝑅! − 𝑅𝑅is" not the typical set up. Fig.16. LTspice schematic for simulating the cut/boost filter in Fig.15. The sharpness of a frequency response peak is measured by the Q provide reasonably independent control can be accounted for by designing for (quality) factor which is defined as f0/ over a different frequency band. The a slightly higher gain. 𝑅𝑅( interaction between the circuits for The responses of adjacent bands need 𝐴𝐴BW, where f0 is the centre frequency and %&' = 𝑅𝑅 + 𝑅𝑅 ( bands is not zero; one effect is to reduce to overlap sufficiently so that if their BW is )* the bandwidth, defined as the the maximum gain/attenuation from gains are set the same the response is frequency range between the points the value applicable for a single band close to flat across those bands. If the where the gain falls to 3dB below the filter with the same R IF and R G. This filters for each band have too sharp a peak peak on 𝑅𝑅)* both 𝑅𝑅)*sides. + 𝑅𝑅( The larger the Q the 𝐴𝐴+,,-' = 1+ = peak. The appropriate Q sharper the 𝑅𝑅( 𝑅𝑅( value for the filters in a graphic equaliser depends on the closeness of the bands (octave, half octave…). For octave bands, values of1 Q around 1.5 to 2 are probably 𝑓𝑓. = suitable (different sources vary on the 2𝜋𝜋√𝐿𝐿𝐿𝐿 values to use). The Q for an RLC filter is given by: 𝑄𝑄 = 2𝜋𝜋𝜋𝜋. 𝐿𝐿 𝑅𝑅 This implies, for the circuit in Fig.16 and Fig.18 that resistor RG controls both the Q and the full cut/boost gain. An equaliser circuit can be designed by finding RG as above, then using the required Q and f0 to find L for each band, and then the LC circuit f0 (resonance) equation to find C in each case. Fig.17. Simulation results for the circuit in Fig.16. RIF RIF Vin RIF RP1 RP2 – RPN Vin RG2 RGN C1 C2 CN L1 L2 LN Fig.18. Graphic equaliser using cut/boost circuit with multiple gain-control filters. 66 – RP Vout + Vout + RG1 RIF C2 R2 C1 – + R1 Fig.19. Cut/boost filter using a gyrator instead of an inductor. Practical Electronics | October | 2023 Fig.20. LTspice circuit for simulating a two-band equaliser circuit. Fig.21. Simulation results from the circuit in Fig.19 with M = 0 (unity gain in 500Hz band). Fig.22. Simulation results from the circuit in Fig.19 with M = 0 (maximum cut in 500Hz band). Using gyrators Finally, we get back to the gyrator. An advantage of the circuits in Fig.16 and Fig.18 is that they have grounded inductors, which means that we can replace the inductors with gyrators, as shown in Fig.19. We do not need a separate RG resistor because the series Practical Electronics | October | 2023 resistor in the gyrator (R2 in Fig.2 and Fig.19) takes on this role. The circuit design follows the procedure just described (with RG replaced with R2). R1 is typically a relatively large value, for example 100kΩ, as used last month. When the inductor value is found this is used to set C1 in the gyrator, using the required L plus R1 and R2 in the inductor equivalence formula. C2 in Fig.19 is found in the same way as C for the RLC version. Fig.20 shows an LTspice circuit for a cut/boost ‘equaliser’ with just two octave bands centred on 250Hz and 500Hz, both with a nominal maximum cut/boost of ±12dB (same values as in Fig.16). The 250Hz band has the same RLC capacitor as in Fig.16 (200nF) and the gyrator capacitor is based on the 2.03H inductor value from that circuit using: C1 = L/(R1 − R2)R2 = 2.03/((100k –1.677k) × 1.677k) = 12.31nF in the gyrator. The 500Hz stage was designed in the same way. The simulation uses parameters for wiper positions for both bands, but currently the 250Hz band is stepped and the 500Hz band is fixed. The specific values in Fig.20 results in unity gain in the 500Hz band (M = 0.5). Fig.21 shows the results, which ideally would be the same as those in Fig.18. There are some differences – in particular the peak gain is lower due to the presence of the other potentiometer and gyrator. Fig.22 shows what happens with M = 0 (full cut fixed on the 500Hz band). Note the ripple in the full-cut-in-both-bands case (green trace at the bottom of Fig.22) – ideally this would be flat from 250 to 500Hz as discussed above. Fig.22 illustrates the wide range of response shapes which can be obtained using multiple cut/boost filter bands. Most real equalisers would have more bands. Simulation files Most, but not every month, LTSpice is used to support descriptions and analysis in Circuit Surgery. The examples and files are available for download from the PE website. 67