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Circuit Surgery Regular clinic by Ian Bell Gyrators Part 3 – Parametric Equaliser Circuits T wo months ago, we looked at gyrators and specifically how a gyrator built from a single op amp can behave like an inductor, by ‘impedance converting’ a capacitor to an inductor (see Fig.1). A key use of this circuit is in filters, where the gyrator provides advantages of size, weight, cost and potentially performance over wound inductors. Last month, we discussed graphic equaliser circuits using a combination of a cut/boost amplifier and a set of RLC filters in which the inductors are implemented using gyrators. Equalisers provide a means of adjusting the volume of an audio signal within a set of different frequency bands, and are widely used in sound processing. Cut/boost recap We discussed the operation of the cut/ boost circuit (Fig.2) in detail last month. To recap very briefly, the potentiometer (RP) can be adjusted from a maximum gain (boost) with the wiper at one end through unity gain (0 dB) with the wiper metric Equaliser Circuits to maximum attenuation at the mid-point at the other end. The maximum boost gain is given by: 𝐴𝐴!""#$ = 1 + 𝑅𝑅%& 𝑅𝑅' where R IF is the input and feedback resistor value (they must be the same) and grounded value. The 𝐿𝐿 = R(𝑅𝑅 −the 𝑅𝑅) )𝑅𝑅 𝐶𝐶 G (is ) 𝐶𝐶 ≈ 𝑅𝑅( 𝑅𝑅)resistor maximum attenuation in cut mode is the reciprocal of this (ACut = 1/ABoost). This means in decibels ACut dB = −ABoost dB, 1 𝑓𝑓* = R2 2𝜋𝜋√𝐿𝐿𝐿𝐿 Vin 𝑄𝑄 = 𝑓𝑓* 𝐵𝐵+ so the circuit has symmetrical maximum cut and boost. If we replace R G with a frequencydependent circuit which has a minimum resistance at a particular frequency then the boost gain, or cut attenuation, will peak at that frequency (depending on the potentiometer position). When the potentiometer is centred the gain will be unity at all frequencies. An RLC series circuit provides the requisite frequency dependence. An advantage of the circuit in Fig.2 is that multiple RP/RG circuits can be wired in parallel, without too much interaction. Using multiple RLC series circuits, with each tuned to a different frequency, provides the basis of equaliser circuits (see Fig.3) where the inductors can be implemented using gyrators (note the inductors are grounded). At the resonant frequency of an RLC series circuit the combined impedance of an (ideal) series inductor and capacitor is zero, so the impedance of the RLC circuit is equal to R. Thus, the peak gain and attenuation with R G in the cut/boot circuit (Fig.2) replaced with an RLC circuit (as in Fig.3) is found using RG = R in the above gain equations. As discussed previously, the effective series resistance of the gyrator ‘inductor’ is equal to R 2 in Fig.1, so this can be used to set the peak gain of the filter. – C + R1 2𝜋𝜋𝑓𝑓* 𝐿𝐿 1– input 𝐿𝐿 to ground behaves like an Fig.1. 𝑄𝑄 = Gyrator = 2 𝑅𝑅 𝐶𝐶 resistance. inductor𝑅𝑅 with series 50 𝑅𝑅 RIF Boost Vin RIF – RP Vout + Cut RG Fig.2. Cut/boost circuit. Parametric equalisers Last month, we concentrated on graphic equalisers, where all the channels have a fixed frequency and the only value the user can change is the level of cut/boost for each channel. Parametric equalisers provide more control over each band than graphic equalisers, with adjustable frequency and possibly bandwidth (often described as variable Q – see discussion on Q later). They may also be switchable between peaked (bandpass/bandstop) and highpass/lowpass response (shelving response). Parametric equalisers generally have far fewer channels than graphic equalisers (typically two to four, rather than up to thirty). This is similar to basic tone controls, but with much more flexibility in setting the frequency response. Parametric equalisers are found in applications such as audio RIF Vin RIF RP1 RP2 – RPN Vout + RG1 RG2 RGN C1 C2 CN L1 L2 LN Fig.3. Graphic equaliser using cut/boost circuit with multiple gain control filters. Practical Electronics | November | 2023 RIF Boost Vin RIF – RP + Cut SW1 Vout C RD L Fig.4. Cut/boost filter with parametric controls. processing systems, modular synthesisers Gyrators Part 3 – Parametric Equaliser Circuits Fig.6. Simulation results showing resistance variation with frequency for RLC and RL circuits. and effects units. The combined gyrator and cut/boost can provide more control over its behaviour the total ‘resistance’ of the RLC or RL Q factor than is required for a graphic equaliser. circuit (strictly, we should We mentioned Gyrators Partsay 3 –‘impedance Parametric Equaliser Circuits 𝑅𝑅%& filter Q (quality factor) last = 1 +but did not discuss it in much This concept is illustrated in Fig.4 where, magnitude’). We conclude that the RL 𝐴𝐴!""#$ month, 𝑅𝑅' in addition to the cut/boost control (RP) the circuit will result in a low-pass filter detail. Fig.7 shows a bandpass filter with the circuit in boost mode (low response – the gain peak is at the centre inductance (L) and series resistance (RD) 𝑅𝑅%&drops off on both sides. resistance at low frequencies produces frequency (f0) and can be varied. There is also a switch across 𝐴𝐴 =1+ higher gain). In cut mode, the RL circuit the capacitor. Shorting out the capacitor We !""#$ define filter cutoff (where it is 𝐿𝐿 = (𝑅𝑅 − 𝑅𝑅) )𝑅𝑅) 𝐶𝐶 ≈ 𝑅𝑅( 𝑅𝑅)𝑅𝑅𝐶𝐶' will act as a high-pass filter (low resistance ( considered results in an RL circuit instead of an RLC to block rather than pass at low frequencies produces lower gain, one. Last month, we saw that the RLC signals) as some gain value relative to the or more attenuation). As always, with series circuit has an impedance which peak (3dB below the peak is commonly the potentiometer centred, the gain 𝐿𝐿 = reached a minimum equal to the resistor used), so𝑅𝑅)we two cutoff frequencies (𝑅𝑅( −1 )𝑅𝑅)have 𝐶𝐶 ≈ 𝑅𝑅 ( 𝑅𝑅) 𝐶𝐶 will be unity at all frequencies. For the 𝑓𝑓*(f1=and f2). The bandwidth (Bw) of the filter value at the resonant frequency. With 2𝜋𝜋√𝐿𝐿𝐿𝐿 RLC circuit, as we saw last month, the only the inductance present there is no is the difference between the cutoff minimum resistance results in maximum resonance. Inductor impedance increases frequencies (f2 − f1). Q is defined as the boost gain or cut attenuation at the LC with frequency and is (ideally) zero at DC, ratio of centre 1 frequency to bandwidth: 𝑓𝑓* = resonant frequency. so the RL circuit has a total impedance 𝑓𝑓* 2𝜋𝜋√𝐿𝐿𝐿𝐿 Changing L in the circuit in Fig.4 will which increases from a minimum of R at 𝑄𝑄 = 𝐵𝐵 + change the centre frequency of the cut/ low frequencies towards infinity at very Gyrators Part 3 – Parametricboost Equaliser Circuits with the RLC circuit notch/peak high frequencies. (SW1 open), or the cutoff frequency of A similar definition can be used for 𝑓𝑓* 𝑄𝑄 = the high-pass/low-pass filter with the bandstop/notch filters. A high Q implies RLC and RL circuits 𝐵𝐵 RL circuit (SW1 closed). When the a sharp peak +(or notch). Fig.8 illustrates RLC and RL circuits are shown in the 𝑅𝑅 %& inductor is implemented using a gyrator, the LTspice schematic in Fig.5. The results 2𝜋𝜋𝑓𝑓*difference 𝐿𝐿 1 𝐿𝐿 between relatively high-Q 𝐴𝐴!""#$ =1+ = 2 𝑅𝑅' the effective inductance is set by the𝑄𝑄 = and filters. The Q for an in Fig.6 are plots of the total resistance 𝑅𝑅 low-Q 𝑅𝑅 bandpass 𝐶𝐶 resistor values (as discussed in the RLC filter is given by: (applied voltage divided by current) previous articles), and we have: for the two circuits. Recalling that the 2𝜋𝜋𝑓𝑓* 𝐿𝐿 1 𝐿𝐿 boost gain is inversely proportional to 𝑄𝑄 = = 2 𝐿𝐿 = (𝑅𝑅( − 𝑅𝑅) )𝑅𝑅) 𝐶𝐶 ≈ 𝑅𝑅( 𝑅𝑅) 𝐶𝐶 RG in Fig.2 – where RG corresponds to 𝑅𝑅 𝑅𝑅 𝐶𝐶 𝑅𝑅 𝑓𝑓, = 2𝜋𝜋𝐿𝐿 The approximation is Gain based on the fact that in Gyrators Part 3 – Parametric Equaliser Circuits Peak a1typical implementation 𝑅𝑅 𝑓𝑓* = 𝑓𝑓, = R1 is much larger than 2𝜋𝜋√𝐿𝐿𝐿𝐿 2𝜋𝜋𝐿𝐿 Cutoff R2. Given that the value of R2 is also the effective 𝑅𝑅%& resistance of the 𝐴𝐴series 1+ !""#$ = g𝑓𝑓*y r a t o r i n 𝑅𝑅 d'u c t o r, i t 𝑄𝑄 = makes sense to vary the 𝐵𝐵+ effective inductance using R 1 . The centre 𝐿𝐿 = (𝑅𝑅frequency ≈ 𝑅𝑅resonant ( − 𝑅𝑅) )𝑅𝑅) 𝐶𝐶(LC ( 𝑅𝑅) 𝐶𝐶 f0 Frequency frequency) of an RLC circuit is given by: f1 f2 2𝜋𝜋𝑓𝑓* 𝐿𝐿 1 𝐿𝐿 𝑄𝑄 = = 2 𝑅𝑅 𝑅𝑅 𝐶𝐶 1 Fig.7. Bandpass filter response. Fig.5. LTspice circuit to compare RLC and RL circuits. 𝑓𝑓* = 2𝜋𝜋√𝐿𝐿𝐿𝐿 Practical Electronics | November | 2023 51 𝑅𝑅 Gain Low Q High Q Frequency Fig.8. High and low Q bandpass filter responses. Fig.11. Simulation results from the circuit in Fig.10. RIF Boost An ideal LC circuit has no resistance and therefore has infinite Q (from the above equations). Of course, this is not achievable in practice because all components and wiring have some resistance. However, LC circuits can achieve large Q values due to the fact that they exhibit resonance. Fig.10. LTspice schematic for simulating the gyrator-based parametric equaliser in Fig.9. and potential energy. Capacitors and inductors both store energy (in electric and magnetic fields) and transfer the energy easily via current flow. Circuits containing just R and C, or just R and L do not show resonant behaviour because there is only one type of energy storage present. A child on a playground swing being swung by someone is pushed once on each cycle exactly as the swing reaches its peak amplitude. If the pushing is stopped the swing will continue oscillating for a while with decaying amplitude – this is similar to the oscillation which occurs when a bell is struck. If the pushing continues we have what is known as a ‘forced oscillator’. This corresponds to applying a sinewave to an LC circuit. In the case of a swing, if it is pushed at the right moments then the forcing frequency is at the resonant frequency and the result is a large amplitude oscillation (and hopefully a happy child). If one attempts to push a swing at the wrong times (ie, not at the resonant frequency) the oscillations will be much smaller in amplitude. This corresponds with high gain or attenuation at the resonant frequency of an LC circuit, which diminishes away from resonance. The rate at which the amplitude of a swing’s movement decreases when it is not pushed depends on factors such as friction and air resistance. This process is referred to as ‘damping the oscillation’. If these were not present the swing could continue forever from a single push. In the RLC circuit the amount of damping depends on R – the larger the value of R the more damping. Damping is the reciprocal of Q – the 52 Practical Electronics | November | 2023 – RP Vin RIF Vout + Cut SW1 C2 Response type Q factor R3 gain Resonance and damping R2 – C4 + C1 Frequency R1 Fig.9. Gyrator-based parametric equaliser (single channel shown). Resonance does not only occur in circuits; it is something we are aware of in everyday situations and popular culture, even if we do not always name it as such. Perhaps the most well-known, and most often quoted, examples are bells, glasses shattered by singers and children’s playground swings. These can illustrate various aspects of resonant systems. The physics of resonance involves the efficient transfer of energy between different forms. For a swing (or pendulum) these are kinetic peak/notch or low/highpass response type. Varying R 1 changes effective inductance and hence the centre/cutoff frequency. The Q/damping could be controlled by the gyrator’s effective series resistance, which is equal to R2. Unfortunately, R2 also controls the effective inductance (hence frequency) and the gain, which would make the circuit difficult to adjust. Therefore, the circuit in Fig.9 has an additional series resistance R3 which can be used to control Q, this also affects the gain, butCircuits not the frequency. This means that Gyrators Part 3 – Parametric Equaliser the effective resistance of the RLC circuit is R1 + R3. Simulations 𝑅𝑅%& 𝐴𝐴!""#$ = 1 + circuit for simulating the An LTSpice 𝑅𝑅' gyrator-based parametric equaliser is shown in Fig.10. The three resistance values in the gyrator are configured as investigating 𝐿𝐿 = (𝑅𝑅(parameters, − 𝑅𝑅) )𝑅𝑅) 𝐶𝐶 ≈ 𝑅𝑅to( 𝑅𝑅facilitate ) 𝐶𝐶 the effect of varying them. The default value of RG3 is set very small compared with RG2 so that it has little effect in the simulation unless it is stepped to a 1 𝑓𝑓* = different value (see later). The default 2𝜋𝜋√𝐿𝐿𝐿𝐿 circuit is effectively the same as the 250Hz channel in the graphic equaliser discussed last month. The response for a range𝑓𝑓 of potentiometer settings (N) * is 𝑄𝑄shown in Fig.11. Implementation = 𝐵𝐵+ of the potentiometer in LTspice was discussed last month. Restimulating the circuit with the capacitor shorted out results in the response shown in Fig.12. As discussed Fig.13. Simulation results from the circuit in Fig.10 showing the effect of changing the 2𝜋𝜋𝑓𝑓* 𝐿𝐿this1produces 𝐿𝐿 above, a high-pass or loweffective inductance via gyrator resistor RG1. 𝑄𝑄 = = 2 pass 𝑅𝑅 response 𝑅𝑅 𝐶𝐶 depending on R P. The standard equation for the cutoff frequency (fc) of an RL filter is: Fig.12. Simulation results from the circuit in Fig.10 with C2 shorted. 𝑓𝑓, = 𝑅𝑅 2𝜋𝜋𝐿𝐿 The effective inductance of the gyrator is 2.03H (see last month) with R = 1.677k (= RG2, the series resistance since RG3 is set very small), we get a cut-off frequency value of 131Hz. This is the –3dB frequency of the circuit at full boost or full cut (top and bottom traces), but not at other potentiometer settings. Parameter stepping Fig.14. Simulation results from the circuit in Fig.10 showing the effect of changing the effective inductance via damping resistor RG3. equation above shows that Q is inversely proportional to R in the RLC circuit. In Fig.4, therefore, RD controls the amount of damping and hence the Q of the filter. Gyrator parametric equaliser Fig.9 shows a gyrator-based parametric equaliser based on Fig.4 and the Practical Electronics | November | 2023 preceding discussion. One channel is shown, but multiple channels can be implemented using multiple copies of the potentiometer and RLC circuitry, as shown in Fig.3. The gyrator circuit replaces the inductor in Fig.4. R P is the cut/boost control and operates as previously discussed. SW1 selects the Fig.13 shows how the frequency of the filter can be varied using gyrator resistor R G1 to vary the effective inductance. This is achieved by adding the following SPICE directive to step RG1 (in this case from 50kΩ to 250kΩ in 20kΩ steps). .step param RG1 50k 250k 20k Note that we comment out the stepping of the potentiometer setting, so that 53 it takes the default value (N = 1, for full boost). ;.step param N 0 1 0.1 Other values of N could easily be used if required. The plot shows that the centre frequency is varied as expected, with constant peak gain, but the Q of each response is not the same. Similarly, the plot in Fig.14 shows how the Q of the filter can be varied using the additional series (damping) resistor RG3. Again, this is achieved by adding a SPICE directive to step the value of the resistor (in this case from 1mΩ (effectively zero) to 10kΩ in 1kΩ steps). Fig.15. Simulation results from the circuit in Fig.10 stepping both RG1 and RG2. .step param RG3 1m 10k 1k The previous .step statement (for RG1) is commented out. The responses in Fig.14 show that varying the Q also varies the gain, but the centre frequency is constant. By choosing the right combinations of RG1, RG3 and N it is possible to create a set of responses with two parameters (Q, f, gain) constant, while the other is varied, but this is not necessarily easy and may only be possible over a limited range of values. means they are more difficult to use effectively than graphic equalisers. If you run both the R G1 and R G2 parameters stepping together then the resulting plot (Fig.15) looks a bit psychedelic and is probably not very useful as a graph, except that it illustrates the wide variety of responses which can be achieved – and this is a fixed cut/boost value, which of course can also be varied as needed. As noted last month, the very wide range of responses from parametric equalisers Simulation files Most, but not every month, LTSpice is used to support descriptions and analysis in Circuit Surgery. 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