

Back in the 1970s when I started earning pocket money to buy vinyl records I had a bicycle that I made from parts of other 'bikes' which I used for my paper round and getting to the library in the next town
I often rode when it was dark so I fitted a 6 volt dynamo similar to the one pictured to the rear wheel of the bike that powered a red rear light and dim front one for free No matter how fast I rode or turned the wheel with the bike upsidedown the lights would not get any brighter ~ the only way to set the light output was to choose specific wattage 6V bulbs Upsidedown and spinning the wheel really fast you could get a shock from the dynamo with no load connected ~ the voltage was far higher than 6V 

A bicycle dynamo uses permanent magnets moving relative to a fixed coil of wire to generate a sine wave voltage proportional to the speed of rotation ~ The output from a dynamo is higher than its rated voltage when not loaded but with the correct bulbs attached the light is fairly constant above an initial slow speed because the source impedance is mainly inductive and this impedance increases as the speed or frequency of current increases Alternators provide a fixed voltage for a wide range of load conditons unlike the fixed load the bicycle dynamo requires ~ They do not use magnets but use 'field windings' fed by a variable direct current to create a variable magnetic field ~ A voltage feedback regulator changes the field winding current to control the output voltage independent of rotation speed or load 

Pictured is a schematic of a bicycle dynamo and 2 bulbs ~ L_{D} is the inductance of the fixed iron cored coil and is also the a.c. voltage source ~ R_{D} is the d.c. resistance of the coil winding ~ The ratio of L_{D }to R_{D} is large and the impedance Z between b and chassis is referred to as 'mainly inductive'  
The dynamo impedance Z is the reactance X_{L} of the inductance L_{D} added to the resistance R_{D} ~ Reactance is the opposition to a rate of change of current through an inductor or capacitor ~ No other components exhibit reactance but all will have an impedance due to elements of L or C like the dynamo or a wire wound resistor
For an ideal dynamo the the intention is to have only a coil of wire generating a.c. but a coil has inductance L_{D} the wire has resistance R_{D }and winding it produces capacitance across it ~ Knowing the total impedance of what's inside will be useful when analysing the dynamo looking into point b relative to chassis 

The schematic above is commonly used to demonstrate dynamos and other permanent magnet a.c. generators and magnetic pickup cartridges but it is not accurate because the voltage source in series with the inductor gives an output at d.c. and does not give a voltage proportional to frequency ~ The schematic shown on the right is more representative when the inductor is also the generator  
A current source ~ capable of operation from d.c. to a.c. ~ placed across the generator inductor gives a more accurate model ~ The bulbs are replaced with a single load resistor R and the unavoidable winding capacitance is added across the inductor which is required for magnetic pickup cartridge and magnetic microphone analysis A point not to be missed is that all permanent magnet generators have a maximum current limit ~ From stationary or zero Hz the voltage developed across L_{D} is now modelled as truely proportional to speed of rotation or frequency ~ As the frequency increases there is a point where X_{L }= R_{D }+ R and the current in R is √0.5 x the current limit The current limit is set by physical properties of the generator like the strength and coupling of the permanent magnets and losses in the magnetic path and the coil winding method ~ The frequency where X_{L }= R_{D }+ R is also the point where the power in R is half the maximum it can be and is the –3dB frequency often referred to when defining frequency responses and filter responses ~ Where is this heading ? Standard wire wound resistors will have some series inductive reactance X_{L} and to a lesser extent some parallel capacitive reactance X_{C} but they should be mainly resistive ~ The reactive parts of the resistor or L_{D} in the dynamo do not dissipate any power but they will influence the flow of a.c. as frequency changes Reactances X_{L} and X_{C} are measured in ohms Ω and like resistors oppose the flow of current but only alternating current ~ The reactive value of X_{L} increases and of X_{C} decreases with frequency ~ 

Although at the first schematic above does not correctly represent a dynamo we can treat the circuit as a LR potential divider with the sereies voltage source as a fixed a.c. supply V_{IN} and the combined R_{D}+Bulbs = R = 1Ω as the output load ~ The dynamo inductor L_{D} is now simply L with a chosen reactance 

The white trace below represents the current flowing in the 1Ω resistor R in series with inductor L that has reactance The white trace also represents the voltage across the 1Ω resistor because the voltage across and current through a resistor are always in phase ~ 1V peak (pk) across the resistor produces 1A pk through it and the real power in the resistor is 1W peak or 0.5W average ~ Click links to see I_{R}×V_{R} ~ You can almost feel the red heat 

The current through the inductor is the same phase as the resistor current because they are in series but the voltage is 90˚ out of phase ~ the peaks of voltage across the inductor occur at the zero current crossing points where the rate of change of current is maximum ~ When the inductor voltage is peak its current is zero and vice versa
Where the resistor current or voltage are zero 3 times per cycle there is no power generated ~ In the inductor this occurs 5 times per cycle ~ But what happens between these zero points ? 

Inductors oppose the flow of a.c by generating a 'back emf' which is maximum at the maximum rate of current change (crossing zero) ~ In a dynamo L_{D} generates its own rotating magnetic field and because of this a dynamo with a fixed load cannot produce more power beyond a certain speed simply by turning faster even though the higher speed generates a higher voltage (inside the dynamo) The formula for back emf is V_{L} = –L(∂I/∂t) when the current through the inductor falls the voltage across it rises and multiplying I_{L}×V_{L }at any time we get the instantaneous power in the inductor ~ Power is generated between the 5 zero crossing points but unlike the resistors real power half of it is negative so the average per cycle is zero If the circuit is suddenly broken ~ especially when the current is at a peak ~ the rate of fall of current to zero will be much greater than at the sine wave zero crossings ~ often so much more that V_{L} = –L(∂I/∂t) now produces a back emf greater than the normal peak voltage and we may see an arc at the disconnection point If the resistive power is real the inductive or reactive power (half of which is negative) can be considered as apparent or imaginary ~ The current through a capacitor is also 90˚ out of phase with the voltage across it and there are also 5 zero crossings and imaginary reactive power when I_{C}×V_{C} is integrated over a whole number of cycles Ideal inductors have only inductance and no resistance or capacitance ~ In practice capacitors can be made 'more ideal' than inductors with very low inductance and losses and very high resistance ~ Practical inductors will have winding resistance and capacitance plus losses due to the magnetic field leaking and the core material if used At any frequency including zero Hz d.c. an ideal reactance does not dissipate any power ~ This condition for zero power despite current flowing only occurs when the voltage and current through a component are 90˚ apart and that component will be an ideal (not a real or practical) inductor or capacitor Any deviation from an ideal reactance can be accounted for by including additional reactance and resistance in the component model ~ Dielectric and other losses in a capacitor can be represented by series and or parallel resistors and sometimes by multiple CR paths to model the change of losses with frequency The proportionally higher losses in some inductors can also be represented by series and sometimes parallel resistors ~ The core material where used may have losses due to hysterisis and saturation due to high flux density ~ The coil winding resistance is an obvious loss which also increases with frequency due to skin effect 

The R–L potential divider circuit normalised to 1Ω described above or the dynamo or a magnetic pickup can be expressed on a graph where the relationship between R L and Z is easier to envisage ~ This graph is known as an impedance diagram
R and L are in series so the x axis can indicate the value of R or the voltage across R or the current in the circuit ~ The y axis shows that the inductive reactance or voltage across L is leading the current through L and R by 90˚ as in the waveforms above We can calculate the voltage V_{IN} across the series R L circuit and the impedance of the circuit ~ In order to have V_{L} leading V_{R} by 90˚ V_{IN} must lead V_{R }by 45˚ and lag V_{L} by 45˚ in this case where R and X_{L }are the same ~ The magnitude of the impedance Z is given by √R^{2}+X_{L}^{2} and Z = Z/Θ 

The impedance Z has to be expressed using 2 terms ~ magnitude and phase ~ so Z = 1.414Ω/45˚ ~ Applying V_{IN }= 1.414Vpk at the frequency where X_{L }=1Ω the input current will be 1Apk and the circuit waveform relationships will look like those shown above with V_{IN} leading V_{R }(and I_{R}) by 45˚ and lagging V_{L} by 45˚ On an impedance diagram X_{L }is strictly X_{L}/90˚ and X_{C} would be X_{C}/270˚ or X_{C}/–90˚ ~ We can determine 

An impedance diagram is a specific form of Phasor Diagram where it is usual to show resistive vectors R or V_{R }or I_{R} along the positive x axis as a 0˚ phase reference and reactive vectors ±90˚ as shown and impedances in the quadrants The negative x axis indicates a 180˚ phase shift relative to values on the positive x axis ~ We can replace –R by j^{2}R where j = √–1 which Leonhard Euler called an 'imaginary unit' As we have defined j^{2}R = R/180˚ it is reasonable to define X_{L}/90˚ as jX_{L }from which it follows that X_{C}/–90˚ becomes defined as –jX_{C }As shown The term jX_{L} means that X_{L} is multiplied by j and –jX_{C} that X_{C }is multipled by –j or j^{3} ~ Although j is an imaginary concept each multiplication by j rotates a vector by 90˚ 

Above the series L–R potential divider impedance was calculated as Z = 1.414Ω/45˚ because X_{L} R and Z follow the sides of a right angle triangle where Z^{2} = X_{L}^{2} + R^{2 }and in polar form Z_{L} = R_{1} +jX_{L} is the simplest unambiguous form to define a resistor and inductor in series ~ For a capacitor and resistor in series Z_{C} = R_{2} –jX_{C} ~ The terms R_{1} +jX_{L} and R_{2} –jX_{C} are known as complex numbers because they require 2 parts to be able to define both amplitude and phase with time or frequency The complex impedances R_{1} +jX_{L} and R_{2} –jX_{C} can be added together to give the impedance of a series RCL circuit ~ The real R values and the imaginary X values are added separately Z=(R_{1}+R_{2})+(jX_{L} –jX_{C}) ~ There will be a frequency where jX_{L} = –jX_{C} and the circuit is said to be resonant and the reactances cancel leaving only (R_{1}+R_{2}) Here and elsewhere on this site the j operator is simply used to indicate multiples of the 90˚ relationship and remind us that X_{C} and X_{L} are resistances that do not dissipate power ~ If a potential divider is made using two identical resistors the output voltage V_{OUT} is 0.5x V_{IN} but if one arm is a reactance equal to R the output is √0.5x V_{IN } Time is Constant ~ But how we perceive time Varies 

In the formula V_{L}= –L(∂I/∂t) the derivatives ∂ indicate there has to be a rate of change of current I with time t for V_{L} to exist ~ If the inductor were fed with a d.c. that increased in value linearly with time the voltage across L would be constant or put another way if the d.c. voltage across L were constant the current would increase linearly forever unless limited by say a series resistor 

Configured as shown above the initial rate of change of current through the circuit at switch on (t=0) is As current starts to flow V_{OUT} across R reduces V_{L} and The currents and voltages follow the curves defined and at time interval t1 V_{OUT} is 0.632 of V_{IN} ~ From t1 to t2 V_{OUT} increases 0.632 of its t1 value and from t2 to t3 . . . yeah it's exponential 

The natural response of an L↴R circuit to a step change of input is an exponential curve and has been for a long time ~ Differentiating the curve gives the same curve ~ The rate of rate of change of current is the same as the rate of change of current ~ If When the rate of change of current is kept linear at V_{IN}/R/t until V_{OUT }= V_{IN} and I limits at V_{IN}/R as shown dotted the time taken to reach V_{IN} is known as the LR circuit time constant and is given the greek letter tau τ which is also know as the exponential decay constant where e^{–1} = 0.368 

The L↴R circuit (L in series with the input and output across R) is a low pass filter which can also be made with a R↴C circuit (R in series with the input and output across C) ~ In both cases the output will be 0.993V_{IN} after 5τ and for most practical purposes when t>5τ 

Configured as shown above the circuit current at t=0 is The currents and voltages follow the curves defined and at time interval t1 V_{OUT} is 0.632 of V_{IN} ~ From t1 to t2 V_{OUT} increases 0.632 of its t1 value and from t2 to t3 . . . If the charging current were kept constant the capacitor would charge faster as V_{OUT} increases linearly (shown dotted) until 

The inductor equivalent of the charge Rearranging Similarly for any CR circuit rearranging 

Still Under Construction March 2022
While waiting for the next installment check out how not to make an RIAA preamp ~ this is a good example of the rubbish published by HiFi World 



A pole at 3.18µs (50.05kHz) "fits in" with the other RIAA time constants but why not choose 45kHz or 55kHz ? ~ For "normal" audio recording there is very little energy above 10kHz and the recording chain of microphones mixing consoles and tape machines ensures that if there were any energy above 20kHz it would be 10s of dB lower before it reached the cutting head and would most likely be distortion products
Even if some record cutting lathes have a pole above 20kHz ~ which most do ~ attempting to correct for this at replay would be a waste of time — The signal to the cutting head falls naturally above 20kHz due to drive circuitry limitation and there is often a "formal pole" beyond 40kHz which is 2nd order and corrects for phase up to 20kHz 

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Passive RIAA EQ or Active feedback EQ ?  

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Separate time constants or an all in one EQ Block ?  
The circuit above is a direct copy from the National Semiconductor (now Texas Instruments) data sheet for their "Hi–Fi Audio operational amplifier" the low noise and very low distortion LME49710 and LME49860 etc. and although it is only offered as a typical application it demonstrates how little thought goes into these designs It looks like the capacitor and resistor values have been carefully crafted to provide the most accurate RIAA EQ ~ The resistors are chosen from the E96 series and as stated are 1% tolerance but I have seen an implementation that recommends 0.1%! ~ The capacitor type and tolerance 5% puts paid to any precision 26.1kΩ+909Ω is 27.009kΩ and 3.83kΩ+100Ω is 3.93kΩ so why not use single E24 series 1% values 27kΩ and 3.9kΩ ? ~ Calculated using 27kΩ the capacitor 47nF∥33nF should be 81nF not 80nF so 68nF∥13nF or 51nF∥30nF ~ The other RIAA capacitor 27nF∥4.7nF∥500pF should be 27.769nF so rather than 5% polypropylene why not use 27nF∥750pF 1% polystyrene which are readily available and I would say are better capacitors for this application 



The same circuit shown above using standard E12 values (E24 1% in practice) gives a more accurate RIAA equalisation than the Texas Instruments "design" with its peak–peak deviation of 0.16dB using perfect components ~ The age old standard values of 47k 6k8 16n and 47n have a peak–peak deviation from perfect RIAA of only 0.05dB This circuit has a higher impedance passive lumped RIAA than the TI circuit but this does not affect the noise performance which is dominated by the source S/N and opamp input noise voltage ~ Both circuits as shown have the same overall gain which is about 530x or a 3mV cartridge input gives about 160mV equalised output The open loop or A_{o} gain and GBW of the LME49710 will allow the gain of each stage to be increased without affecting the S/N greatly ~ If R4 in my circuit were changed to say 100Ω a 3mV cartridge input would give 320mV output with theoretically better S/N ~ Also changing R6 to 1k would give about 0.7V out for a 3mV input Both circuits are d.c. coupled which is not desirable for a phono stage ~ Rather than use coupling capacitors which would also require additional resistors for d.c. biasing of the opamps it is possible to place capacitors in series with R4 or R6 or both ~ the values should be calculated to provide a high pass response say 3dB at 20Hz or lower With capacitors in series with both R6 and R4 the final slope of the the high pass response will be 12dB/oct and will provide a reasonable rumble filter which may not be appreciated by some critics even though it keeps the through signal path d.c. coupled ~ The input offset voltages of the opamps will only be amplified by 1x so d.c. at the output is near zero 

Comparison between the 2 circuits was made with PSpice computer modelling using an Analogue Behavioural Model (ABM) block to provide a perfect inverse RIAA source with a 0Ω output as shown on the right
The circuits were modelled with various op–amps and the other components shown in the schematic so the impedances around the RIAA EQ section were accounted for 



Whether the RIAA EQ is achieved in a single block or in separate sections or is passive or in a feedback loop the source and load impedances will always have an affect on the accuracy of the calculated values When the source or load impedances around passive RIAA sections or an EQ block ~ as shown above and left ~ are incorporated into the calculation not only is the overall response correct but gain wasted due to coupling can be minimised The valve circuit on the left uses the same lumped passive EQ as my LME49710 circuit above (RIAA1 in ref.3) but configured for current drive ~ The output of the triode can be considered as a current source with an internal shunt resistance r_{a} of about 31kΩ for I_{a} = 1mA The unbypassed cathode resistor R11 raises r_{a} to r_{a'} ≈ 126kΩ so the resistance affecting the EQ is 126kΩ∥75kΩ which is the required 47kΩ for this lumped RIAA block ~ Any load of the next stage must also be incorporated in the EQ calculation but as the total value of R1 is the only parameter that needs adjustment the calculation is simple and the response of this EQ block predictable 

In practice a valve stage configured as shown will require a pre stage for best S/N and for sufficient output level and will need a high input impedance following stage to prevent loading of the EQ block ~ C2+R2 and C1 would best be connected to ground and depending on the HT supply used the impedance of the EQ block could be made higher for more gain provided the S/N is not compromised by the Johnson noise of R1
Using a separate current source as the valve anode load with the EQ block returned to ground offers little or no advantage and would introduce semiconductors and or additional noise ~ because some amplification is required before the valve EQ stage for best S/N this may lead to the conclusion that separate time constant gain stages would be easier to use with valves 3 or more stages are often required to amplify the range of signal levels from magnetic cartridges and the RIAA TCs could be split across 2 or 3 of them but each stage has to provide correct loading for the EQ elements ~ When a single EQ block is used on the 2nd stage the 1st stage can be made a very high g_{m} valve with high I_{a} for best S/N High g_{m} valves tend to have a low r_{a} which is never well defined and varies with the slightest change of heater or HT voltage and with age ~ An unbypassed cathode resistor gives a higher r_{a'} but at the expense of gain ~ The 1.8kΩ cathode resistor of the ECC81 stage above makes r_{a'} very predictable and stable but the 1kHz gain is only about 6dB By placing all 3 (there are not 4 or 5) RIAA time constants in a single EQ block after a flat response high gain 1st stage the construction of a good RIAA pre amplifier is actually easier than using separate EQ sections whether using opamps or other devices with large amounts of feedback


References and further reading: ref.1 ~ Peter M. Copeland ~ BBC ~ Analogue Sound Restoration ref.2 ~ J. D. Smith ~ W.H. Livy (EMI Studios Abbey Road London) ~ Wireless World Nov 1956 & Jan 1957 ref.3 ~ Keith Snook 1982 ~ RIAA Lumped CR equalisation calculations ref.4 ~ E. A. Faulkner ~ The design of Lownoise audio frequency amplifiers ref.5 ~ Editor S.W. Amos ~ BBC ~ Radio TV and Audio Reference Book published by NewnesButterworth Ltd ref.6 ~ Allen Wright ~ Secrets of the phono stage ref.7 ~ Stanley Kelly ~ Stereo Gramophone Pickups (The State of the Art at the end of 1969) 
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