~ Bicycle Dynamos and RIAA Amplifiers ~

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 upside-down the lights would not get any brighter ~ the only way to set the light output was to choose specific wattage 6V bulbs

Upside-down 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

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

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

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 ~ X_L∝ƒ and X_C∝1/ƒ ~ The actual values at any frequency are given by X_L = 2πƒL Ω and X_C = 1/2πƒC Ω but these Ωs do not produce any power

The white trace below represents the current flowing in the 1Ω resistor R in series with inductor L that has reactance X_L = 1Ω ~ If the fixed frequency were say 1kHz then the inductance must be 1/(2πƒ) = 159µH ~ 159 crops up a lot when 'normalising' reactive circuits ~ A capacitor with 1Ω reactance at 1kHz has a value of 159µF

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

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

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 √R²+X_L² 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 |Z|/Θ for other frequencies and for other values of R and L ~ This x y representation is easier than drawing waveforms but X_L needs to be calculated for each fixed frequency so its use is limited

The negative x axis indicates a 180˚ phase shift relative to values on the positive x axis ~ We can replace –R by j²R where j = √–1 which Leonhard Euler called an 'imaginary unit'

As we have defined j²R = 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 j³ ~ 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 Z² = X_L² + R² and in polar form |Z|/Θ can be calculated knowing the relationship between X_L and R is orthogonal (90˚) ~ In the expression Z = R + jX_L ~ j indicates the 90˚ relationship

Z_L = R₁ +jX_L is the simplest unambiguous form to define a resistor and inductor in series ~ For a capacitor and resistor in series Z_C = R₂ –jX_C ~ The terms R₁ +jX_L and R₂ –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 R₁ +jX_L and R₂ –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=(R₁+R₂)+(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 (R₁+R₂)

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 V_IN/R/t as shown dotted ~ At t=0 V_L=V_IN and ∂I/∂t must be V_IN/R/t

As current starts to flow V_OUT across R reduces V_L and (∂I/∂t) reduces proportionally ~ As time progresses V_L approaches zero and V_OUT approaches V_IN

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

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

Configured as shown above the circuit current at t=0 is V_IN/R ~ As the capacitor charges V_OUT increases as shown because V_R falling reduces the rate of charge current V_R/R

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 V_OUT = V_IN at time interval 1 and this is the definition of the time constant τ for a CR circuit

The inductor equivalent of the charge Q = It = CV stored in a capacitor is magnetic flux Φ = Vt = IL but unlike a capacitor we cannot remove an inductor from circuit without disrupting the current and loosing Φ

Rearranging Φ = Vt = IL we get t = IL/V and subsituting I/V with 1/R we get t = L/R and from the definition above this is the time constant for any LR circuit so τ = L/R seconds

Similarly for any CR circuit rearranging Q = It = CV we get t = CV/I and substituting V/I with R we get t = CR and from the definition of the time constant above we have τ = CR seconds

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

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

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 op-amp 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 op-amps 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 op-amps will only be amplified by 1x so d.c. at the output is near zero

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

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 (RIAA-1 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

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 op-amps 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 Low-noise audio frequency amplifiers

ref.5 ~ Editor S.W. Amos ~ BBC ~ Radio TV and Audio Reference Book published by Newnes-Butterworth 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)