# output resistance and damping factor

### Internal Resistance

different names for the same

Internal resistance and source voltage determine the efficiency of a voltage source. On the basis the example of a battery, the influence to Hifi amplifier technology is to be exlained.

### Introduction

Every battery has an internal resistance.

 Why? Fig. 1 the Ohm's law applied by the example 1.5 V of a Mignon battery. Load e.g. 10 ohms of constant of source tension (open circuit voltage) of 1.5 V. If we attach now a very large load (very small resistance 0.01 ohms) at the battery, then we expect a large current. The small battery cannot supply the large current. The Ohm's law applies nevertheless also to batteries? Naturally, it applies also to batteries as to all voltage supplies. What happened? The actual maximum current is smaller than expected, i.e. it must be positioned still another resistance in series to the load. Yes, that is output resistance. Output resistance determines the maximally possible current. The maximally possible current is called short circiut current.
 The effective resistance is the sum of load resistance and internal resistance. Internal resistance has chemical causes and additionally Ohm's resistances inside the battery. With the transformer it is the Ohm's resistance of the copper coil and the saturation of the magnetic core. Each voltage supply could supply its specific physical causes only a limited current. Fig. 2 of R internal resistance limits the maximum current of a voltage supply. Simplified said the more largely a battery, the smaller the internal resistance. A car battery has a small internal resistance. An electronic Piezo lighter has a very high voltage, but an extremely large internal resistance and therefore only very small rivers flowing. A power station generator generates also relatively high voltages in combination with a very low internal resistance, thus it can make very high power available.

### Equivalent circiut diagram

Fig. 3 shows the battery, internal resistance Ri, the source voltage U and the load resistance RL - here a lamp. Drawn in also the current  I, which is equivalent large by all elements of the equivalent circuit diagram.

The short circiut current of a circuit can be determined with high impedance sources, in which this are short circuit and which the current is measured. Ri = U/I short-circuit. For low impedance sources this method is not only inaccurate, but also lethal (spark, burns), inaccurately and mostly also harmful for the source. E.G. with a car battery this should be never tried, the developing currents, amounts of heat and health risks (sparks and burns) is enormous.

In order to measure the internal resistance of a low impedance source, first a measurement of the source voltage without load can take place. In the second measurement the source is loaded, with a load, which corresponds to a usual operating condition.

Fig. 4 output resistance can be computed in this way simply. First a measurement of the source voltage without attached load. Afterwards with a normal load. The delta of the two voltages divided by the current calculates the internal resistance.

### Static Output Resistance

A static internal resistance meant, the resistance is constant for all load conditions. This condition does not occur  in nature, a dependence on the load is always available, sometimes more, sometimes less. For many voltage supplies, e.g. batteries may be spoken of a static internal resistance, which remains constant in many operating points. With voltage supplies, which are not regulated, a "static" internal resistance can be defined

### Dynamic Output Resistance and Amplifier

Here it acts over an amplifier with negative feedback

Output resistance in the case of negative feedback under closed loop conditions.

A regulation is based on the principle of the negative feedback and a much higher open loop gain than a closed loop gain. Also a Hifi amplifier has an output resistance. Since a closed loop is present, I speak now of a dynamic output resistance. Dynamically therefore, the amplifier tries to regulate the output voltage, since it accomplishes continuously an actual value with desired value comparison. In other words, an amplifier is a DC source voltage with a very fast adjustable internal resistance. Since a constant continuous (following the signal) dynamic adjustment of internal resistance happens here, I call this internal resistance: dynamic internal resistance in the respective operating point. The measurement of dynamic internal resistance effected as in fig. 4 described, as signal source is set an alternating voltage on the amplifier. Dynamic internal resistance has validity for this operating point only.

In a coordinate system output resistance is represented as y axis and as x axis e.g. the pertinent operating point (e.g. the respective output voltage). Still the first derivative could be formed by this function, in order to meet further statements.

In electronics and audio world many names were found for output resistance.

• internal resistance

• dynamic output resistance

• Dämpfungsfaktor (german)

• Damping Factor

### Damping Factor definition

In principle all listed designations describe dynamic internal resistance in the unit ohm. A privileged position assumes thereby the damping factor, it into relation to loudspeaker impedance (4 ohms or 8 ohms) is set, the unit shortens itself ohms. The damping factor corresponds therefore also to output resistance, only in transformed way of writing.

 Fig. 5 shows the mathematical definition of the damping factor. Quotient of attached loudspeaker impedance and output resistance in the respective operating point of the hifi amplifier. The damping factor is not a constant size, as it is often written. It stands in dependence to variable internal resistance and loudspeaker impedance. From technical view the indication of a damping factor is meaningful only if the boundary conditions of the measurement are indicated. with which frequency dynamic output resistance was measured. with which amplitude dynamic output resistance was measured with which load impedance dynamic output resistance was measured It would result in for example at all no sense, it would be even wrong to determine a very good dynamic internal resistance during small load and to calculate a quotient with 4 ohms, in order to receive a good damping factor. Very meaningfully it is to be measured output resistance with that load, for which also the damping factor is indicated later. Without data of the boundary conditions the indication of a damping factor from technical view is not sufficient.

The output resistance and the open loop of an amplifier depends essentially on the frequency, temperature and the load. It lets general say, a very low dynamic output resistance is a good condition for a low distortion factor. Dynamic output resistance depends to 100% on the open loop. If a dynamic output resistance rises during very high load conditions, then the cause lies in the fact that in this operating condition the open loop was drastically reduced.

The dynamic output resistance is a "regulatedoutput resistance" and can take thereby very small values. Of course only validity for the range of the still adjustable load. Thus it is clear why also a small operational amplifier can have a small output resistance.

 Fig. 6 shows a somewhat unusual model for further discussion for a common used amplifier type.  To see: two source voltages, with in each case an adjustable output resistance. Now if a positive input signal applies against the automatic controller (control Amp), then upper resistance is setted to a very high impedance value, so that only very small current can flow by the upper source. At the same time to it from the automatic controller the lower resistance reduced to the desired current by the loudspeaker flows so long. More exactly said, until the desired voltage at the output adjusts itself. For negative voltages conditions are changed. The feedback line is a part of constant control (regulation) whether two  resistances were setted also correctly. And the more precisely control happens, the more highly is the open loop. If the desired voltage is zero V, then both resistances are adjusted to the same value during same source voltage. However to which value? Good question - until the desired idle current of the output stage is reached. The transistors in the amplifier takes over the role of adjustable internal resistances.

Large amplifiers small output resistance? - small amplifier large output resistance?

A large hifi amplifier - that means not that dynamic internal resistance is automatically very small. No, a very well made small amplifier can be quite better than a less well developed large amplifier. The substantial difference is: the small amplifier cannot follow with rising load any longer, its dynamic output resistance rises strongly. While with the large one only with larger load dynamic output resistance rises somewhat.

#### Effects on a bad amping factor

Keep in mind, an indication of a damping factor is technically only meaningfully under indication boundary condition only if for the DUT's the damping factors are present, which were measured under similar boundary conditions, then becomes a number comparison possible. Effects on bad damping factor are, the output signal correspond less to the input signal. The effects are linear and also nonlinear distortions.

Also a loudspeaker can generate voltages. An amplifier with low dynamic output resistance can compensate these disturbances better than one with a high damping factor. Loudspeakers generate inadvertent voltages to their connectors, similarly like the function mode of a microphone. The amplifier must reject these. A measure for it like generated disturbances being suppressed, is e.g. the load Rejection Ratio. Damping factors affect the sound, loudspeaker manufacturer have thereby experience.

A relatively linear controlled system reduces also the effect of the generation of nonlinear distortions. With which we concerned at the open loop amplifier without feedback. An open loop amplifier meant, the amplifier does not have an overall negativefeedback of the output signal. Only individual amplifier stages using negative feedback, e.g. through emitter resistances or through cathode resistances with a tube amplifier.

A feedback free amplifier works in a steered mode and not as regulation. During directly comparable achievement of both amplifiers types, the steered type has a higher output resistance than the regulated amplifier. The developer of a steered only amplifier directs his special attention to linearize his circuit. This method helps to keep distortions low.