GB2298983A - A power amplifier for driving an output transformer - Google Patents

A power amplifier for driving an output transformer Download PDF

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Publication number
GB2298983A
GB2298983A GB9505428A GB9505428A GB2298983A GB 2298983 A GB2298983 A GB 2298983A GB 9505428 A GB9505428 A GB 9505428A GB 9505428 A GB9505428 A GB 9505428A GB 2298983 A GB2298983 A GB 2298983A
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Prior art keywords
power
voltage
transformer
primary
power amplifier
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GB9505428A
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GB9505428D0 (en
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Martin Leach
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    • HELECTRICITY
    • H03ELECTRONIC CIRCUITRY
    • H03FAMPLIFIERS
    • H03F3/00Amplifiers with only discharge tubes or only semiconductor devices as amplifying elements
    • H03F3/181Low frequency amplifiers, e.g. audio preamplifiers
    • H03F3/183Low frequency amplifiers, e.g. audio preamplifiers with semiconductor devices only
    • H03F3/185Low frequency amplifiers, e.g. audio preamplifiers with semiconductor devices only with field-effect devices
    • HELECTRICITY
    • H03ELECTRONIC CIRCUITRY
    • H03FAMPLIFIERS
    • H03F3/00Amplifiers with only discharge tubes or only semiconductor devices as amplifying elements
    • H03F3/26Push-pull amplifiers; Phase-splitters therefor
    • H03F3/265Push-pull amplifiers; Phase-splitters therefor with field-effect transistors only

Abstract

A power amplifier includes a power device (Q5) which has a control terminal (GT1) and two current carrying terminals (DT1) and (ST1). The amplifier has a power supply with upper and lower voltage levels (Vcc) and (G). The power device is coupled to a transformer (T2) which has a primary (T2P) and a secondary (T2S) and the arrangement is such that the power device and the primary form a follower configuration between the upper and lower voltage levels, the current carrying terminal which follows the control terminal (GT1) being connected via the primary to one of the voltage levels. When a load (40) is connected across the secondary (T2S) and a voltage signal applied to the control terminal (GT1) a current substantially linearly dependent on the voltage signal flows through the primary and variations in the current are linearly coupled to the load (40). The figure shows a differential form of the amplifier.

Description

POWER AMPLIFIER The present invention relates to a power amplifier.
It has particular utility as an audio power amplifier in a high-fidelity reproduction system.
Most conventional high quality audio reproduction systems comprise one or more source components( e.g. a CD player, a tuner, a cassette deck or turntable), a preamplifier, a power amplifier and loudspeakers. In operation, the source component provides a low voltage signal, which is subsequently amplified to a desired voltage level by the pre-amplifier stage (the voltage level being controlled by the volume control), the amplified voltage signal then being used to drive the power amplifier which, in turn, delivers the required current through the loudspeaker coils.
If the music is to be faithfully reproduced, the dynamic output/input characteristic of each component must be linear. It has been found particularly difficult to provide such a linear characteristic in the power amplifier component of hi-fi reproduction systems.
One type of conventional power amplifier is shown schematically in Figure 1.
The amplifier comprises an upper voltage supply rail (Vcc), a lower voltage supply rail (-VEE), a PNP follower 1, an NPN follower 2, and a loudspeaker 3.
In operation, the low power audio signal from the pre-amplifier stage is supplied to the input terminal 4 of the PNP follower 1. Although the voltage of the signal has been amplified by the pre-amplifier stage, the signal current must be amplified in order to drive the loudspeaker 3. In the present case, a first power amplification is provided by the PNP follower 1. The output voltage signal from the PNP follower 1 (audio signal voltage plus an offset voltage) is connected to the input of the NPN follower 2. Provided transistor Q2 is operating in its active region, the voltage of the emitter 5 of transistor Q2 follows the audio signal voltage. Two conditions must be met in order for the transistor Q2 to remain in the active region.Firstly, the peak voltage of the audio signal must be at least a few tenths of a volt below Vcc Secondly, at the minimum voltage of the audio signal, the current through resistor R1 must be slightly greater than the current flowing from ground through the loudspeaker 3. Assuming that R1 is of a similar resistance to the loudspeaker 3, this means that, at the minimum voltage of the audio signal, the power dissipated in the resistor R1 must be slightly greater than the power dissipated in the loudspeaker 3.
Hence, the minimum power dissipated by the resistor R1 is required to be greater than the maximum power output of the loudspeaker 3.
At the minimum voltage of the input signal, most of the current required to flow through the resistor R1 is drawn through the loudspeaker 3. However, as the input signal rises towards ground, not only does the current through R1 increase, the proportion of this current drawn through the loudspeaker from ground decreases. This means that a large current is drawn through the transistor, even when no input signal voltage is applied to the circuit. As the input signal voltage rises still further towards its maximum voltage, the current flows in the opposite direction through the loudspeaker whilst the current flowing through R1 continues to increase.
This means at the peak of the signal swing, a very large current is drawn through the transistor Q2.
Therefore, in the above example, the quiescent power dissipation in the power amplifier may be as much as twenty times the maximum power supplied to the loudspeaker 3.
An amplifier (such as that described above) in which all the output power devices remain on at all times is said to be operating as a Class A amplifier.
The term 'power device' includes bipolar transistors, FET's and valves.
Semiconductor Class A amplifiers are typified by an average power dissipation which is many times greater than the maximum power output by the load (a loudspeaker in the above example).
In order to overcome the problems presented by such large power dissipation, present day Class A amplifiers must be built with a large number of power devices operating in parallel, each having a high power rating and being provided with bulky heatsinks.
This has the result that semiconductor Class A amplifiers are bulky, heavy and expensive. In addition to these disadvantages, the power consumed by the amplifier can significantly increase the electricity bills of the user.
A further problem with semiconductor Class A amplifiers is that the performance of the components they contain is often degraded by the large power dissipation within those components. In particular, the gain characteristics of the power devices may become significantly non-linear when they are dissipating large amounts of power.
The above problems have been exacerbated by developments in loudspeaker design, wherein loudspeakers have been modified to reduce colouration of their audio output and larger diaphragms have been used to improve the bass performance. Both of these developments have meant that the power required to be input to the loudspeakers in order for them to output a given level of sound has been increased.
The vast majority of conventional power amplifiers obviate the above problems by utilising complementary power devices in their output stages. Typically, in the operation of such amplifiers, one power output device sources current to the load in one half of the input signal cycle and another, complementary, power device sinks current from the load in the other half of the input signal cycle. This has the result that less power need be dissipated in the components of the power amplifier in order to achieve the same power output by the load.
However, such amplifiers (said to be operating as Class B amplifiers) present an additional problem in that the output of each power device is typically non-linear in the transition region i.e. when one power device switches off and the other power device switches on.
Hence, the benefits of lower power dissipation in the components at full output power, i.e. reducing nonlinearities and improving the faithfulness of the reproduction, are offset by the introduction of nonlinearities at low and moderate output power levels. In order to correct these non-linearities, most modern power amplifier designs employ large amounts of negative feedback from the output to the input of the power amplifier circuit.
Additionally, the power devices are often arranged to conduct over a range slightly greater than half the input signal cycle. This is known as class AB operation.
Other conventional amplifiers divide the output stage into a low power Class A and high power Class B section. Nevertheless, such systems still utilise negative feedback.
Although negative feedback alleviates the harmonic distortion associated with Class B operation, it itself introduces further problems into the amplifier performance. For example, the delay involved in feeding the signal back to the input of the circuit may result in the phase of the output signal not faithfully following the phase of the input signal. Also, the "slew-rate" of the amplifier may be decreased such that sounds having sharp attacks (such as drum beats) may be poorly reproduced.
In other words, it has not been possible to provide a Class B amplifier which provides the quality of reproduction given by a Class A amplifier.
Hence, there is a need for an amplifier which can operate as a Class A amplifier, but in which the above-mentioned drawbacks are mitigated or avoided altogether.
An additional problem with many conventional power amplifiers is that they represent a safety hazard. In direct drive semiconductor amplifiers, the voltage across the loudspeaker must be increased in order to increase the power supplied to the loudspeaker. Hence, high power direct drive semiconductor amplifiers often use high voltage supply levels which may be lethal if contacted by the user. The risk of electric shock is further increased by the presence of the large electrolytic capacitors used to smooth the rectified AC power supply.
These can maintain the high voltages even after the amplifier is switched off.
Another problem associated with these high voltages is that the cost of the electrolytic capacitors increases as their voltage specification increases.
Potentially lethal voltages are also found in conventional valve power amplifiers, where voltages many hundreds of volts are needed for the operation of the valves within the amplifier. The above-mentioned problems apply to these amplifiers as well.
It is an object of the present invention to provide a power amplifier which operates in Class A and which provides one or more of the following benefits: a) It can be manufactured at a reasonable cost; b) It is not affected by the non-linear gain characteristics of power devices operating at higher power levels; c) It is not affected by the distortion provided by negative feedback; d) It can provide high power output levels without having dangerously high voltages within its circuitry.
According to the present invention, there is provided a power amplifier comprising: a power device means having a control terminal and two current carrying terminals; means for connecting the amplifier to a power supply to provide in use an upper voltage level and a lower voltage level; and a transformer means having a primary and a secondary; wherein said power device means and said primary are arranged into a follower configuration between said upper voltage level and said lower voltage level, the current carrying terminal which follows the control terminal voltage being connected via the primary to one of said voltage levels;; the arrangement being such that on a load being connected across the secondary of said transformer and a voltage signal being applied to said control terminal, a current substantially linearly dependent on said voltage signal flows through said primary, variations in said current being linearly coupled to said load.
The present invention provides the advantage that the power output to the load could be increased without increasing the voltage supply levels used within the power amplifier. This is because the power can instead be increased by increasing the current through the primary of the transformer means. This can simply be done by reducing the resistance of the transformer primary.
This also has the advantage of providing a Class A amplifier in which, in comparison to known power amplifiers, a greater proportion of the power supplied between the voltage levels is dissipated in the load than is dissipated in the components of the power amplifier.
Preferably, said power device is a semiconductor power device, this has the advantage that the high potentials required to operate thermionic vacuum tubes are absent from the amplifier.
Preferably, the secondary of the transformer is matched to drive the load.
Advantageously, the secondary of the output transformer is centre-tapped in order to enable the load to be driven differentially. This has the advantage that the electromagnetic field radiating from the means connecting the transformer to the load is reduced. In addition, interference with the signal from external sources is also reduced.
According to a second aspect of the present invention, there is provided a power amplifier comprising: means for connecting the amplifier to a power supply to thereby provide in use an upper voltage level and a lower voltage level; a transformer comprising: a primary having a first part and a second part; a secondary; a first power device having a first control terminal; a second power device having a second control terminal; wherein said first power device and said first part of said primary are arranged into a first follower configuration between said voltage levels; and said second power output device and said second part of said primary are arranged into a second follower configuration between said voltage levels; the arrangement being such that on a differential voltage signal being applied across said first and second control terminals the current through the first follower causes a flux in said secondary in the opposite direction to the flux induced by the current through said second follower, the net flux in said transformer being proportional to said differential voltage signal, any variations in the differential voltage signal being linearly coupled to said load.
Preferably, the input signal is transformer-coupled to the power amplifier. This has the advantage that the amplifier input presents an isolated low impedance input which prevents any ground current loops between the power amplifier and any pre-amplifier stages. Furthermore, the transformer coupling enables the input impedance of the power amplifier stage to be matched to a cable impedance, thereby providing a better termination characteristic and reducing or eliminating cable reflections. This in turn has the advantage of allowing a long interconnection between the pre-amplifier stage and the power amplifier stage, thereby enabling the power amplifier stage to be situated close to the load. The low impedance input provided by such transformer coupling also has the benefit of not producing loud hums from electromagnetic interference. For example, in a domestic Hi-Fi audio system incorporating this preferred feature, touching the input by hand does not result in the loudspeaker producing loud hums owing to mains pick-up.
Advantageously, the input transformer is arranged to produce a differential signal around a reference voltage. This has the advantage that the reference voltage need not be carefully controlled as any variations in that voltage will be added equally to either side of the differential input signal.
Following is a description by way of example only, with reference to and as illustrated in the accompanying drawings, of methods of carrying the invention into effect.
In the Figures: Figure 1 is a schematic circuit diagram of a known single-ended power amplifier; Figure 2 is a schematic circuit diagram of a preferred embodiment of the present invention; Figure 3 is a schematic circuit diagram of a singleended embodiment of the present invention.
With reference to Figure 2, a power amplifier circuit is provided with an upper voltage supply rail Vcc and a ground rail G. A primary T1P of an input transformer T1 has an upper terminal 16 and a lower terminal 18. The secondary T1S of the input transformer T1 is centre-tapped and connected thereby to a voltage regulator means 20.
The voltage regulator means 20 comprises a resistor R8 in series with a reversed-biased Zener diode D3, the resistor R8 and diode D3 being connected between the upper voltage supply rail Vcc and the ground rail G. A smoothing capacitor C1 is provided in parallel with the Zener diode D3.
The upper terminal 22 of the input transformer secondary T1S is connected to the lower terminal 24 by a resistance R9.
The upper terminal 22 is also connected to a first end of a resistor R10. The other end of the resistor R10 is connected to a power device means, in this case an nchannel power MOSFET Q5. The connection is to the gate terminal GT1 of the MOSFET Q5. The drain terminal DT1 of MOSFET QS is connected to the upper voltage supply rail Vcc and the source terminal ST1 of MOSFET Q5 is connected to an upper terminal 26 of an output transformer primary T2P. A protective Zener diode D4 is connected between the source terminal ST1 and the first end of the resistor R10. An upper half 30 of the output transformer primary T2P extends from the upper terminal 26 to a centre-tap 27 which is connected to the ground rail G.
It will be seen that the MOSFET QS and the upper half 30 are thereby arranged into a first follower configuration, the follower operating between the upper voltage rail Vcc and the ground rail G. The input terminal of the follower is provided by the gate terminal CTl and the output is provided by the source terminal ST1.
A lower half 32 of the output transformer primary T2P extends from the centre-tap 27 to a lower terminal 28.
The lower terminal 24 of the input transformer secondary T1S is also connected to a first end of a resistor Rill. The other end of the resistor Rii is connected to the gate terminal GT2 of a second n-channel power MOSFET Q6. The drain terminal DT2 of the MOSFET Q6 is connected to the upper voltage supply rail Vcc and a protective Zener diode D5 is connected between the source terminal ST2 and the first end of the resistor Rill. The source terminal ST2 of the MOSFET Q6 is also connected to the lower terminal 28 of the output transformer primary T2P.
It will be seen that the MOSFET Q6 and the lower half 32 are thereby arranged into a second follower configuration, the follower operating between the upper voltage rail Vcc and the ground rail G. The input terminal of the follower is provided by the gate terminal GT2 and the output is provided by the source terminal ST2.
The windings of the lower half 32 of the output transformer T2P primary are wound in the same sense around the transformer core to those of the upper half 30.
The secondary T2S of the output transformer T2 is centre-tapped, the centre-tap 35 being connected to the ground rail G. Upper 34 and lower 36 terminals of the output transformer secondary T2S are connected across a loudspeaker 40.
In operation, the upper voltage rail Vcc and the ground rail G are connected respectively to the upper voltage terminals and lower voltage terminals of a power supply (not shown). Current thereby flows in reverse direction through Zener Diode D3 to provide a reference voltage (Vref) to the centre-tap 23 of the input transformer secondary T1S. Any ripples present on the power supply are smoothed by the capacitor C1.
If no audio signal is present across the input transformer primary T1P, the gates of each of the MOSFETS Q5 and Q6 are at the reference voltage (Vref). Since this reference voltage is at least a few volts higher than the ground potential, current flows from the upper voltage supply rail Vcc through each of the power MOSFETS Q5 and Q6 and the associated halves (30,32 respectively) of the output transformer primary T2P to the ground voltage rail G. Because the current flowing through the lower half 32 of the output transformer primary T2P flows in the opposite direction to the current flowing through the upper half 30, the net flux in the transformer core is zero. Therefore, no voltage is applied across the loudspeaker 40 and the speaker is silent.
On a signal being applied to the primary T1P of the input transformer T1, an e.m.f. magnified by the turns ratio of the input transformer T1 is induced in the secondary T1S of the input transformer. Hence, the voltage at the upper terminal 22 rises a given voltage (V) above the reference voltage (Vref) and the voltage at the lower terminal 24 falls below the reference voltage (Vref) by a similar voltage (at). The current through the first MOSFET Q5 therefore increases by an amount linearly related to the given voltage (AV) and the current through the MOSFET Q6 decreases by a similar amount.The flux flowing through the upper half 30 of the output transformer primary T2P therefore rises in proportion to the increased current, whereas the flux flowing through the lower half 32 falls by a similar amount. As a result, the net flux through the output transformer primary increases in linear relation to the input signal voltage and induces a linearly related potential across the terminals (34,36) in the secondary T2S of the output transformer T2. Thus, the loudspeaker 40 is driven by a current that varies linearly with the voltage of the input signal across the input transformer primary T1P. Hence, the sound produced by the loudspeaker 40 is a faithful reproduction of the lowpower audio signal input across the input transformer primary T1P.
The resistor R9 across the secondary T1S of the input transformer T1 produces a roll-off at high frequency and is chosen to filter the input signal so that it is below the resonance point of the input transformer T1.
With reference to the above-mentioned embodiment, the total output power available to drive the loudspeaker 40 is determined by the design of output transformer T2.
Higher output power from the loudspeaker 40 requires a larger transformer with a lower impedance primary 30,32, higher current power MOSFETS Q5,Q6 with larger heat sinking, and a higher current power supply.
It will be seen that the present embodiment provides the following advantages: 1) The operating voltage of the amplifier need not be raised in order to increase the power output by the loudspeaker 40. This advantage is not realised by known direct drive semiconductor amplifiers.
2) The input signal is amplified by the turns ratio of the transformer T1 without recourse to active devices.
3) As non-linearities in the circuit are small the amplifier is operated without overall feedback.
Hence, the power output is directly derived from the input signal. This represents an advantage over known power amplifiers.
Figure 4 shows a single-ended version of the above described embodiment of the present invention.
The single-ended power amplifier (Figure 3) is provided with an upper voltage supply rail Vcc and a ground rail G. A primary T3P of an input transformer T3 has an upper terminal 40 and a lower terminal 42. The secondary T3S of the input transformer T1 has an upper terminal 44 and a lower terminal 46. The lower terminal 46 is connected to a voltage regulator means 50.
The voltage regulator means 50 is similar to that already described with reference to the differential power amplifier (Figure 2).
The upper terminal 44 of the input transformer secondary T1S is connected to the lower terminal 46 by a resistance R14.
The upper terminal 44 is also connected to a first end of a resistor R12. The other end of the resistor R12 is connected to a power device means, in this case an n-channel power MOSFET Q7. The connection is to the gate terminal GT3 of the MOSFET Q7. The drain terminal DT3 of MOSFET Q7 is connected to the upper voltage supply rail Vcc and the source terminal ST3 of MOSFET Q7 is connected to an upper terminal 48 of an output transformer primary T4P. A protective Zener diode D6 is connected between the source terminal ST3 and the first end of the resistor R12. The output transformer primary T2P extends from the upper terminal 48 to the ground rail G.
It will be seen that the MOSFET Q7 and the output transformer primary T4P are thereby arranged into a follower configuration, the follower operating between the upper voltage rail Vcc and the ground rail G. The input terminal of the follower is provided by the gate terminal GT3 and the output is provided by the source terminal ST3.
The secondary T4S of the output transformer T3 is connected across a loudspeaker 60.
In operation, power is supplied between the upper voltage rail Vcc and the ground rail G.
If no signal is superimposed on the reference voltage (Vref) output by the voltage regulator means 50 then the voltage Vref is applied to the gate of the MOSFET Q7. The steady voltage difference between the gate of the MOSFET Q7 and the ground rail G results in a steady DC current flowing through the MOSFET Q7 from the upper voltage level Vcc to the lower voltage level G.
However, since only time-varying currents induce an e.m.f. in the secondary T4S of the output transformer T4, no voltage signal is applied across the loudspeaker 60 and the loudspeaker is silent.
On an audio signal being applied across the input transformer primary T3P, the current through the output MOSFET Q7 varies linearly with the voltage difference between the input audio signal voltage. The timevarying component of the current is coupled to the secondary of the output transformer T4S and is in turn presented across the loudspeaker 60. The currents through the loudspeaker 60 are therefore linearly related to the variations in the input audio signal and hence the loudspeaker provides a faithful reproduction of the input audio signal.
The resistor R14 across the secondary T1S of the input transformer T1 produces a roll-off at high frequency and is chosen to filter the input signal so that it is below the resonance point of the input transformer T3.

Claims (8)

CLAIMS:
1. A power amplifier comprising: a power device means having a control terminal and two current carrying terminals; means for connecting the amplifier to a power supply to provide in use an upper voltage level and a lower voltage level; and a transformer means having a primary and a secondary; wherein said power device means and said primary are arranged into a follower configuration between said upper voltage level and said lower voltage level, the current carrying terminal which follows the control terminal voltage being connected via the primary to one of said voltage levels;; the arrangement being such that on a load being connected across the secondary of said transformer and a voltage signal being applied to said control terminal, a current substantially linearly dependent on said voltage signal flows through said primary, variations in said current being linearly coupled to said load.
2. A power amplifier according to claim 1, wherein said power device is a semiconductor power device.
3. A power amplifier according to claim 1 or claim 2, wherein the secondary of the transformer is matched to drive the load.
4. A power amplifier according to any preceding claim, wherein the secondary of the output transformer is centre-tapped in order to enable the load to be driven differentially.
5. A power amplifier comprising: means for connecting the amplifier to a power supply to thereby provide in use an upper voltage level and a lower voltage level; a transformer comprising: a primary having a first part and a second part; a secondary; a first power device having a first control terminal; a second power device having a second control terminal; wherein said first power device and said first part of said primary are arranged into a first follower configuration between said voltage levels; and said second power output device and said second part of said primary are arranged into a second follower configuration between said voltage levels; the arrangement being such that on a differential voltage signal being applied across said first and second control terminals the current through the first follower causes a flux in said secondary in the opposite direction to the flux induced by the current through said second follower, the net flux in said transformer being proportional to said differential voltage signal, any variations in the differential voltage signal being linearly coupled to said load.
6. A power amplifier according to claim 5, wherein the input signal is transformer-coupled to the power amplifier.
7. A power amplifier according to claim 5 or claim 6, wherein the input transformer is arranged to produce a differential signal around a reference voltage.
8. A power amplifier substantially as hereinbefore described with reference to and as shown in Figure 2 or Figure 3 of the accompanying drawings.
GB9505428A 1995-03-17 1995-03-17 A power amplifier for driving an output transformer Withdrawn GB2298983A (en)

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Application Number Priority Date Filing Date Title
GB9505428A GB2298983A (en) 1995-03-17 1995-03-17 A power amplifier for driving an output transformer

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Application Number Priority Date Filing Date Title
GB9505428A GB2298983A (en) 1995-03-17 1995-03-17 A power amplifier for driving an output transformer

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GB9505428D0 GB9505428D0 (en) 1995-05-03
GB2298983A true GB2298983A (en) 1996-09-18

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Citations (6)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
GB840666A (en) * 1957-03-01 1960-07-06 British Telecomm Res Ltd Improvements in or relating to amplifiers employing transistors
GB1001390A (en) * 1962-05-28 1965-08-18 Westinghouse Electric Corp Semiconductor amplifier
GB1032224A (en) * 1964-05-22 1966-06-08 Philco Corp Improvements in and relating to electric signal amplifiers
GB1342109A (en) * 1970-05-05 1973-12-25 Westinghouse Electric Corp Thin film transistor phongraph amplifier
GB1351589A (en) * 1972-03-17 1974-05-01 Philips Electronic Associated Multistage transistor amplifier for broad-band information signals
GB2099634A (en) * 1981-05-21 1982-12-08 Marconi Co Ltd Impedance matching circuits

Patent Citations (6)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
GB840666A (en) * 1957-03-01 1960-07-06 British Telecomm Res Ltd Improvements in or relating to amplifiers employing transistors
GB1001390A (en) * 1962-05-28 1965-08-18 Westinghouse Electric Corp Semiconductor amplifier
GB1032224A (en) * 1964-05-22 1966-06-08 Philco Corp Improvements in and relating to electric signal amplifiers
GB1342109A (en) * 1970-05-05 1973-12-25 Westinghouse Electric Corp Thin film transistor phongraph amplifier
GB1351589A (en) * 1972-03-17 1974-05-01 Philips Electronic Associated Multistage transistor amplifier for broad-band information signals
GB2099634A (en) * 1981-05-21 1982-12-08 Marconi Co Ltd Impedance matching circuits

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