四象限乘法器设计

Design of an BJT analog multiplier1. IntroductionThe analog multiplier is a wildly used circuit that can take two analog inputs and produce an output proportional to their product. In modern IC industry, the analog multiplier is a very useful building block for the design of analog signal processing and the communication systems. Its application fields include signal modulation/demodulation, gain control, phase lock loops, frequency translation and waveform generation (Graeme, 1971). The purpose of this report is to explain the design method of a 4 quadrant analog multiplier using bipolar transistor. In this report, the theory of the analog multiplier is introduced. And then a suggested circuit based on the Gilbert multiplier provided. After that, the circuit simulation is processed with LTSPICE. In the last, the limitation and improvement of this design is being discussed.2. Principle of analog multiplier2.1 Characteristic of bipolar transistorThe basic cell of an analog multiplier is bipolar transistor. Here is a LTSPICE schematic diagram of a biased BJT.Fig.1. Schematic diagram of bipolar transistor Fig.2. I-V characteristic of bipolartransistorBecause of the semiconductor physical characteristic, the base-emitter voltage has an exponential relationship with the collector current (Ashburn, 2003). In most situations, the I-V characteristic can be written in the following equation:2exp()exp()nb i BE BE E C S th B AB qAD n V qV I I I V W N kT(Eq.1)Here, 2nb i S B AB qAD n I W N is saturation current of the base. th kT V q is thermal voltage, which can be considered equal to 26mV in room temperature. I C is the collector current. I E is the emitter current. V BE is the base-emitter voltage. I S is a constant which is determined by the geometric size of base and its doping density. V th is a constant which is only determined by the temperature. It can be seen from the equation that the collector current will exponential increase with the base-emitter voltage. Using LTSPICE DC simulation, the figure 2 can show this relationship directly. From the graph it can be seen that the I C equals nearly to 0 when V BE is less than 520mV. Also I C has a maximum value when V BE arrival 700mV. So the threshold voltage V T is in this range. And the exponential relationship only works in this range.2.2 Simple multiplierBased on the characteristic of bipolar transistor discussed above, a simple multiplier circuit can be developed. The structure of this circuit is shown in figure 3 and 4.Fig.3. the differential amplifier circuit Fig.4. 2 quadrant analog multiplier circuit In figure 3, Vx is the differential input. Io is the current source. Q1, Q2 are two NPN transistors. It can be seen that:And:According to these 2 equations, . From the Eq.1., we canobtain . In mathematics, when(about 2*26mV), . Thus we get:(Eq.2) The different current and can be used as the output, so we can get an output voltage which is proportional to the input voltage (Gray, 2001). This is:(Eq.3) The circuit of figure 4 is established on the circuit of figure 3. In this circuit, there isSubstitute this formula into the Eq.3, we have:(Eq.4) If, Eq.4 is approximately equal to:(Eq.5) In this equation, can be either positive or negative. But in order to ensure the Q3 is turned on, must be larger than , which means must be positive. Thus this analog multiplier can only work in 2 quadrants, the first quadrant and the second quadrant.2.3 The Gilbert CellThe basic Gilbert analog multiplier cell is presented in the figure 5.Fig.5 the circuit of Gilbert cellIn this circuit, the input voltage Vx and Vy can generate pairs of differential currents, which results in the collector currents of the transistors. According to Eq.2, there areThus, the combination of these 3 equations above gives the difference of two output currentsBecause the derivation of this derivation is based on the Eq.2, it must satisfy the same requirement, which is and . At this time, the output voltage isFrom this equation it can be seen that the output voltage is proportional to the product of Vx and Vy. However, different from the 2 quadrant multiplier, here the input signal Vy can be either positive or negative. So, an analog multiplier can be realised.3. Circuit design3.1 basic analog multiplier circuitThe circuit of the practical basic 4 quadrant multiplier build on the previous analysis is presented in the following figure.Fig.6 the 4 quadrant multiplier based on the Gilbert cellThe compositions of this circuit are resisters and transistors. For the rough design, current source is used directly. And the structure of two current sources is more sensitive than that of one current source, so the circuit will get a better balance characteristic. According to the analysis in 2.1, the exponential relationship between base-emitter voltage and collector current only remain in the range of 520mV-700mV. So the static base-emitter current can be chosen at 610mV, the middle point of thisrange. From figure 4, the corresponding collector current is about 0.2mA. In addition, the emitter current is approximately equal to the collector current. So 0.2mA can be set as the value of the current resources.In this circuit, the npn transistors are all 2N2222. It can be seen from the circuit that the function of the transistor pair Q7 and Q8 is transforming the voltage input signal into current input signal, so is the transistor pair Q5 and Q6. Thus, the multiplier operation is based on the variation of the currents. In addition, the four transistors Q7, Q8, Q9, Q10 and the resistor R2 consist of a block which can generate a signal of function. This current can compensate the function generated by the next transistors Q1, Q2, Q3 and Q4. Thus, the output current signal has no relationship with function, that means there is no limitation of (Gray, 2001). This feature significantly expands the input signal magnitude. The transistors Q5, Q6 and resistor R1 have the same feature and the limitation is also cancelled. The output of the nonlinear compensation block in the front of Gilbert cell isFor Ic7 and Ic8, there areSubstitute Ic7 and Ic8 into Eq.9, the output of the nonlinear compensation block is given byIn the Gilbert cell block, the emitter resistors of Q5 and Q6 are much smaller than R1, so we have . So Ic5 and Ic6 is given byUse Eq.11, Eq.12 and Eq.10 to calculate the output voltage based on the Eq.6, Eq.7 and Eq.8. Finally we can obtainFrom this equation and the above derivation, it can be seen that:When R1 and R2 is big enough, the output voltage Vo is proportional to theproduct of Vx and Vy. And the circuit will get the nearly ideal multiplierfunction.Both Vx and Vy can be positive and negative, so it is a 4 quadrant multiplier.The gain coefficient , which is determined by the paramount Rc,R1, R2 andK has no relationship with Vth, so this circuit has a nice thermal stability.In this circuit, because the current sources are all 0.2mA, a several independentcurrent sources based on the current mirror can be used.From Gray and Hurst, there are mainly 3 types of current mirrors: the simple current mirror, simple current mirror with beta () helper, and the Wilson current mirror (Gray, 2001). For the first one, the output current . For the third one, the outputcurrent is . It can easily be seen that the Wilson current mirror can geta much more precise output current. Thus, in this report, the Wilson current mirror isused as the current source. The whole structure is shown in figure 7.Fig.7 the circuit structure with mirror current sourceIn order to get a precise value of the current , a resistor is added between the Vdd and the current mirror transistor. The aimed current is 0.2mV, so the value of this resistor .Use LTSPICE to simulate the circuit. Here is the DC sweep for the circuit.Fig.8. The DC sweep simulation resultThe settings of this simulation are as the follows: sweep Vx from -13V to 13V, and the increment is 1V; sweep Vy from 13V to -13V, and the increment is 1V. During the design, it can be noticed that the magnitude of input signals have a proportional relationship with the product of and R1 (or R2). Besides, to get high amplitude of Vo, Eq.13 must be considered. In this report, the circuit has a nice linear range of output signal when the input signal varies from -13V to +13V. And the output signal has the same range with the input signals.In order to obtain this amplitude of output signal, the V dd and V EE are set up at +32V and -15V respectively.Transient simulation:Fig.8 simulation result for AC inputs Fig.10 simulation results for squareoperationThe settings of the simulation of figure 8 are as the follows:Vx: amplitude=13V; frequency=10 kHz; phase=0. Vy: amplitude=13V; frequency=10 kHz; phase=90o. The Vx input is a sin(t) signal, the Vy input is a cos(t) signal. In this simulation, the Vx and Vy contain all 4 quadrant inputs combinations in one period, which are (Vx>0, Vy>0), (Vx>0, Vy<0), (Vx<0, Vy<0), (Vx>0, Vy>0). So the results proved the circuit can work in 4 quadrants. Furthermore, the frequency of the output signal is twice as the input signals and the amplitude is about half of the input ones. These features satisfy the functionof. So the result proved the circuit can work as an analog multiplier.One useful application of analog multiplier is square operation. As the figure 10 shows, make the two pulse input signal same, then we can obtain a parabola output signal. So the results of can be calculated.Frequency response simulation:Fig.11 Simulation result of frequency responseThis figure shows the frequency response of the circuit. When theAmplitude-frequency characteristic decrease from -22dB to -25dB (increased by -3dB), the relative operation frequency is 500.4 kHz. Below this frequency, the circuit can work nicely.Another useful application of analog multiplier is modulator. In the modulator, the low frequency signal is used to modulate, and the high frequency signal acts as a carrier.Fig.12. modulator simulation: Vx has no DC offset Fig.13 modulator simulation: Vx has a 1.5V DC offsetIt can be seen from the above 2 figures, the frequency of the output signal can be changed by change the magnitude of the input signal.3.2 improvement of output signal amplifyThe output signal of the 3.1 circuit has double ends, which is not so convenient to the following circuits. Besides, for some applications, people need large output signals and a wide band width. This can be realised by adding a Double-ended input single-ended output circuit (shown in figure 14). This additional circuitconsist of a differential amplifier and a basic common collector circuit (shown infigure 15 and 16).Fig.14 2 ended inputs 1 ended output circuit Fig.15 LTP with activeloadFig.16 basic commoncollector circuitThe practical circuit with this addition output circuit is shown below.Fig.17 the circuit followed with 2 input ended 1 output ended circuit The DC sweep simulation is as follow (same setting as 3.1):Fig.18 the relative DC simulationCompared with figure 8, the output voltage increased, but the amplitude of output signal decreased. Furthermore, the common mode signal was amplified by the differential amplifier circuit, so the average value of output voltagebecomes to 22V. When Vx=0 or Vy=0, the output voltage equals to 22V. Thiscircuit can not operate as a multiplier. In a word, this additional circuit did not get the expected result.4. DiscussionAs the analysis showed in 3.1, the circuit of figure 6 can cancel the influence caused by temperature. This is because the differential and balance structure of the nonlinear composition block can remove the common mode signal. The temperature is physical paramount, which change by the same value in a small electric circuit. So noise caused by the temperature can be considered as the common mode signal. The circuit design in this report has a nice thermal stability.Use LTSPICE simulation, the power consumption of each and total elements can be detected easily. In order to reduce the power of the circuit, the current source should relatively be at a small value. But at the same time, the current should ensure the operation of the transistors.The limitation of this design is the same of the bipolar transistor. As the figure 2 shows, bipolar transistor has a limited exponential operation range. This feature asks for a highly accurate collector current control. To solve this problem, one way is to change the physical paramount of the transistor. For example, reduce the cross-section area. Another way is to use MOS transistors. Because the CMOS operation is based on the movement of majority carriers, there is no exponential relationship between drain current and drain-source voltage.5. ConclusionIn this report, an analog multiplier is proposed. Firstly, the principle of the design is introduced. Then a practical circuit is proposed. At the same time, the methods of how to change the paramount to achieve a suitable output signal are described. During the design, theory analyses and explains are also presented. The simulation is processed with LTSPICE. For this report, the results of simulation can satisfy the principle derivation. The designed circuit achieves its porpours.ReferenceGraeme.J.G., Tobey.G.E. and Huelsman. L.P. 1971. Operational Amplifiers: Design and Application. New York: McGRAW-HILL Book Company. (p.397-398)Peter Ashburn. 2003. SiGe Heterojunction Bipolar Transistors. West Sussex: John Wiley & Sons Ltd.GRAY Paul R. Analysis and design of analog integrated circuits. New York: John Wiley & Sons Ltd.。