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CACD Group updated 2000.03.30 Author Janez Puhan |
Active Low-pass RC FilterThe circuitLet us consider the optimisation of the active low-pass RC filter. We will use the constrained simplex optimisation method. The circuit topology is shown in the figure below. The filter is built around three standard operational amplifiers UA741. All capacitances are set to the same value of 2.7nF. The values of all resistances are left undetermined and are subject to optimisation process. Here is the netlist of the circuit (part of active_low-pass_rc_filter.cir file): *** active low-pass rc filter *** vcc 12 0 dc 15V vee 14 0 dc -15V vin 1 0 dc 0 ac 1V 0 r01 20 1 50k r02 20 3 500k r03 3 4 2.5k r04 3 5 250k r05 5 7 500k r06 6 9 50k r07 4 0 50 r08 8 9 500 r09 9 25 500 r10 10 8 250k r11 25 0 500 r12 10 15 250k r13 17 11 50k r14 16 11 50 r15 11 0 2.5k c01 20 3 2.7nF c02 5 9 2.7nF c03 7 6 2.7nF c04 4 6 2.7nF c05 10 11 2.7nF c06 15 17 2.7nF c07 25 17 2.7nF x1 0 20 3 12 14 ua741 x2 7 8 8 12 14 ua741 x3 15 16 16 12 14 ua741 .subckt ua741 11 5 4 6 2 vof 5 10 dc 1mV ibn 5 0 dc 100nA ibp 11 0 dc 80nA r1 7 1 5k r2 8 7 50k rid 10 11 200k rin 5 0 10meg rip 0 11 10meg rl1 3 4 1k ro 3 0 75 rp 2 6 10k rxx 0 2 10meg c1 7 0 13pF c2 8 3 2.7pF cin 5 0 1.4pF cip 11 0 1.4pF cl1 4 0 3pF l1 3 4 30uH ea 1 0 10 11 1 ga 3 0 8 0 2700S .ends .end The cost function is defined by the boundaries in the figure above. We tolerate a 0.5dB ripple, further we demand at least 60dB damping and a don't care band is from 750Hz to 1.35kHz. At the same time we would like to have at least 20dB gain at low frequencies. So we are looking for the resistances r1 to r15, which will give the most appropriate frequency response. We set the explicit constraints of the all resistances to: 10kW < r1 < 100kW Now we have to mathematically define the cost function. Its value has to increase with ripple greater than 0.5dB, damping less than 60dB and gain less than 20dB. There are no implicit constraints, so all we have to do is to run the optimisation process and wait for the results. The optimisation is performed by optimize command. It stops after 3433 iterations and the value of the cost function equals to 1.26. That means that not all the requests determined by the cost function were fulfiled. Our optimised filter has 0.59dB ripple, 60dB damping and 19.6dB gain. The figure below shows the ac responses of the initial and optimised circuit. |
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