lecture_09

# lecture_09 - MIT OpenCourseWare http/ocw.mit.edu 2.161...

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MIT OpenCourseWare http://ocw.mit.edu 2.161 Signal Processing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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U ( s ) v o u t v + v - 1 Massachusetts Institute of Technology Department of Mechanical Engineering 2.161 Signal Processing - Continuous and Discrete Fall Term 2008 Lecture 9 1 Reading: Class Handout: Introduction to the Operational Ampliﬁer Class Handout: Op-amp Implementation of Analog Filters Operational-Ampliﬁer Based State-Variable Filters We saw in Lecture 8 that second-order ﬁlters may be implemented using the block diagram structure s 1 s 1 X ( s ) s X ( s ) s X ( s ) 2 a - a 1 0 + - a 0 0 + + a 1 a Y 4 ( s ) ( b a n d - s t o p ) Y 3 ( s ) ( h i g h - p a s s ) Y 2 ( s ) ( b a n d - p a s s ) Y 1 ( s ) ( l o w - p a s s ) and that a high-order ﬁlter may be implemented by cascading second-order blocks, and possibly a ﬁrst-order block (if the ﬁlter order is odd). now look into a method for implementing this ﬁlter structure using operational am- pliﬁers. 1.1 The Operational Ampliﬁer What is an operational ampliﬁer? It is simply a very high gain electronic ampliﬁer, with a pair of diﬀerential inputs. Its functionality comes about through the use of feedback around the ampliﬁer, as we show below. G a i n : A + - 1 copyright ± c D.Rowell 2008 9–1
v v o u t i n R i n i i n i f The op-amp has the following characteristics: It is basically a “three terminal” ampliﬁer, with two inputs and an output. It is a diﬀerential ampliﬁer, that is the output is proportional to the diﬀerence in the voltages applied to the two inputs, with very high gain A , v out = A ( v + v ) where A is typically 10 4 –10 5 , and the two inputs are known as the non-inverting ( v + ) and inverting ( v ) inputs respectively. In the ideal op-amp we assume that the gain A is inﬁnite. In an ideal op-amp no current ﬂows into either input, that is they are voltage-controlled and have inﬁnite input resistance. In a practical op-amp the input current is in the order of pico-amps (10 12 ) amp, or less. The output acts as a voltage source, that is it can be modeled as a Thevenin source with a very low source resistance. The following are some common op-amp circuit conﬁgurations that are applicable to the active ﬁlter design method described here. (See the class handout for other common conﬁg- urations). The Inverting Ampliﬁer: s u m m i n g j u n c t i o n R f ( v i r t u a l g o u n d ) + - In the conﬁguration shown above we note Because the gain A is very large, the voltage at the node designated summing junc- tion is very small, and we approximate it as v = 0 the so-called virtual ground assumption. We assume that the current i into the inverting input is zero. Applying Kirchoﬀ’s Current law at the summing junction we have v in v o i 1 + i f = + =0 R 1 R f from which R f v out = v in R in 9–2

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± v v o u t i n R i n i i n i f The voltage gain is therefore deﬁned by the ratio of the two resistors. The term inverting ampliﬁer comes about because of the sign change.
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lecture_09 - MIT OpenCourseWare http/ocw.mit.edu 2.161...

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