LCS_3_notes - 1 LINEAR CIRCUITS AND SIGNALS Basic Circuits...

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Unformatted text preview: 1 LINEAR CIRCUITS AND SIGNALS Basic Circuits Department of Electrical and Computer Engineering University of Miami I. OPERATIONAL AMPLIFIERS (OP-AMPS) Table I shows the 5 basic terminals of an op-amp. TABLE I BASIC TERMINALS OF AN OP-AMP Terminal Symbol Remarks Non-inverting input v + Inverting input v- Output v o Cannot exceed the bias voltage, i.e., | v o | ≤ v b Positive bias voltage + v b v b ≈ 10 V Negative bias voltage- v b When | v o | = v b , op-amp is in saturation; otherwise, it is operating within its linear region. Table II shows the three most important parameters of an op-amp: TABLE II PARAMETERS OF AN OP-AMP Parameter Symbol Remarks Input resistance R i Very high ∼ 10 6 Ω Output resistance R o Very low ∼ 1- 10 Ω Gain A Very high ∼ 10 6 A simple equivalent cct of an op-amp operating within its linear region: v o = A ( v +- v- )- R o i o , (1) 2 where | v o | = | A ( v +- v- )- R o i o | < v b . (2) So, under output open conditions (i.e., i o = 0 ), linear operation of op-amp requires that | v +- v- | < v b A . (3) So, when operating within the linear region, the op-amp possesses the features in Table III. TABLE III FEATURES OF OP-AMP OPERATING WITHIN ITS LINEAR REGION Parameter Symbol Typical value Differential input v +- v- Very small ∼ 10 V/ 10 6 Ω = 10 μV Input voltages v + and v- v + ≈ v- Input currents i + and i- Very small ∼ 10 μV/ 10 6 Ω = 10 pA These features give rise to the ideal op-amp conditions: • Infinite input resistance: R i → ∞ . • Infinite gain: A → ∞ • No voltage drop across the input terminals: v + = v- . • No current through the input terminals: i + = i- = 0 . For more accurate results, use the equivalent cct in (1). II. BASIC OP-AMP CIRCUITS A. Voltage Follower See Figure 1(a). Note that v o = v- = v + = v s . (4) So, the output ‘follows’ the input source voltage irrespecive of the load R L . Contrast this with the case when the op-amp voltage follower is not used: v o = R L R s + R L v s . (5) 3- + _ + R s v s v + v- v o R L i + i- (a) Ideal op-amp. _ + R s v s v + v- =v o R L i + i L +- R i R o A(v +-v- ) =AR i i + (b) Equivalent circuit. Fig. 1. Voltage follower. a) Linear Range Operation: Assume that | v s | < v b . • v o ≥ v b = ⇒ v +- v- > (because v = A ( v +- v- ) ). But we also must have v +- v- = v s- v o = v s- v b < because | v s | < v b . This is impossible. • v o ≤ - v b = ⇒ v +- v- < . But we must also have v +- v- = v s- v o = v s + v b > because | v s | < v b . This is also impossible. We conclude that, | v s | < v b = ⇒ | v o | < v b , i.e., the op-amp operates in its linear region. (6) 1) Non-Ideal Case: See Figure 1(b). Let us write the nodal equation for v- = v o : v-- v + R i + v-- A ( v +- v- ) R o + v- R L = 0 . (7) This yields v + = 1 R i + A + 1 R o + 1 R L 1 R i + A R o v- . (8) Then A R o >> 1 R L = ⇒ v + ≈ 1 R i + A + 1 R o 1 R i + A R o v- ≈ v- , (9) with the typical values for A , R i , and R o . 4 Next, consider the nodal equation for...
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LCS_3_notes - 1 LINEAR CIRCUITS AND SIGNALS Basic Circuits...

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