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Lecture 321 Linearity
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Lecture 322 Introduction Linearity is a mathematical property of circuits that makes very powerful analysis techniques possible: –LaPlace transforms –Fourier transforms (Bode plots) –Stability analysis
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Lecture 323 Linearity Linearity leads to many useful properties of circuits: –Superposition: the effect of each source can be considered separately. –Equivalent circuits: Any linear network can be represented by an equivalent source and resistance (Thevenin’s and Norton’s theorems).
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Lecture 324 Linearity Linearity leads to simple solutions: –Nodal analysis for linear circuits results in systems of linear equations that can be solved by matrices. –Nodal analysis for non-linear circuits results in equations that must be solved numerically.
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Lecture 325 Linearity Linear circuits (and more generally linear systems) have behavior that is predictable- small perturbations stay small. Sinusoidal sources in a linear circuit result in sinusoidal responses with the same frequency.
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Lecture 326 Non-linearity Non-linear circuits or systems may have chaotic behavior-small perturbations result in large changes. –Turbulent fluid flow –Weather –The 3 body problem –Electrical processes within the heart
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Lecture 327 Models and Linearity Linearity is a mathematical property of a model. We use linear models whenever possible. Real circuit elements are never exactly linear, but many are close enough for practical purposes.
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Lecture 328 Definition of Linearity For ECE 301, a circuit element is characterized in terms of its current-voltage relationship. Linear elements: –Resistors: v(t) = R i(t) –Inductors: v(t) = L di(t)/dt –Capacitors: v(t) = 1/C i(x) dx
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Lecture 329 Determining Linearity The relationship between current and voltage for a linear elements satisfies two properties: –Homogeneity –Additivity
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Lecture 3210 Homogeneity Let v(t) be the voltage across an element with current i(t) flowing through it. In an element satisfying homogeneity, if the current is increased by a factor of K, the voltage increases by a factor of K.
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Lecture 3211 Examples Resistor: V = R I –If current is KI, then voltage is R KI = KV Squarer: V = I 2 –If current is KI, then voltage is (KI) 2 = K 2 V
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Lecture 3212 Additivity Let v 1 (t) be the voltage across an element with current i 1 (t) flowing through it, and let v 2 (t) be the voltage across an element with current i 2 (t) flowing through it In an element satisfying additivity, if the current is the sum of i 1 (t) and i 2 (t), the voltage is the sum of v 1 (t) and v 2 (t).
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Lecture 3213 Examples Resistor: V = R I –If current is I 1 + I 2, then voltage is R(I 1 + I 2 ) = RI 1 + RI 2 = V 1 + V 2 Squarer: V = I 2 –If current is I 1 + I 2, then voltage is (I 1 + I 2 ) 2 = I 1 2 + 2 I 1 I 2 + I 2 2
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Lecture 3214 Consequences of Linearity Superposition Source transformation Thevenin’s and Norton’s theorems
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