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Lecture 321 Linearity. Lecture 322 Introduction Linearity is a mathematical property of circuits that makes very powerful analysis techniques possible:

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Presentation on theme: "Lecture 321 Linearity. Lecture 322 Introduction Linearity is a mathematical property of circuits that makes very powerful analysis techniques possible:"— Presentation transcript:

1 Lecture 321 Linearity

2 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

3 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).

4 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.

5 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.

6 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

7 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.

8 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

9 Lecture 329 Determining Linearity The relationship between current and voltage for a linear elements satisfies two properties: –Homogeneity –Additivity

10 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.

11 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

12 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).

13 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

14 Lecture 3214 Consequences of Linearity Superposition Source transformation Thevenin’s and Norton’s theorems


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