Ch02 Network Theorems 1s09 - Chapter 2 Linear Element A...

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Network Theorems Network Theorems Chapter 2 Artemio P. Magabo Artemio P. Magabo Professor of Electrical Engineering Department of Electrical and Electronics Engineering University of the Philippines - Diliman Department of Electrical and Electronics Engineering Linear Element A linear element is a passive element whose voltage-current relationship is described by a linear equation; i.e. if the current through the element is multiplied by a constant k, then the voltage across the element is likewise multiplied by k. Consider a resistor R with current i=i 1 . From Ohm’s Law, we get + R v i - 1 1 Ri v v = = Suppose the current is increased by a factor k; i.e. i 2 =ki 1 . The new voltage is 1 1 2 2 kv Rki Ri v = = = (R is a linear element) Department of Electrical and Electronics Engineering L + v i - Consider an inductor R with current i=i 1 . The inductor voltage is dt di L v v 1 1 = = Suppose the current is increased by a factor k; i.e. i 2 =ki 1 . The new voltage is 1 1 2 2 kv dt ) ki ( d L dt di L v = = = Thus,L is a linear element. Note: Following the same analysis, we can show that a constant capacitor is a linear element. Department of Electrical and Electronics Engineering Linear Dependent Source A linear dependent source is a current or voltage source whose output current or voltage is proportional only to the first power of some current or voltage variable in the circuit, or to the sum of such quantities. Linear Dependent Voltage Sources + - kv x + - ki y Linear Dependent Current Sources kv w ki z
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Department of Electrical and Electronics Engineering Linear Electric Circuit An electric circuit is linear if it consists of independent sources linear dependent sources linear elements The response of a linear circuit is proportional to the sources; that is, if all independent sources are multiplied by a constant k, all currents and voltages will likewise increase by the same factor k. Department of Electrical and Electronics Engineering Principle of Superposition In an electric circuit containing N independent sources, the current (or voltage) in any branch is equal to the algebraic sum of N components, each of which is due to one independent source acting alone. 1. For a voltage source, remove the source and replace with a short circuit; Note: Reducing an independent source to zero: 2. For a current source, remove the source and replace with an open circuit. Department of Electrical and Electronics Engineering Example : Find I 1 and I 2 using nodal analysis. R 1 REF R 2 + V x I 1 I 2 V s + - I s From KCL, we get 2 x 1 s x s R V R V V I + - = or s 2 1 2 s 2 1 2 1 x V R R R I R R R R V + + + = s 1 s 2 1 x V R 1 I R 1 R 1 V + = + which can be simplified as Department of Electrical and Electronics Engineering s 2 1 2 s 2 1 1 I R R R V R R 1 I + - + = Thus, we get 1 x s 1 R V V I - = s 2 1 1 s 2 1 2 I R R R V R R 1 I + + + = and 2 x 2 R V I =
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Ch02 Network Theorems 1s09 - Chapter 2 Linear Element A...

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