# Chapter 2 Circuit Elements

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1 Chapter 2 Circuit Elements 2.1 Voltage and Current Sources 2.2 Electrical Resistance (Ohm s Law) 2.3 Construction of a Circuit Model 2.4 Kirchhoff s Laws 2.5 Analysis of a Circuit Containing Dependent Sources 1

2 Key points Microscopic view of Ohm s law. Physics of Kirchhoff s laws. 2

3 Section 2.1 Voltage and Current Sources 1. Ideal sources 2. Real sources 3. Dependent sources 3

4 Ideal voltage and current sources Ideal voltage source: maintains a prescribed voltage regardless of the current in the device. Ideal current source: maintains a prescribed current regardless of the voltage in the device. 4

5 Real voltage sources The output voltage V o will change with the (1) the output current I o, (2) the load resistance R L. The entire source is modeled by an ideal one plus a series resistor R S. 5

6 Real current sources The output current I o will change with the (1) the output voltage V o, (2) the load resistance R L. The entire source is modeled by an ideal one plus a parallel resistor R S. 6

7 Independent vs. dependent sources Independent sources: The output voltage (current) is not influenced by any other voltage or current in the circuit. All the cases shown above belong to this category. Dependent sources: The output voltage (current) is determined by some other voltage or current in the circuit. This happens when the circuit has active devices (e.g. transistors). 7

8 Symbols of dependent sources Voltagecontrolled voltage source voltage gain transconduct ance Voltagecontrolled current source Currentcontrolled voltage source transresistance current gain Currentcontrolled current source 8

9 Section 2.2 Electrical Resistance (Ohm s Law) 1. Physics 2. Electric power 9

10 Microscopic view of conductors A conductor is made of (1) immobile ions, and (2) free electrons that move fast (Fermi speed, v F ~ 10 6 m/s) and randomly. When an external E-field is applied, electrons have slow (~mm/s) drift velocity v d statistically. 10

11 Ohm s law The drift velocity v d ( net current i) is proportional to the applied electric field E ( applied voltage v), resulting in: v ir if passive sign convention is held: 11

12 Comments The Ohm s law is valid for good conductors, instead of every material. Counter examples: vacuum tubes (I V 1.5 ), transistors (I e V ). Resistance depend on material, shape, and the way of connection. E.g. washer resistor: 12

13 Electric power consumption Microscopically, frequent collisions between free electrons and immobile ions transfer energy from electric field to thermal vibration. Power in a resistor in terms of current: p vi ( ir) i i 2 R Power in a resistor in terms of voltage: p v 2 R 13

14 Example 2.3 v a p 1 8 vi 8 V W v c p ir ( 1) V vi ( 20) ( 1) 20 W vi > 0, power is always dissipated in a resistor. 14

15 Section 2.4 Kirchhoff s Laws 1. Nodes & loops 2. Kirchhoff s laws & the physics 15

16 Nodes and loops Electric circuits consist of connected basic circuit elements. A node is a point where two or more circuit elements join. A loop is a closed path, starting and ending at the same node without passing through any intermediate node more than once. 16

17 Example: Loop Node, 4 elements join Not a node 17

18 Kirchhoff s current law (KCL) The algebraic sum of all the currents at any node in a circuit equals zero. Microscopic equivalent: Conservation of charges. 18

19 Kirchhoff s voltage law (KVL) The algebraic sum of all the voltages around any loop in a circuit equals zero. Microscopic equivalent: Static electric field is conservative E dl 0, which is not true in time-varying cases! 19

20 Example 2.6: KCL into out-of Apply KCL over Node b: ( 3 i ) i ( i ) i ( i ) 1 2 a b 0 20

21 Example 2.7: KVL rise drop Apply KVL over Loop a: ( 3 v ) v v ( v ) ( v ) b 0 21

22 Section 2.5 Analysis of a Circuit Containing Dependent Sources 22

23 Step 1: Specify unknowns 3 branch currents: {i ab = i, i bc = i o, i cb = 5i }, only 2 independent unknowns, need 2 independent equations. 23

24 Step 2: Apply KVL/KCL KVL for Loop cabc: 500 = 5i +20i o...(1) KVL for Loop cbc: -20i o +(??) = 0, failed for voltage across a current source is undetermined. i = 4 A, KCL for Node b: i +5i = i o (2) i o = 24 A. 24

25 Practical Perspective: Electrical Safety Burns is not the major electrical injury. Current can disturb the electrochemical signals of nerves that control oxygen supply to the brain or regulate heartbeat. Barely perceptible: 5 ma. Heart stoppage: 500 ma. 25

26 Key points Microscopic view of Ohm s law. Physics of Kirchhoff s laws. 26

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