Voltage is the energy per unit charge created by the separation, which can be expressed as

joule/coulumb( )Voltdwvdq

where = the voltage in volts = the energy in joules = the charge in coulombs

vwq

1.2 Voltage

The rate of flow of charges is called the current which is expressed as

coul( )umb/secondAm peredqidt

where = the current in amperes = the charge in coulombs = the time in seconds

iqt

1.3 The Current

Power

Power is defined as the time rate of expanding or absorbing energy

W dwPdt

where power in Wattts Energy in Joules Time in Seconds

Pwt

1 J 1

1 Ws

= dqdw dwP vidt dq dt

This shows that the power is simply the product of the current in the element and the voltage across the element

Passive Sign Convention

Figure 1.10Charging a discharged automobile battery to illustrate the concept of power delivered to or absorbed by an element and the passive sign convention.

Figure 1.11 Illustration of the power delivered to (absorbed by) an element and the power delivered by the element.

Figure 1.12 Examples of the computation of power delivered to or by an element.

Figure 1.13Illustration of an electric circuit as a particular interconnection of circuit elements.

Electric Circuit

1i2i

3i

Current entering the node is positive and leaving the node is negative

( ) 0 1 2 3i i i 0 1 2 3i i i

Current entering the node is negative and leaving the node is positive

( 0) ( ) 1 2 3i i i 0 1 2 3i i i

Note the algebraic sign is regardless if the sign on the value of the current

Kirchhoff's Current Law ( KCL):

The algebraic sum of all the currents at any node in a circuit equals zero.

Figure 1.14 Illustration of Kirchhoff

’s current law (KCL).

( ) ( ) 0 1 4 52 3i i i i i

Entering currents a nodecur Leavrents a no in de g

1 5 43 2 i i i i i

KCL also applies to larger and closed regions of circuit called supernodes

2 6 9 10i i i i

Example 1.3: Determine the currents ix, iy and iz

KCL at node dix+3=2ix = 2-3 = -1A

KCL at node aix+ iy +4 = 0iy = -3A

KCL at node b4 + iz + 2 = 0iz = -6A

We could have applied KCL at the supernode to getiy + 4A + 2A = 3AThus iy = -3

Figure 1.17Example 1.4.

Kirchhoff Voltage Law (KVL)

The algebraic sum of all the voltages around any closed path in a circuit equals zero.

First we have to define a closed path

+

a b c

def

A closed path or a loop is defined as starting at an arbitrary node, we trace closed path in a circuit through selected basic circuit elements including open circuit and return to the original node without passing through any intermediate node more than once

abea bceb cdec aefa abcdefa

Kirchhoff Voltage Law (KVL)

The algebraic sum of all the voltages around any closed path in a circuit equals zero.

The "algebraic" correspond to the reference direction to each voltage in the loop.

Assigning a positive sign to a voltage rise ( to + )

Assigning a negative sign to a voltage drop ( to )

5 V +

2 W 3 W

6 W 5 W

+ 1v +

3v+

2v

+

4v

Assigning a positive sign to a voltage drop ( to )

Assigning a negative sign to a voltage rise ( to )

OR

Example

5 V +

2 W 3 W

6 W 5 W

+ 1v +

3v+

2v

+

4v

Loop 1 0 1 2 5v v

Loop 2 0 43 2v v v

We apply KVL as follows:

Figure 1.23 Another example of the application of KVL.

Ex 1.8: Determine vx, vy, vz by KVL

zv 2 5 ( 4) 7

yv 3 6 5 8

x zv 6 2 v 4 11 15

Ex 1.9: Determine voltage v and current i

KVL around loop E,H,B,A,G gives v 3 2 1 4 4

KCL at the supernode gives 1+4+ix=3, thus ix =-2A

KCL at node e gives i+3=-2, thus i=-5A

1.7 Conservation of Power

The sum of powers delivered to all elements of a circuit at any time equals to zero

i i iallelements allelements

p v i 0

Ex 1.10 Verify conservation of power for the circuit

Element power

A 1A x 1V=1W

B -4A x 2V=-8W

C -3A x 3V=-9W

D -5A x 1V=-5W

E -3A x (-4V)=12W

F 5A x (-1V)=-5W

G 2A x 4V=8W

H -(-2A) x 3V=6W

ic = -3A, vc = 3V if = 5A, vf = -1Vid = -5A, vd = 1V ih = ix = -2Ave = v = -4V

1.8 Series and Parallel Connection of Elements

A B Cv v v A B C

A B C

v v v

i i i

Figure 1.32

series connection of elements, parallel connection of elements

KCL KVL

A B Cv v v A B Ci i i

Figure 1.33Example to illustrate to proper classification of series and parallel connections

Figure E1.19Exercise Problem 1.19.

Determine which elements are connected in series and which elements are connected in parallel

Figure E1.20Exercise Problem 1.20.

Determine which elements are connected in series and which elements are connected in parallel

HW 1 is due now