7/31/2019 lec3_2003

1/16

EECS 40 Spring 2003 Lecture 3 W. G. Oldham and S. Ross

1

Lecture 3

Definitions: Circuits, Nodes, Branches

Kirchoffs Voltage Law (KVL)

Kirchoffs Current Law (KCL)

Examples and generalizations

RC Circuit Solution

7/31/2019 lec3_2003

2/16

EECS 40 Spring 2003 Lecture 3 W. G. Oldham and S. Ross

BRANCHES AND NODES

Branch: elements connected end-to-end,nothing coming off in between (in series)

Node: place where elements are joinedentire wire

7/31/2019 lec3_2003

3/16

EECS 40 Spring 2003 Lecture 3 W. G. Oldham and S. Ross

NOTATION: NODE VOLTAGES

The voltage drop from node X to a reference node(ground) is called the node voltage Vx.

Example:a b

ground

Va Vb

+

_

+

_

_

+

7/31/2019 lec3_2003

4/16

EECS 40 Spring 2003 Lecture 3 W. G. Oldham and S. Ross

KIRCHOFFS VOLTAGE LAW (KVL)

The sum of the voltage drops around any closed loop is zero.

We must return to the same potential (conservation of energy).

+

-

V2

Pathrise or step up(negative drop)

+

-

V1

Pathdrop

Closed loop: Path beginning and ending on the same node

Our trick: to sum voltage drops on elements, look at the first sign

you encounter on element when tracing path

7/31/2019 lec3_2003

5/16

EECS 40 Spring 2003 Lecture 3 W. G. Oldham and S. Ross

KVL EXAMPLE

Path 1: 0vvv b2a Path 2: 0vvv c3b

Path 3:0vvvv c32a

vcva+

+

3

21

+

vb

v3v2+

+

-

Examples ofthree closedpaths:

a b c

7/31/2019 lec3_2003

6/16

EECS 40 Spring 2003 Lecture 3 W. G. Oldham and S. Ross

UNDERLYING ASSUMPTIONS OF KVL

Assume no time-varying magnetic flux through the loop

If there was, Faradays Law induced emf (voltage)

Avoid these loops!

How do we deal with antennas (EECS 117A)?

Include a voltage source as the circuit representation of theemf or noise pickup.

We have a lumped model rather than a distributed (wave) model.

Antennas are designed to pick up

electromagnetic waves

Regular circuits often do the same thing not desirable!

)t(B

)t(v+

7/31/2019 lec3_2003

7/16

EECS 40 Spring 2003 Lecture 3 W. G. Oldham and S. Ross

ALTERNATIVE STATEMENTS OF KIRCHHOFFSVOLTAGE LAW

1) For any node sequence A, B, C, D, , M around aclosed path, the voltage drop from A to M is given by

LMCDBCABAM vvvvv

2) For all pairs of nodes i and j, the voltage drop from i to j is

where the node voltages are measured with respect to

the common node.

jiij vvv

7/31/2019 lec3_2003

8/16

EECS 40 Spring 2003 Lecture 3 W. G. Oldham and S. Ross

MAJOR IMPLICATION

KVL tells us that any set of elements which are connected atboth ends carry the same voltage.

We say these elements are in parallel.

KVL clockwise,

start at top:

Vb Va = 0

Va = Vb

7/31/2019 lec3_2003

9/16

EECS 40 Spring 2003 Lecture 3 W. G. Oldham and S. Ross

KIRCHOFFS CURRENT LAW

Circuit with several branches connected at a node:

3i2i

4i1i

(Sum of currents entering node) (Sum of currents leaving node) = 0

Charge stored innode is zero (e.g. entire capacitor is part of a branch)

KIRCHOFFs CURRENT LAW KCL:

7/31/2019 lec3_2003

10/16

EECS 40 Spring 2003 Lecture 3 W. G. Oldham and S. Ross

USING KCL

Kirchhoffs Current Law (KCL)

Formulation 1:

Sum of currents entering node = sum of currentsleaving node

Use/write reference directions to determineentering and leaving currents--no concernabout actual current directions

7/31/2019 lec3_2003

11/16

EECS 40 Spring 2003 Lecture 3 W. G. Oldham and S. Ross

ALTERNATIVE KCL FORMULATIONS

Formulation 2:Algebraic sum of currents entering node = 0

where algebraic sum means currents leaving are

included with a minus sign

Formulation 3:

Algebraic sum of currents leaving node = 0

currents entering are included with a minus sign

7/31/2019 lec3_2003

12/16

EECS 40 Spring 2003 Lecture 3 W. G. Oldham and S. Ross

MAJOR IMPLICATION

KCL tells us that all of the elements in a single branch carrythe same current.

We say these elements are in series.

Current entering node = Current leaving node

i1

= i2

EECS 40 S i 2003 L 3 W G Oldh d S R

7/31/2019 lec3_2003

13/16

EECS 40 Spring 2003 Lecture 3 W. G. Oldham and S. Ross

KIRCHHOFFS CURRENT LAW EXAMPLE

Currents entering the node: 24 A

Currents leaving the node: 4 A + 10 A + i

Three formulations of KCL:

A18i0i10424:3

A18i0i10)4(24:2

A18ii10424:1

i10 A

24 A -4 A

EECS 40 S i 2003 L t 3 W G Oldh d S R

7/31/2019 lec3_2003

14/16

EECS 40 Spring 2003 Lecture 3 W. G. Oldham and S. Ross

GENERALIZATION OF KCL

Sum of currents entering/leaving a closed surface is zero

Note that circuit branches could be inside the surface.

Could be a big chunkof circuit in here, e.g.,could be a Black Box

The surface can enclose more than one node!

EECS 40 S i 2003 L t 3 W G Oldh d S R

7/31/2019 lec3_2003

15/16

EECS 40 Spring 2003 Lecture 3 W. G. Oldham and S. Ross

KIRCHOFFS CURRENT LAW USING SURFACES

Example

iA2A5

entering leaving

surface

i must be 50 mA

50 mA

i?

Another example

i=7A

5A

2A i

EECS 40 Spring 2003 Lecture 3 W G Oldham and S Ross

7/31/2019 lec3_2003

16/16

EECS 40 Spring 2003 Lecture 3 W. G. Oldham and S. Ross

KCL at node X:

Current into X from the left:

(Vin - Vx) / R

Current out of X down to ground:

C dVx / dt

KCL: (Vin Vx) / R = C dVx / dt

KCL EXAMPLE: RC CIRCUIT

X

ground

R

CVin Vx+

_

+

-

)VV(RC

1

dt

dVoutin

out

Vx(t) = Vin + [ Vx(t=0) Vin ] e-t/(RC)

Solution: