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

1

Lecture 3

� Definitions: Circuits, Nodes, Branches

� Kirchoff¶s Voltage Law (KVL)

� Kirchoff¶s Current Law (KCL)

� Examples and generalizations

� RC Circuit Solution

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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 joined²entire wire

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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

VaVb

+

 _ 

+

 _  _ +

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

KIRCHOFF¶S 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

Path³rise´ or ³step up´

(negative drop)

+

-

V1

Path

³drop´

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

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

KVL EXAMPLE

Path 1: 02a !Path 2:

c !

Path :c2a!

ca

+

+

3

21

+

vb

v3v2 +

+

-

Examples of three closed

paths:

a b c

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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, Faraday¶s Law p induced emf (voltage)

Avoid these loops!

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

Include a voltage source as the circuit representation of the

emf or ³noise´ pickup.

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

 Antennas are designed to ³pick up´

electromagnetic waves

³Regular circuits´ often do the same thing p

not desirable!

)t(BX

)t(v+

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

ALTERNATIVE STATEMENTS OF KIRCHHOFF¶S

VOLTAGE LAW

1) For any node sequence A, B, C, D, «, M around a

closed path, the voltage drop from A to M is given by

LMCDBC AB AM 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 !

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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 at

both ends carry the same voltage.

We say these elements are in parallel.

KVL c

lockwi

se,start at top:

Vb ± Va = 0

Va = Vb

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

 

KIRCHOFF¶S 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 in node is zero (e.g. entire capacitor is part of a branch)

KIRCHOFF¶s CURRENT LAW ³KCL´: 

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

USING KCL

Kirchhoff¶s Current Law (KCL)

Formulation 1:

Sum of currents entering node = sum of currents

leaving node

Use/write ref erence directions to determine³entering´ and ³leaving´ currents--no concer n

about actual current directions

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

ALTERNATIVE KCL FORMULATIONS

Formulation 2:

³Algebraic sum´ of currents entering node = 0

where ³algebraic sum´ means currents leaving are

included with a minus sign

Formulation 3:

³Algebraic sum´ of currents leaving node = 0

currents entering are included with a minus sign

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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

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

KIRCHHOFF¶S CURRENT LAW EXAMPLE

Currents entering the node: 24 Q A

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

Three formulations of KCL:

 A18i 0i1044 :

 A18i 0i10)4(4 :

 A18i i1044 :1

Q!!

Q!!

Q!!

i  Q A

  Q A - Q A

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

GENERALIZATION OF KCL

Sum of currents entering/leaving a closed sur f ace is zero

Note that circuit branches could be inside the surface.

Could be a big chunk

of circuit in here, e.g.,

could be a ³Black Box´

T he surface can enclose more than one node!

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

KIRCHOFF¶S CURRENT LAW USING SURFACES

Example

i A2 A5 !QQ

entering leaving

surface

i must be 0 mA

0 mA

i?

 Another example

i=7Q A

Q A

2Q A i

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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+

 _ 

+

-

)(t

tit

!

Vx(t) = Vin + [ Vx(t=0) ± Vin ] e

-t/(RC)

Solution: