Lecture 4

Kirchoff: The Man, The Law, The Field

EC2Polikar

Ohm: The law

The amount of current passing through a conductor is directly proportional to the voltage across the conductor and inversely proportional to the resistance of the conductor.

R

VI IRV

+

_

V RI

V: The voltage difference between two locations, aka potential difference. It is analogous to pressure. I: The amount of current flowing between these two points. It is analogous to flow. R: Resistance between the two points.

So how did Ohm come with this law…?

Power

A conductor’s resistance to electric current produces heat. The greater the current passing through the conductor, the greater the heat. Also, the greater the resistance, the greater the heat. A current of I amps passing through a resistance of R ohms for t seconds generates an amount of heat equal to joules (a joule is a unit of energy equal to 0.239 calorie).

Energy is required to drive an electric current through a resistance. This energy is supplied by the source of the current, such as a battery or an electric generator. The rate at which energy is supplied to a device is called power, that is power is energy supplied per unit time. Power is often measured in units called watts. The power P supplied by a current of I amp passing through a resistance of R ohms is given by

RtI 2

VIR

VRIP

22

ivT

E1

Gustav Robert Kirchoff

Born: 12 March 1824 in Königsberg, Prussia (now Kaliningrad, Russia)Died: 17 Oct 1887 in Berlin, Germany

Kirchoff’s Laws

Two sets of rules define circuit analysis: Kirchoff’s current laws Kirchoff’s voltage laws

KCL: The algebraic sum of all currents in any given node is zero KVL: The algebraic sum of all voltages in any given loop is zero

What is a node? What is a loop? What is an algebraic sum…?

Assumptions & Definitions

Wires: We connect the components to each other by wires. Wires, just like any other component in a circuit, has an

internal resistance. However, since this resistance is so small, it is usually ignored. We will “model” the wire as a zero-resistance device, just like we model the ideal current meter as a zero-resistance device.

Nodes: A node is defined as the point where two or more components are connected to each other. You must show care, however, in locating and counting the nodes.

+

-vA

RC

RD

RF

RE

RB

How many nodes are there in this circuit?

Definitions

Loops: Any trace that starts and ends at the same point is a “closed loop”. A loop typically follows components, but it can, jump

across “space”. We need to understand how to locate, identify and

count loops.

Short / open circuit: A short circuit issimply connection of wires with noother components in between. Open circuitis a lack of connection between twonodes.

How many loops are there in this circuit?

+

-

v1

R1

R2

R4

R3

R5

+

va

Can you identify short and open circuits ?

-

Kirchoff’s Current Law(KCL)

Algebraic sum of all currents at any given node must be zero. This is equivalent to saying sum of all currents entering

a node (let’s call these negative currents) and the sum of all current leaving a node (and let’s call these positive currents), must be equal.

In other words, no current (or no charge) may accumulate at a node. Current must flow!

A

B+V

_i1 i3 i5 i7

i2 i4 i6

i11

i9i8 i10

For node A:-i1+i2+i3=0 i1=i2+i3

How about other nodes?

Kirchoff’s Voltage Law(KVL)

The algebraic sum of all voltages in a loop must sum to zero! The law states that energy must be conserved.

This is where things may get little tricky, since you must follow a norm for defining polarities: You may follow this simple rule: A voltage source

produces energy (power), so there is an increase in potential energy when you move from its – to + terminal.

Resistors consume energy / dissipate power, so there is a potential drop across the terminals of a resistor, equivalent to the value of the resistor times the current passing through that resistor (by Ohm’s law).

+

_

V R1I

R2For the loop on the left: +V- I.R2 – I.R1 = 0, or

V=I(R1+R2)

More on KVL

How about this circuit? A good practice is always to choose your current

directions first, and then follow all potential drops and rises. Do not be concerned about choosing the right direction. If at the end you find out that the results for a current is negative, that means you picked the polarity wrong.

A

B+V

_

i1 i3 i5 i7

i2 i4 i6

i11

i9i8 i10

Open & Short Circuits

Remember: No current passes through an open circuit, however,

there can, and usually is a potential difference (voltage) across the terminals of the open circuit.

The potential difference between two points on a short circuit is, be definition, zero; however, there can, and usually is a nonzero current flowing through a short circuit.

A

B+

V

_

iCD=0+vCD-

iBE≠0vBE=0

C

DE