eecs 42 fall 2004 lecture 5 9/10/2004 ee 42 lecture 5 equivalent resistance of a passive linear...

EECS 42 fall 2004 lecture 5 9/10/2004

EE 42 lecture 5

Equivalent resistance of a passive linear circuit Thevenin’s Theorem for a single source linear

circuit Norton’s Theorem for a single source linear

circuit Measurements Nodal analysis


Linear equations-> straight line

We know it is a straight line because the circuit gives us a set of linear equations, and so they have a linear solution.

V

I

+V-

I


Equivalent resistance for a linear passive circuit.

If we have a circuit which includes only wires and resistors, the relationship between the voltage and current are related by a straight line, through the origin.

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+V-

I


Why it must pass through the origin if there are no sources of power

The function of current vs voltage must pass through the origin, otherwise it would pass through a quadrant where we could extract power—and we said there were no sources of power!

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+V-

I


Circuits with sources

If we have any circuit which includes DC sources (voltage or current) and linear resistances, then the I-V curve is still linear, but it does not have to pass through the origin.

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I

+V-


Models and measurements of circuits with sources

Since the I-V curve is a straight line, we can characterize it with two parameters, for example: The current intercept and slope. The voltage intercept and slope. Both the current and voltage intercept.

If we measure any two points, we can calculate any of these parameters.


Thevenin’s Theorm

We can make an equivalent circuit for any circuit which includes only resistors and linear sources with a single voltage source and a resistor. This translates the voltage intercept and slope equation into a circuit.

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~


Thevenin’s Theorm

The point where the current is equal to zero gives us the voltage put out by the voltage source in the equivalent circuit.

(1/slope) gives us the resistance R of the resistor in the equivalent circuit.

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I

I

+V-

~+V-

R


Thevenin’s Theorm

A large slope means a small value of R

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I

I

+V-

~+V-

R


Norton’s Theorem

We can also make a different equivalent circuit, for any circuit which contains only linear sources and resistors, with a current source

The needed value of current from the current source is the amount of current put out by the actual circuit when the voltage is zero (the current intercept)

This model is called a Norton equivalent circuit.


Norton’s Theorem

The point where the voltage is equal to zero gives us the current put out by the voltage source in the equivalent circuit.

(1/slope) gives us the resistance R of the resistor in the equivalent circuit.

V

I

I

+V-

RI


Taking Measurements To measure voltage, we use a two-terminal

device called a voltmeter. To measure current, we use a two-terminal

device called a ammeter. To measure resistance, we use a two-terminal

device called a ohmmeter. A multimeter can be setup to function as any of

these three devices. In lab, you use a DMM to take measurements,

which is short for digital multimeter .


Measuring Current To measure current, insert the measuring

instrument in series with the device you are measuring. That is, put your measuring instrument in the path of the current flow.

The measuring devicewill contribute a verysmall resistance (like wire)when used as an ammeter.

It usually does not introduce serious error into your measurement, unlessthe circuit resistance is small.

i

DMM


Measuring Voltage To measure voltage, insert the measuring

instrument in parallel with the device you are measuring. That is, put your measuring instrument across the measured voltage.

The measuring devicewill contribute a verylarge resistance (like air)when used as a voltmeter.

It usually does not introduce serious error into your measurement unlessthe circuit resistance is large.

+ v -

DMM


Example

For the above circuit, what is i1?

Suppose i1 was measured using an ammeter with internal resistance 1 Ω. What would the meter read?

9 Ω 27 Ωi1 i2 i3

54 Ω3 A


Measuring Resistance To measure resistance, insert the measuring

instrument in parallel with the resistor you are measuring with nothing else attached.

The measuring deviceapplies a voltage to theresistance and measuresthe current, then uses Ohm’slaw to determine resistance.

It is important to adjust the settings of the meter for the approximate size (Ω or MΩ) of the resistance being measured so appropriate voltage is applied to get a reasonable current.

DMM


Measurements to derive a Thevenin equivalent circuit

Measure the open circuit voltage. The voltage for the voltage source in the Tevenin model

is this open circuit voltage. Measure the short circuit current. The resistance to put into the Thevenin model is

R=Voc/Iss Alternatively, you can turn off all of the sources within the

black box and measure its resistance Turn off -> voltage sources to 0 volts (replace them with a short

circuit), current sources to zero current (open)


Measurements to derive a Norton equivalent circuit

Measure the short circuit current The current for the current source in the Norton model is

this open circuit voltage. Measure the open circuit voltage. The resistance to put into the Norton model is R=Voc/Iss Alternatively, you can turn off all of the sources within the

black box and measure its resistance Turn off -> voltage sources to 0 volts (replace them with a short

circuit), current sources to zero current (open)


Solution to Example:

By current division, i1 = -3 A (18 Ω)/(9 Ω+18 Ω) = -2 A When the ammeter is placed in series with the 9 Ω,

Now, i1 = -3 A (18 Ω)/(10 Ω+18 Ω) = -1.93 A

9 Ω 27 Ωi1 i2 i3

54 Ω3 A 9 Ω 18 Ωi1

3 A

9 Ω 27 Ωi1 i2 i3

54 Ω3 A 10 Ω 18 Ωi1

3 A1 Ω

eecs 42 fall 2004 lecture 5 9/10/2004 ee 42 lecture 5 equivalent resistance of a passive linear...

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