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4Loading effects

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Impedance Characteristics

What is impedance ?The term impedance is a general expression which can be applied to any electrical entity which impedes the flow of current. Thus this expression could be used to denote a resistance, a pure reactance, or as is most likely in the real world, a complex combination of both reactance and resistance.

Inductive Reactance XL = 2 * pi * f * L f = frequency in hertz and L = inductance in Henries

Capacitive Reactance XC = 1 / (2 * pi * f * C)f = frequency in hertz and

C = capacitance in Farads

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Z = R ± jX           |Z| = (R2 + X2)½           ϕ = tan-1(X/R)           Y = 1/Z

Impedance Calculation

M" is the mutual inductance between

inductors."ω" is frequency in radians/second, and is equal to 2π times frequency in

cycles/second.

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Impedance Calculation

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Impedance Calculation

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Example:Assume you have available these 4 items on your bench:

(a) A series of eight fresh AA type 1.5 volt cells to create a total of 12 volts supply. (b) A 12 volt heavy duty automotive battery - fully charged. (c) a small 12v bulb (globe) of very, very low wattage. and; (d) a very high wattage automotive high-beam headlight.

loading effect:

Now if we connect the extremely low wattage bulb to the series string of AA cells we would expect all to work well. Similarly if we connect the high wattage, high-beam headlight to the heavy duty automotive battery all will be well. Well for a time anyway. Both of these sets are "sort"of matched together. Light duty to light duty and heavy duty to heavy duty.

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Now what do you think would happen if we connect the high beam head light to the series AA cells and conversely

the low wattage bulb to the automotive battery.?

loading effect:

In the first case we could imagine the high beam headlight would quickly trash our little tiny AA cells.

In the second case our min-wattage bulb would glow quite happily at its rated wattage for quite a long time. Why

The heavy duty battery is capable of delivering relatively large amounts of power (low impedance source )

 but the series string is capable of delivering only relatively minimal power (high impedance source) .

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loading effect:

On the other hand the high beam headlight is capable of consuming relatively large amounts of power (low impedance load ) but the miniature bulb is capable of consuming only minimal amounts of power ( high impedance load)

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loading

Also inserting a thermometer at room temperature into a hot water to measure its temperature changes the temperature of the water which leads to error in the temperature measurement.

inserting an ammeter into a circuit to measure the current changes the value of the current due to the ammeters own resistance which changes the total resistance of the circuit.

Examples:

When components such as sensors and transducers interconnected with signal conditioning hardware it is necessary to match impedances properly , One adverse effect of improper impedance matching is the loading effect.

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Thevenin’s Theorem

An active network having two terminals A and B to which an electrical load may be connected, behaves as if the network contained a single internal source ETh in series with a single internal impedance ZTh as shown in the figure, network have been replaced by their internal impedances.

the equivalent Thevenin circuit

Active network

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Electrical loading

Connecting a load ZL across the output terminals of an active network is equivalent to connecting ZL across the equivalent

Thevenin circuit, as in shown in the figure .

The current i through ZL is thus

LTh

Th

ZZE

i

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The potential difference across the load VL is given by

ThLTh

LLL E

ZZZ

iZV

ThL EV

The effect of connecting the load across the network is to change the potential difference from ETh to VL

Discussion:

The value of VL will approach that of ETh when ZL>>ZTh. Therefore, the condition for maximum voltage transfer is to select ZL to be too much grater than ZTh.

The condition for maximum power transfer is that

ThL ZZ Prove this at home!

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Therefore, the effect of connecting a load across the network yields a loading error of magnitude

LTh VE Error Loading

LTh

LTh

ZZZ

E 1

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Loading a voltmeter

Assume that a voltmeter of resistance Rm is connected across the shown active circuit to measure the voltage

between terminals A and B .

Rm

Active network

Voltmeter

Therefore, the reading indicated by the instrument (voltmeter) is

ThThm

mm E

RRR

V

Voltage before the meter was connected

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This means that the error in the reading is

mTh VE Error

mTh

mTh

RZR

E 1

and the accuracy of the voltmeter is

%100Accuracy Th

m

EV

%100

mTh

m

RZR

Note that if Rm is very large, the error goes to zero and the accuracy goes to 100%

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Loading a potentiometer

If the potentiometer track has a uniform resistance per unit length, then the open loop circuit voltage between terminals A and B (Thevenin voltage) is

STh VLxE

To find the Thevenin impedance, we set VS to be zero (make short circuit) and calculate the impedance between A and B as shown

ppTh RLxLRLxR ]/)[(1

)/(11

where Rp is the total track resistance

Rp

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The previous equation yields

LxLx

RR pTh 1

Now…Rp

i

The current in the circuit is given by

Lp

S

LTh

Th

RLxLxRVLx

RRE

i

)1)(/(

)(

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This yields that the potential difference across the load is

Lp

SLLL

RLxLxRVLxR

iRV

)1)(/(

)(

1)1)(/)(/()(

LxLxRRVLx

Lp

S

This is nonlinear relationship between VL and x. Therefore the effect of the loading is to give a non-linearity error, such that

LThl VEe

1)1)(/)(/(

11)/(

LxLxRRVLxe

LpSl

Non-linearity error:

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1)1)(/)(/(

)1)(/)(/()/(

LxLxRR

LxLxRRVLxe

Lp

LpSl

If RL>>Rp, the above equation approximates to

The maximum value of this error occurs when

i.e., we can reduce the error by selecting RL to be too much greater than Rp

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Loading of a Wheatstone bridge

The Thevenin voltage for a Wheatstone bridge is the open circuit voltage when I = 0 (there is no load). This means that

VS : Supply voltage

)( 211 RRIVS

)( 432 RRIVS Also

)( 211

RRV

I S

)( 432

RRV

I S

Therefore, ADBATh VVE

4211 RIRI

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4

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1

RRR

RRR

VS

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The Thevenin Resistance between B and D can be obtained by setting VS to be zero (i.e., make short circuit between A and C)

=

43

43

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21

RRRR

RRRR

RTh

Now, if there is a load resistance across the circuit, then

)( ThLTh RRIE ThL

Th

RRE

I

B

D

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Now we are interested in the potential difference across the load VL (this is the output of the Wheatstone bridge)

ThL

LThLL

RRRE

IRV

)]/()/([)]/()/([

43432121

434211

RRRRRRRRRRRRRRRVR

L

SL

43214321

4231

))(()][

RRRRRRRRRRRRRVR

L

SL

Note that if RL is very large in comparison to RTh, then it can be seen from equation (**) that the output voltage becomes

(**)

ThL EV )the error goes to zero(

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Loading of elements in a measurement system

Consider the measurement system consisting of a transducer, an amplifier and an indicator. The transducer has an open circuit output voltage of Vt and a resistance Rt . The amplifier has an input resistance Rin which represents a load across the transducer

int

intinin

RRRV

RIV

1

The potential voltage difference across the load is

)As dial gauge (

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Loading of elements in a measurement system

Iout

IinII

RRRGV

RIV

2

Assume that the amplifier has a transfer function of G, then the open circuit output from

the amplifier is GVin .

Since the amplifier has an output resistance of Rout, and the indicator has a resistance of RI, the output potential difference from the indicator (reading of the indicator) is

))(( Ioutint

Iint

RRRRRRGV

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tI GVV

Note that if the amplifier and the indicator has very large input resistances (Rin and RI are large) , then the previous equation yields

Which means that the output (Vt) is proportional to the input signal (VI). And there is no need to know the values of the

resistances .

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Impedance Matching

From the analysis given in the preceding section, it is clear that the signal-conditioning circuitry should have aconsiderably large input impedance in comparison to the output impedance of the sensor-transducer unit inorder to reduce loading errors.

Usually an impedance matching amplifier (impedancetransformer) would be needed between the two components.

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When connecting a device to a signal source, loading problems can be reduced by making sure that the device

has a high input impedance .

Unfortunately, this will also reduce the level (amplitude, power) of the signal received by the device. In fact, a high impedance device may reflect back some harmonics of the source signal. A termination resistance may be connected in parallel with the device in order to reduce this problem.

Impedance Matching

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Consider a dc power supply of voltage Vs and output impedance (resistance) Rs. It is used to power a load ofresistance Rl, as shown in Figure 4.3. What should be the relationship between Rs and Rl if the objective is to

maximize the power absorbed by the load?

Impedance Matching Example:

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Impedance Matching Example:

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Impedance Matching Example:

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Impedance Matching Example:

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Impedance Matching Example:

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Impedance Matching in Mechanical Systems

The concepts of impedance matching can be extended to mechanical systems and to mixed and mechatronicsystems in a straightforward manner. The procedure follows from the familiar electro-mechanical analogies

ElectricalQuantity

Mechanical Analog I(Force-Current)

MechanicalAnalog II

(Force Voltage)

Voltage, eVelocity, vForce, f

Current, iForce, fVelocity, v

Resistance, RLubricity, 1/B

(Inverse friction)Friction, B

Capacitance, CMass, MCompliance, 1/K(Inverse spring

constant)

Inductance, LCompliance, 1/K(Inverse spring

constant)Mass, M

Transformer, N1:N2

Lever, L1:L2Lever, L1:L2

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electro-mechanical analogies

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electro-mechanical analogies

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Electrical to Mechanical 1 (Force-Current).

electro-mechanical analogies

The important relationship when converting from a circuit to the Mechanical 1 analog is that between Kirchoff's Current Law and D'Alemberts Law (with inertial forces included)

One deficiency in this analogy is that it only works easily for capacitors that are grounded   .

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Procedure for Conversion from Electrical to Mechanical 1

The conversion from an electrical circuit to a mechanical 1 analog is easily accomplished if capacitors in the circuit are grounded.  If they are not, 

the process results in a mechanical system where positions must be chosen very carefully

, and the process can be much more difficult.

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Example:

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Example Visual Method:

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Example The capacitor is not grounded:

We can rewrite the equations using analogous quantities.

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