Campbell's Modification of Heaviside Bridge

 Campbell's Modification of Heaviside Bridge

fig shows a modified Heaviside bridge. This modification is due to Campbell. This is used to measure self-inductance. In this case an additional balancing coil L, R is included in arm ad in series with inductor under test. An additional resistance r is put in arm ab . Balance is obtained by varying M and r. A short-circuiting switch is placed across the coil R2, L2 under measurement. Two sets of readings are taken one with the switch being open and the coil R2, L2 under measurement. Two sets of readings are taken one with the switch being open and the other with the switch being closed. Let values of M and r, Be M1 and R1 with the switch open and M1 and r2 with the switch closed.

we have L2+L = ( M(R3 + R4) + R4.L1 )/R3

and               L = ( M2 ( R3 + R4) + R4.L1)/R3

  Therfore     L2 = (M1 - M2)(1 + R4/R3)

similarly  we can write: R2+R = (R1 + r1).R4/R3  and R = (R1 + R2).R4/R3

                       R2= (r1 - r2).R4/R3

This method is a good example of the method adopted to eliminate the effects of leads etc.

if R3 = R4

            L2 = 2(M1 - M2)

            R2 =  r1 - r2

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Heaviside Mutual Inductance Bridge.

  Heaviside Mutual Inductance Bridge.

This bridge uses known self-inductance to measure mutual inductance. The same bridge slightly modified by Campbell is used to measure self-inductance in terms of known mutual inductance.

M = unknown mutual inductance,

L1 = self-inductance of secondary of mutual inductance,

L2 = known self-inductance,

and R1 , R2 ,R3 , R4 = non-inductive resistors.

At balance voltage drop across between b and c must equal to the voltage drop across between d and c. Also, the voltage drop across ahc must equal to the voltage drop across adc. Thus we have the following equations at balance.

                    I1R3 = I2.R4

               (I1+I2)(jwM)+I1(R1+R3+jwL1) = I2(R2+R4+jwL2)

Solving them we get:

                R1.R4 = R2.R3

and           M = (R3.L2 - R4.L1)/(R4+R3)

if R3 = R4    ,  we get,  M = (L2 - L1)/2

                           and     R1 = R2

This method can also ussed for measurement of self inductance. Suppose if we have to determine L2 then

 we can write ,                L2 = (M(R3+R4) + R4.L1 )/R3

                                             = M ( 1 + R4/R3) + ( R4/R3 ).L1

                        and R2 = R1.(R4/R3)

In case             R4 = R3, we have

                        L2 = 2M +L1

 and                 R2 = R1 

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Anderson's Bridge

 Introduction

This Bridge, in fact, is a modification of maxwell's inductance capacitance bridge in this method we measure unknown self-inductance in terms of a standard capacitor this method is applicable for the precise measurement of self-inductance over a wide range of values.

Let L1=Self inductance be measured,

      r1= resistance connected in series with-inductor,

r, R2, R3, R4= known non-inductive resistances,

R1= resistance of the self inductor,

C= fixed standard capacitor.

At balance , I1=I3 and I2 = Ic + I4

Writing and solving balance equations gives us the result 

and

Two obtain easy convergence of balance alternate adjustments of r1 and r are done.

Advantages:

• for a low-value Q coil it is easy to obtain a balance point because there is no sliding occurs like maxwell's bridge.
•  A fixed capacitor is used instead of a variable capacitor like in maxwell's bridge.
• This bridge may be used for accurate measurement of capacitance in terms of inductance

Disadvantages:

• The bridge is more complex than maxwell's bridge, due to more parts it is difficult to manipulate the bridge for different measurements and also to set up . The balance equations are not simple and in fact tedious.

Considering the above complications in the cases where a variable capacitor is permissible maxwell's bridge is used.

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Hay's Bridge

  Hay's Bridge

It is the modification of maxwell's bridge. This bridge uses resistance in series with the standard capacitor.

(unlike the Maxwell's Bridge which uses resistance in parallel with the capacitor).

Let L1 = unknown inductance having resistance R1,

R2, R3, R4= known non-inductive resistances, and C4=standard capacitor.

At balance,

  (R1+jwL1)(R4-j/wC4)=R2.R3

Separating real and imaginary terms and solving we get :

                L1=(R2.R3.C4)/(1+w^2.C^2.R4^2)

and          R1= (w^2.R2.R3.R4.C4^2)/(1+w^2.C4^2.R4^2)

The Q factor of the coil is : Q= (wL1)/R1 = 1/(wC4R4)

there is a frequency term in the above expressions so it seems that it must be accurately known. This is not true for inductance when a high Q coil is being measured. Because:

we can write 

L1= (R2.R3.C4)/(1+(1/Q)^2)

for Q value higher than 10  the equation reduces to L1= R2.R3.C4

we can see the eq. becomes the same as maxwell's bridge.

Advantages:

• This bridge gives a very simple expression for unknown inductance for high Q coils and is suitable for coils having Q>10.
• This Bridge gives a simple expression for the Q factor.
• If we examine the expression for Q factor: Q = 1/(w.C4.R4)

R4 appears in denominator means for a high value of Q we have a small value of R4 thus bridge requires a low value of R4 whereas The Maxwell's bridge requires a parallel resistor, R4 of a very high value.

Disadvantages:

• The bridge is not suitable for the measurement of inductance Q <10.

Reference:

Swerny 

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Measurement of Self Inductance , Maxwell's Bridge

Maxwell's Inductance Bridge: The bridge circuit measures unknown inductance by comparison with a variable standard self-inductance.

Let L1=unknown inductance of resistance R1,

      L2=variable inductance of fixed resistance r2,

      R2= variable resistance connected in series with inductor L2,

      R3, R4= known non-inductive resistances.

By solving balance equations of the bridge we get

                          L1=(R3/R4).L2

                          R1=(R3/R4)(R2+r2)

Resistors R3 and R4 normally a selection of values from 10,100,1000,10,000ohm ,r2 is a decade resistance box. In some cases, an additionally known resistance may have to be inserted in series with an unknown coil in order to obtain balance.

Maxwell's Inductance-Capacitance Bridge:

In this bridge, inductance is measured by comparison with a variable standard capacitance.

Let L1=unknown inductance,   R1= effective resistance of inductor L1,

      R2, R3, R4=known non-inductive resistances, and C4=variable standard capacitor.

 

Writing an equation for balance

(R1+jwL1)(R4/(1+jwC4R4))=R2.R3

Separating the real and imaginary parts and solving we get:

                                R1=(R2.R3)/R4

                        and  L1=R2.R3.C4

  the two variables R4 and C4 appear in different equations so the equations are independent.

The expression for Q factor Q= (w.L1)/R1 = w.C4.R4

Advantages 

• The two balance equations are independent if we choose R4 and C4 as variable elements.
• The frequency does not appear in any of the two equations.
• This equation gives a simple expression for unknowns L1 and R1 in terms of bridge elements.
• for example, if we consider R2.R3=10^6 then 
•                                L1=C4.10^6
• thus if balance is achieved then if the value of capacitance is in micrometer then it gives the direct value of inductance in henry.
• Maxwell's inductance-capacitance bridge is very useful for the measurement of a wide range of inductance at power and audio frequencies.

Disadvantages

1.    The variable capacitor is very expensive if calibrated to a high degree of accuracy

Therefore sometimes a fixed capacitor is used which is known for a high degree of accuracy. In this case, balance is obtained by

(a)   either varying R2 and R4 since R2 appears in both equations thus balance adjustment becomes difficult.

(b) putting additional series resistance with inductance and varying this and R4.

2.    The bridge is limited to measurement of low Q (1<Q<10) .Because for high Q value of R4 must be high as 10^5 or 10^6 and the resistance box with such high resistance is very expensive. Thus bridge is unsuitable for Q>10.

The bridge is also unsuitable for low Q (Q<1) . Because for low Q obtaining balance condition is difficult due to the sliding effect.

 ( The capacitance is generally fixed in the measurement of balance point thus we change values of R2 and R4 and this comes in both equations so if we change one then another variable changes and balance disturbs so we need to do many manipulations) 

Thus we see from the above discussion Maxwell's bridge is suitable for the measurement of medium Q values only.

Reference:

-Swerny

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AC bridge Sources and Detectors

 AC bridge Sources and Detectors

Measurement at low frequency: power line may act as a source of supply to the bridge circuits.

Measurement at higher frequency: Electronic oscillators are universally used as bridge source supplies.

Electronic oscillators and higher frequency

1. Frequency is constant, easily adjustable, and determinable with accuracy.

2. The waveform is very close to a sine wave.

3. Their power output is sufficient for most bridge measurements 

(A typical oscillator has a frequency range of 40Hz to 125Hz with a power output of 7 W.)

Detectors

Commonly used detectors for ac bridges are Headphones, Vibration galvanometers, and Tuneable amplifier detectors. 

Headphones are widely used as detectors at frequencies of 250Hz and over up to 3 or 4kHz.They are the most sensitive detectors for this frequency range.

At a single frequency: Tuned detectors normally give the greatest sensitivity and discrimination against harmonics in the supply.

Vibration galvanometers are extremely useful at power and low audion\ frequency ranges. Vibration galvanometers are manufactured to work at various frequencies ranging from 5Hz to 1000Hz but are most commonly used below 200Hz as below this frequency they are more sensitive the headphones.

Tuneable amplifier detectors are the most versatile of the detectors.

The transistor amplifier can be tuned electrically and thus can be made to respond to a pointer-type instrument.

This detector can be used, over a frequency range of 10 Hz to 100KHz

For ordinary a.c.bridge measurements of Inductance and Capacitance.

 A fixed frequency oscillator of 1000Hz and output of about 1W is adequate. But for more specialized work variable oscillators are preferable with output up to 5W. In practice power is supplied is within limits to the bridge but on some occasions, a higher power may be necessary. 

Usually, an untuned amplifier is used as a detector. The balance detection is sensed both orally by headphones and visually by a pointer galvanometer having a logarithmic deflection (to avoid damage to the galvanometer which may be caused by unbalancing).

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Laboratory Standards of Emf(Zener controlled voltage sources)

In the past standard, cells were used exclusively as laboratory standards but in recent years. Semiconductor devices such as Zener-controlled reference sources have replaced standard cells in practically all industrial applications.

Silicon diodes have voltage-current characteristics such that an extremely sharp reverse current occurs at a point on the voltage curve known as "Zener voltage". 

The point indicates a breakdown of the diode under reverse voltage application, but the process is reversible if safe current and heating limits are not exceeded. 

The Zener voltage may be controlled over a wide range by the processing techniques used during the manufacture of the Zener diodes.

In the fig, the circuit is of Zener diode in a typical application as a laboratory standard.

• he supply voltage E is usually much higher than the Zener voltage of the diode assuming the breakdown of the diode.
• resistor R usually has a high value and is placed in series with the diode and the battery. It serves to limit the current through the diode during breakdown to a safe value.
• In addition, when the supply voltage varies for any reason, most of this variation is taken up in the voltage drop across R with a very small change in diode current and as a result, the reference voltage ( output voltage) remains practically the same.

        

Commercial voltage standards using Zener diodes essentially consist of :

(i)    A Zener controlled voltage source placed in a temperature-controlled environment to improve its long-term stability.

(ii)    A precision output voltage divider. The temperature is controlled to +0.03 degree Celcius over an ambient temperature range of 0 degree Celcius to 50 degree Celcius providing output stability of the order of 10 ppm/month.

The four available outputs are :

(a) a 0 -1000 microVolt source with 1 microVolt precision,

(b) a 1.00 V reference for potentiometer measurements with ratio box,

(c) a 1.018+(delta) reference for saturated Weston cells comparisons.

(d) a 1.019+(delta) reference for unsaturated Weston cell comparison.

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