The range of various instruments can be
extended using multipliers. Similarly the range of electrostatic instruments is
also extended using multipliers. The multipliers used for electrostatic
instruments are of two types,

1.Resistance potential divider 2.Capacitance multipliers

__1.1 Resistance Potential Divider__

The use of resistance potential divider for
extending the range of an electrostatic instrument is shown in the Fig 1.

Fig. 1 Use of resistance potential divider |

R = Total resistance of potential divider

V = Voltage to be measured

r = The resistance whose voltage drop is
applied to an electrostatic voltmeter

v= The voltage across an electrostatic voltmeter

C= The capacitance of an electrostatic voltmeter

The resistance r and capacitance C forms a
parallel circuit and the equivalent impedance is,

Thus the equivalent impedance across the
voltage V is,

The factor by which voltage is changed due to
potential divider is called its multiplying power and given by,

The numerical value of multiplying power m is,

If ω, C and r are very small then ω

^{2}C^{2}r^{2}≤1 and can be neglected,
1.At high
voltages, the cost of this method is high.

2.At high
voltages, the power loss and wastage is excessive.

3.At high
voltages, the accuracy is very less due to stray capacitance effects.

4.When used
for a.c. measurements, it should be wound noninductively and capacitor leakage
resistance must be high.

Thus this
method is not suitable for high voltages but used for d.c. measurements as
capacitance potential divider can not be used for d.c. circuits.

__1.2 Capacitance Multipliers__

The
capacitance multiplier method is nothing but the use of capacitance potential
divider. There are two methods of connecting capacitor for potential division.

**Method 1**: In first method, a single capacitor is connected in series with the voltmeter and the voltage to be measured is applied across the combination as shown in the Fig 2.

Fig. 2 Capacitor multiplier method |

Let C = Series capacitor

C

_{v}= Capacitor of voltmeter
v = Voltage
across voltmeter

V = Voltage to
be measured

The total
capacitance across the supply is,

The total
impedance across the supply is,

The impedance
of voltmeter is,

Thus the
multiplying power of the multiplier is,

1. To have high
value of multiplying power, the voltmeter capacitor must be high.

2. The voltmeter
capacitor varies with the deflection of the moving needle hence te voltmeter
must be calibrated alongwith the series multiplier capacitor.

**Method 2**: In many practical cases a set of capacitors connected in series across the voltage be measured as shown in the Fig 3.

Fig.2 Capacitor multiplier - method 2 |

The capacitors C

_{1}and C_{v}are in parallel hence their resultant is C_{1}+ C_{v}. While C_{2}, C_{3}… C_{n}are in series and their equivalent is C_{s }where,
Thus C

_{s }and (C_{1}+ C_{v}) are in series hence the resultant capacitor across the voltage V is,
While across
the voltage v the capacitor is C

_{1}+ C_{v}hence the impedance is,
Thus the
multiplying power is,

If C

_{1}is large with respect to C_{v}, then there is no appreciable change in the multiplying power alongwith the deflection of the pointer.
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