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MODULE 2 β€” Measuring Instruments

🌑️ 2.1 Swamp Resistance

Temperature Compensation

Swamp Resistance

$$ R_{sw} = \frac{R_m + R_{swamp}}{m - 1} $$
$$ R_{total} = R_{copper} + R_{swamp} $$

Current Relationship

$$ I = I_s + I_m $$
$$ \frac{dR_{total}}{dT} \approx 0 $$

Why Needed

Copper coil: HIGH temp coeff (+)
Manganin/Constantan: β‰ˆ ZERO
Series combo β†’ flat response
R_copper (High Ξ±) R_swamp (Ξ± β‰ˆ 0) β†’ Iβ†’ dR_total/dT β‰ˆ 0 (swamp dominates)
25Β°C
Rsw = β€”

⚑ 2.2 Multirange Ammeter β€” Individual Shunts

Range Extension

Shunt Resistance

$$ \boxed{R_{sh} = \frac{R_m}{m - 1}} $$
$$ m = \frac{I}{I_m} $$

Shunt Current

$$ I_s = I - I_m = (m-1)I_m $$
$$ V_{sh} = I_s R_{sh} = I_m R_m $$

Multiple Ranges

$$ R_{sh_1} = \frac{R_m}{m_1-1} $$
Larger range β†’ smaller Rsh
PMMC Rm , Im Rsh1 Rsh2 Rsh3 SW β†’ β†’ Click a range to switch shunts
m = β€”, Rsh = β€”

⚑ 2.3 Universal Shunt (Ayrton Shunt)

Safe Switching

Tapped Shunt

$$ R_1 = \frac{R_m}{m_1 - 1} $$
$$ R_2 = \frac{R_m + R_1}{m_2} - R_1 $$
$$ R_3 = \frac{R_m + R_1 + R_2}{m_3} $$

Voltage Equality

$$ I_m(R_m + R_{remainder}) = I_n \cdot R_{tap} $$
$$ m_i = \frac{I_i}{I_m} $$

Key Advantage

βœ“ No break in meter circuit
βœ“ Meter always protected
βœ— Individual: switch gap β†’ danger
PMMC Rm R₁ Rβ‚‚ R₃ Iβ†’ βœ“ Meter always has a path β€” never open-circuited
R₁ = β€”, Rβ‚‚ = β€”, R₃ = β€”

πŸ“ 2.4 Voltmeter Multiplier (Series Resistance)

Voltage Range Extension

Multiplier

$$ \boxed{R_{se} = (m-1)R_m} $$
$$ m = \frac{V}{V_m} $$

Derivation

$$ I_{FSD} = \frac{V_m}{R_m} = \frac{V}{R_m + R_{se}} $$
$$ R_{total} = m \cdot R_m $$

Sensitivity

$$ S = \frac{1}{I_{FSD}} \;\; \Omega/\text{V} $$
$$ R_{total} = S \times V $$
V Rse (multiplier) PMMC Rm FSD βœ“ S = 1000 Ξ©/V β€” Higher S β†’ Better Voltmeter
10 V
m = β€”, Rse = β€”, S = β€” Ξ©/V

πŸ”΅ 2.5 Moving Iron (MI) Instrument

AC & DC

Deflecting Torque

$$ \boxed{T_d = \frac{1}{2}I^2\frac{dL}{d\theta}} $$
$$ T_c = k\theta $$

Deflection (Square Law)

$$ \boxed{\theta \propto I^2} $$
$$ \theta = \frac{1}{2k}\frac{dL}{d\theta} \cdot I^2 $$

Frequency Error

$$ I_m = \frac{V}{\sqrt{(R_m+R_s)^2+(\omega L_m)^2}} $$
$$ \boxed{C = 0.4\frac{L_m}{R_s^2}} $$
Attraction Type
50%
50 Hz

⚠ Errors in MI Instruments

1. Hysteresis Error
2. Frequency Error (f↑ β†’ θ↓)
3. Eddy Current Error
4. Stray Magnetic Field
5. Temperature Error
Im = β€”, C = β€” F

⚑ 2.6 Electrodynamometer Type Instrument

Wattmeter / Transfer Instrument

Deflecting Torque

$$ \boxed{T_d = I_1 I_2 \frac{dM}{d\theta}} $$
$$ T_{d(avg)} = I_1 I_2\cos\phi\frac{dM}{d\theta} $$

As Wattmeter

$$ \boxed{\theta \propto VI\cos\phi = P} $$
$$ \cos\phi = \frac{P}{VI} = \frac{W}{VI} $$

Two-Wattmeter (3Ο†)

$$ P_{total} = W_1 + W_2 $$
$$ \tan\phi = \sqrt{3}\frac{W_1 - W_2}{W_1 + W_2} $$
Fixed Coil (I₁) Fixed Coil (I₁) Moving (Iβ‚‚) dM/dΞΈ
120V
5A
30Β°

βœ“ Advantages

Ammeter, Voltmeter, Wattmeter
AC & DC accurate
Transfer instrument (lab standard)

⚠ Errors

Frequency error
Eddy current error
Stray field error
Temperature error
P = β€” W

πŸ“Š 2.7 Instrument Comparison: PMMC vs MI vs Electrodynamometer

Parameter PMMC Moving Iron Electrodynamometer
Working Principle Permanent magnet + current coil Magnetization of soft iron by coil field Interaction of two current-carrying coils
Deflecting Torque $T_d = BINA$ $T_d = \frac{1}{2}I^2\frac{dL}{d\theta}$ $T_d = I_1I_2\frac{dM}{d\theta}$
Deflection $\theta \propto I$ $\theta \propto I^2$ $\theta \propto P$ (wattmeter)
Scale Linear Non-linear (IΒ²) Non-linear
Used For DC only AC & DC AC & DC
Accuracy High Moderate Very High
Frequency Effect None High Moderate
Applications Ammeter, Voltmeter, Galvanometer Ammeter, Voltmeter (AC panels) Wattmeter, Ammeter, Voltmeter (lab)
$$ \text{PMMC: } \theta = K_1 I $$
$$ \text{MI: } \theta = K_2 I^2 $$
$$ \text{Edyn: } \theta = K_3 VI\cos\phi $$
50%

⚑ 2.8 Electrodynamometer β€” Complete Analysis

DC Β· AC Β· Wattmeter

General Deflection

$$ \boxed{\theta = i_1 i_2 \frac{1}{k}\frac{dM}{d\theta}} $$
$$ T_c = K\theta $$

DC Operation

$$ T_d = I^2\frac{dM}{d\theta} $$
$$ \boxed{\theta \propto I^2} $$

AC Operation

$$ T_i = I_1 I_2\cos\phi\frac{dM}{d\theta} $$
$$ \boxed{\theta = \frac{I_1 I_2}{k}\cos\phi\frac{dM}{d\theta}} $$

EDM Applications Summary

Application Deflection Scale Key Condition
Ammeter $\theta = \frac{I^2}{K}\frac{dM}{d\theta}$ Non-linear (IΒ²) $\frac{L_{shunt}}{R_{shunt}} = \frac{L_m}{R_m}$
Voltmeter $\theta = \frac{V^2}{kR_s^2}\frac{dM}{d\theta}$ Non-linear (VΒ²) High Rs multiplier
Wattmeter $\theta = \frac{P}{kR_p}\frac{dM}{d\theta}$ Linear (P) $P = VI\cos\phi$
V Fixed Coil (I₁) Current Coil Load R + jX Moving Coil (Iβ‚‚) Pressure Coil + Rp Ο† P = 0 W Mode: Wattmeter β€” ΞΈ ∝ P (Linear Scale)
230V
5A
30Β°
P = β€” W, ΞΈ ∝ P

⏱️ Time Constant Matcher (EDM Ammeter)

Ο„shunt = β€”, Ο„coil = β€”

🌑️ 2.9 Electrothermic Instruments

Hot Wire Β· Thermocouple Β· True RMS

Hot Wire

$$ H = I^2 R t $$
$$ \Delta\ell = \ell_0 \alpha \Delta T $$
$$ \theta \propto I^2 $$

Thermocouple

$$ E_{thermo} = \alpha_{AB}(T_H - T_C) $$
$$ T_H - T_C \propto I^2 R $$
$$ \boxed{\theta \propto I^2_{rms}} $$

True RMS

$$ I_{rms} = \sqrt{\frac{1}{T}\int_0^T i^2\,dt} $$
$$ I_{rms} = \frac{I_m}{\sqrt{2}} = 0.707\,I_m $$
Hot Wire Instrument I→ →I Heater Wire Metal A Metal B E = 0 mV PMMC T_H Thermocouple Instrument
50%

Hot Wire βœ“

RF frequency capable
AC/DC same heating
True RMS reading

Thermocouple βœ“

True RMS (AC + DC)
RF frequencies
Most accurate RMS meter

TC Types

Contact type: wire touches junction
Vacuum type: evacuated glass bulb
Irms = β€”

⚑ 2.10 Electrostatic Instrument

Voltage Only Β· Zero Current

Deflecting Torque

$$ \boxed{T_d = \frac{1}{2}V^2\frac{dC}{d\theta}} $$
$$ T_c = K\theta $$

Deflection

$$ \boxed{\theta = \frac{1}{2k}V^2\frac{dC}{d\theta}} $$
$$ \theta \propto V^2 \quad \text{(non-linear)} $$

Characteristics

I = 0 β†’ Zinput = ∞ (ideal)
No frequency error
No power consumed
Works on AC and DC
Fixed Vanes +V βˆ’V I = 0 A (No current drawn) Electrostatic Voltmeter
50 V

Applications

High Voltage Measurement
Dielectric Constant Measurement
Laboratory Standard Voltmeter
ΞΈ = β€” rad

πŸ“ 2.11 Range Extension β€” Electrostatic Voltmeters

Resistance & Capacitance Dividers

Resistance Divider

$$ V_m = \frac{r}{R}\cdot V $$
$$ \boxed{m = \frac{R}{r}} $$
Power loss: $P = V^2/R$

Capacitance Divider

$$ \boxed{V_m = \frac{C_s}{C_s + C_v}\cdot V} $$
$$ \boxed{m = 1 + \frac{C_v}{C_s}} $$
Zero power loss (ideal)

Comparison

R-div: freq. dependent
C-div: freq. compensated
For higher m: Cs β‰ͺ Cv
V (supply) Rβˆ’r r V Vm = β€” m = R/r = β€” Ploss = β€” V (supply) Cs Cv V Zero Power Loss βœ“ Vm = β€” m = 1 + Cv/Cs = β€”
500V
50 Hz
Parameter Resistance Divider Capacitance Divider
Power loss$P = V^2/R$ (high)Zero (ideal)
FrequencyAffected by stray CFrequency compensated
Used forDC, low freq ACAC, RF
Multiplying m$R/r$$1 + C_v/C_s$
AccuracyModerateHigh (AC)

πŸ“ Resistance Divider Calculator

Vm = β€”, m = β€”

πŸ“ Capacitance Divider Calculator

Vm = β€”, m = β€”
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