Measurements & Instrumentation Formula Book

📊 3.1 Half Wave Rectifier Meter

HW · FF = 2.22 · 45% Sensitivity

Average Output

$$ V_{avg} = \frac{V_m}{\pi} = 0.45\,V_m $$

$$ I_{avg} = 0.45\,I_m $$

Form Factor

$$ FF_{HW} = \frac{V_{rms}}{V_{avg}} = \frac{V_m/\sqrt{2}}{V_m/\pi} $$

$$ \boxed{FF_{HW} = \frac{\pi}{\sqrt{2}} \approx 2.22} $$

Reading & Sensitivity

$$ \boxed{\text{HW reading} = 2.22 \times \text{PMMC}} $$

$$ S_{ac} = 0.45\,S_{dc} $$

Range Extension (Half Wave)

DC: $$ \boxed{R_s = S_{ac}V - R_m - R_d} $$

AC: $$ R_s = 0.45\,S_{dc}V - R_m - R_d $$

PMMC (V): S_dc (Ω/V): V (range):

R_m (Ω): R_d (Ω):

HW reading = —, R_s = —

📊 3.2 Full Wave Rectifier Meter

FW · FF = 1.11 · 90% Sensitivity

Average Output

$$ V_{avg} = \frac{2V_m}{\pi} = 0.9\,V_m $$

$$ I_{avg} = 0.9\,I_{rms} $$

Form Factor

$$ FF_{FW} = \frac{V_m/\sqrt{2}}{2V_m/\pi} = \frac{\pi}{2\sqrt{2}} $$

$$ \boxed{FF_{FW} \approx 1.11} $$

Reading & Sensitivity

$$ \boxed{\text{FW reading} = 1.11 \times \text{PMMC}} $$

$$ S_{ac} = 0.9\,S_{dc} $$

Range Extension (Full Wave)

DC: $$ \boxed{R_s = S_{ac}V - R_m - 2R_d} $$

AC: $$ R_s = 0.9\,S_{dc}V - R_m - 2R_d $$

Half Wave vs Full Wave Comparison

Parameter	Half Wave	Full Wave
$V_{avg}$	$V_m/\pi$	$2V_m/\pi$
Form Factor	2.22	1.11
Sensitivity	$0.45\,S_{dc}$	$0.90\,S_{dc}$
Diode drop	$-R_d$	$-2R_d$
Ripple	High	Low
Accuracy	Lower	Higher
$k_r$ ratio	$k_r(HW)/k_r(FW) = 2.22/1.11 = 2$

PMMC (V): S_dc (Ω/V): V (range):

R_m (Ω): R_d (Ω):

FW reading = —, R_s = —

🔍 3.3 Measurement of Resistance — Classification

Low · Medium · High

Classification	Value of R	Primary Methods
Low Resistance	$R < 1\,\Omega$	Kelvin Bridge, Potentiometer, A-V Method
Medium Resistance	$1\,\Omega < R < 100\,k\Omega$	Wheatstone Bridge, A-V Method, Ohmmeter
High Resistance	$R > 100\,k\Omega$	Megger, Loss of Charge, A-V Method

Ammeter-Voltmeter Principle

$$ \boxed{R_m = \frac{V}{I}} $$

🔌 3.4 Ammeter-Voltmeter Method

Case I · Case II · R_critical

Case I — Ammeter Near R

$$ R_m = R_A + R_x $$

$$ \boxed{\% \text{error} = \frac{R_A}{R_x}\times 100} $$

Use when $R_x \gg R_A$ (High R)

Case II — Voltmeter Near R

$$ R_m = \frac{R_V R_x}{R_V + R_x} $$

$$ \boxed{\% \text{error} = -\frac{R_x}{R_V}\times 100} $$

Use when $R_x \ll R_V$ (Low R)

Optimal Selection

$$ \boxed{R_{critical} = \sqrt{R_A \times R_V}} $$

$R_x > R_{crit}$: Case I

$R_x < R_{crit}$: Case II

Parameter	Case I (A near)	Case II (V near)
$R_m$ formula	$R_A + R_x$	$R_V \\| R_x$
Error formula	$+R_A/R_x$	$-R_x/R_V$
Error type	Always positive (+)	Always negative (−)
Use when	$R_x \gg R_A$ (large R)	$R_x \ll R_V$ (small R)
Best for	High resistance	Low resistance

R_x (Ω): 1 kΩ

R_A (Ω): 10Ω

R_V (kΩ): 1 MΩ

R_x (Ω): R_A (Ω): R_V (Ω):

Case I err = —%, Case II err = —%, R_crit = —

🌉 3.5 Wheatstone Bridge

Balance · PS = QR · Sensitivity

Balance Condition

$$ \frac{P}{Q} = \frac{R}{S} \quad \Rightarrow \quad PS = QR $$

$$ \boxed{S = \frac{QR}{P}} $$

Sensitivity

$$ \delta V = \frac{E}{4}\cdot\frac{\delta R}{R} $$

$$ I_g = \frac{E\cdot\delta R/R}{4(R_g + R_{th})} $$

Thévenin Resistance

$$ R_{th} = \frac{PQ}{P+Q} + \frac{RS}{R+S} $$

Range: $1\,\Omega$ – $100\,k\Omega$

Errors & Limitations

Contact resistance

Thermoelectric EMF

Heating by test current

Lead resistance (low R)

Not suitable for very low R → Use Kelvin

P (Ω): 1 kΩ

Q (Ω): 1 kΩ

R (Ω): 500 Ω

S (Ω): 500 Ω

P (Ω): Q (Ω): R (Ω):

S = QR/P = —, Balance: PS = QR?

🌉 3.6 Kelvin Double Bridge

Low R · Lead Elimination · ±0.1%

General Balance

$$ X = R\frac{P}{Q} + \frac{pq}{p+q+r}\left(\frac{P}{Q}-\frac{p}{q}\right)r $$

Simplified ($P/Q = p/q$)

$$ \boxed{X = R\cdot\frac{P}{Q}} $$

Lead resistance r eliminated!

Condition

$$ \boxed{\frac{P}{Q} = \frac{p}{q}} $$

Inner = Outer ratio arms

Range: $R < 1\,\Omega$

R (Ω): P (Ω): Q (Ω):

X = R·P/Q = —, Condition: P/Q = p/q?

📊 3.7 Resistance Measurement — Methods Summary

All Methods · Decision Tree · Accuracy

Low R (< 1 Ω)

1. Kelvin Bridge: $X = R \cdot P/Q$

2. A-V Method (Case II)

3. Potentiometer

4. Ducter

Medium R (1 Ω – 100 kΩ)

1. Wheatstone Bridge: $S = QR/P$

2. A-V Method (Case I or II)

3. Substitution method

4. Ohmmeter

High R (> 100 kΩ)

1. Megger: $R = V_{test}/I_{leak}$

2. Loss of Charge: $V = V_0 e^{-t/RC}$

3. A-V Method (Case I)

4. Wheatstone (high R arms)

Loss of Charge & Megger

$$ V = V_0 e^{-t/RC};\quad R = \frac{t}{C\ln(V_0/V)} $$

$$ R_{insulation} = \frac{V_{test}}{I_{leakage}} $$

Method	Range	Accuracy
A-V Method	Any	±1% to ±5%
Wheatstone Bridge	1 Ω – 100 kΩ	±0.1%
Kelvin Bridge	< 1 Ω	±0.1% to ±0.5%
Megger	> 1 MΩ	±5%
Potentiometer	Low R	±0.05%

Select R value: 1 kΩ

📉 Loss of Charge Calculator

V₀ (V): V (V): t (s): C (µF):

R = —

🌉 3.8 Thevenin Equivalent of Wheatstone Bridge

Vth · Rth · Ig Analysis

Crossover Resistance

$$ |\varepsilon_1| = |\varepsilon_2| \quad \Rightarrow \quad \frac{R_A}{R_T} = \frac{R_T}{R_V} $$

$$ \boxed{R_T = \sqrt{R_A R_V}} $$

Thevenin Voltage

$$ V_{th} = V\!\left[\frac{P}{P\!+\!Q} - \frac{R}{R\!+\!S}\right] $$

$$ \boxed{V_{th} = \frac{V(PS - QR)}{(P\!+\!Q)(R\!+\!S)}} $$

Thevenin Resistance

$$ \boxed{R_{th} = \frac{PQ}{P\!+\!Q} + \frac{RS}{R\!+\!S}} $$

At balance: $V_{th}=0$ since $PS=QR$

Galvanometer Current (Unbalanced)

$$ I_g = \frac{V_{th}}{R_{th} + R_g} = \frac{V(PS - QR)}{(P\!+\!Q)(R\!+\!S)(R_{th}\!+\!R_g)} $$

P: 100Ω

Q: 100Ω

R: 100Ω

S: 100Ω

🌉 Thevenin Bridge Calculator

P (Ω): Q (Ω): R (Ω): S (Ω):

V (V): Rg (Ω):

Vth = — | Rth = — | Ig = —

📐 3.9 Bridge & Galvanometer Sensitivity

SI · SV · SB · Max Sensitivity

Current Sensitivity

$$ \boxed{S_I = \frac{\theta}{I_g} \;\text{mm/μA}} $$

Deflection per unit galvanometer current

Voltage Sensitivity

$$ \boxed{S_V = \frac{\theta}{V_{th}} \;\text{mm/V}} $$

$$ S_V = \frac{S_I}{R_g} $$

Bridge Sensitivity

$$ \boxed{S_B = \frac{\theta}{\Delta R/R} \;\text{mm}} $$

$$ S_B = \frac{S_I \cdot V}{4(R_g + R_{th})} $$

Maximum Bridge Sensitivity

$$ \boxed{S_{B_{max}} = \frac{V \cdot S_I}{4}} \quad \text{at } P=Q=R=S $$

$$ R_{th(min)} = R, \quad R_g = R_{th} = R $$

SI (mm/μA): 5

V (V): 10

Rg (Ω): 100

ΔR/R: 0.01

—

📐 Sensitivity Calculator

SI (mm/μA): V (V): Rg (Ω): Rth (Ω):

SV = — | SB = — | SBmax = —

🔗 3.10 Kelvin Double Bridge: Complete Analysis

P/Q = p/q · Link Elimination · Potentiometer

Kelvin Balance Equation

$$ R_x = R\!\cdot\!\frac{P}{Q} + \frac{r \cdot pq}{p\!+\!q\!+\!r}\!\left(\frac{P}{Q}-\frac{p}{q}\right) $$

$$ \boxed{\frac{P}{Q}=\frac{p}{q} \;\Rightarrow\; R_x = R\!\cdot\!\frac{P}{Q}} $$

Potentiometer Method

$$ \frac{E_x}{E_s} = \frac{\ell_x}{\ell_s} = \frac{R_x}{R_s} $$

$$ \boxed{R_x = R_s \cdot \frac{\ell_x}{\ell_s}} $$

Accuracy: ±0.01% to ±0.05% (best for low R)

r (link Ω): 0.1

P/Q ratio: 1.0

p/q ratio: 1.0

🔗 Kelvin Bridge Calculator

R (Ω): P: Q: p: q: r (Ω):

Rx = —

📏 Potentiometer R Calculator

Rs (Ω): ℓx (cm): ℓs (cm):

Rx = —

🔌 3.11 Measurement of Medium Resistance

Wheatstone Balance · PS = QR · ±0.1%

Wheatstone Balance

$$ \frac{P}{Q} = \frac{R}{S} \quad (I_g = 0) $$

$$ \boxed{S = \frac{QR}{P}} $$

Balance Derivation

$$ V_{BD}=0 \;\Rightarrow\; I_1 P = I_2 R $$

$$ I_1 Q = I_2 S \;\Rightarrow\; PS = QR $$

Methods (1Ω–100kΩ)

1. Ammeter-Voltmeter (V-I method)

2. Substitution method

3. Wheatstone bridge (±0.1%)

4. Ohmmeter method

P: 100Ω

Q: 100Ω

R: 100Ω

🔌 Wheatstone Bridge Calculator

P (Ω): Q (Ω): R (Ω):

S = QR/P = —

🔋 3.12 Ohmmeter: Series & Shunt Types

Series Scale 0→∞ · Shunt ∞→0 · Non-Linear

Series Ohmmeter

$$ \boxed{I = \frac{V}{R_1 + R_x}} $$

Mid-scale: $R_{mid} = R_1 = \frac{V}{I_{FSD}}$

Scale: 0Ω right → ∞ left

Shunt Ohmmeter

$$ \boxed{I_m = \frac{V \cdot R_x}{R_1(R_m\!+\!R_x)\!+\!R_m R_x}} $$

Scale: 0Ω left → ∞ right

Suitable for low R measurement

Scale Behavior

$R_x=0$: $I=I_{FSD}$ (full)

$R_x=R_1$: $I=I_{FSD}/2$ (mid)

$R_x→∞$: $I→0$ (zero)

Series vs Shunt Comparison

Parameter	Series	Shunt
Open (Rx=∞)	I=0, LEFT (∞)	I=I_FSD, RIGHT (∞)
Short (Rx=0)	I=I_FSD, RIGHT (0Ω)	I=0, LEFT (0Ω)
Scale dir	0Ω right → ∞ left	0Ω left → ∞ right
Best for	Medium-High R	Low R
Common	✓ Standard	Less common

Rx: 100Ω

R₁: 100Ω

V: 3V

🔋 Series Ohmmeter Calculator

V (V): R₁ (Ω): Rx (Ω):

I = — | θ = —

🔋 Shunt Ohmmeter Calculator

V (V): R₁ (Ω): Rm (Ω): Rx (Ω):

Im = —

MODULE 3 — Rectifier Instruments & Resistance Measurement

📊 3.1 Half Wave Rectifier Meter

Average Output

Form Factor

Reading & Sensitivity

Range Extension (Half Wave)

📊 3.2 Full Wave Rectifier Meter

Average Output

Form Factor

Reading & Sensitivity

Range Extension (Full Wave)

Half Wave vs Full Wave Comparison

🔍 3.3 Measurement of Resistance — Classification

Ammeter-Voltmeter Principle

🔌 3.4 Ammeter-Voltmeter Method

Case I — Ammeter Near R

Case II — Voltmeter Near R

Optimal Selection

🌉 3.5 Wheatstone Bridge

Balance Condition

Sensitivity

Thévenin Resistance

Errors & Limitations

🌉 3.6 Kelvin Double Bridge

General Balance

Simplified ($P/Q = p/q$)

Condition

📊 3.7 Resistance Measurement — Methods Summary

Low R (< 1 Ω)

Medium R (1 Ω – 100 kΩ)

High R (> 100 kΩ)

Loss of Charge & Megger

📉 Loss of Charge Calculator

🌉 3.8 Thevenin Equivalent of Wheatstone Bridge

Crossover Resistance

Thevenin Voltage

Thevenin Resistance

Galvanometer Current (Unbalanced)

🌉 Thevenin Bridge Calculator

📐 3.9 Bridge & Galvanometer Sensitivity

Current Sensitivity

Voltage Sensitivity

Bridge Sensitivity

Maximum Bridge Sensitivity

📐 Sensitivity Calculator

🔗 3.10 Kelvin Double Bridge: Complete Analysis

Kelvin Balance Equation

Potentiometer Method

🔗 Kelvin Bridge Calculator

📏 Potentiometer R Calculator

🔌 3.11 Measurement of Medium Resistance

Wheatstone Balance

Balance Derivation

Methods (1Ω–100kΩ)

🔌 Wheatstone Bridge Calculator

🔋 3.12 Ohmmeter: Series & Shunt Types

Series Ohmmeter

Shunt Ohmmeter

Scale Behavior

Series vs Shunt Comparison

🔋 Series Ohmmeter Calculator

🔋 Shunt Ohmmeter Calculator