MODULE 5 β High Resistance Measurement & AC Bridge Theory
β‘ 5.1 Measurement of High Resistance: Overview
R > 100 kΞ© Β· Insulation Β· LeakageMethods for High R (>100 kΞ©)
$$ \text{1. Loss of charge method} $$
$$ \text{2. Megger (Insulation tester)} $$
$$ \text{3. Direct deflection method} $$
$$ \text{4. Mega ohm bridge} $$
Applications
$$ \text{Insulation resistance of cables} $$
$$ \text{Bushing insulation} $$
$$ \text{Leakage resistance of capacitors} $$
$$ \text{Surface resistance of insulators} $$
π 5.2 Loss of Charge Method
Exponential Decay Β· Ο = RC Β· Capacitor DischargeDischarge Equation
$$ \boxed{V_c(t) = Ve^{-t/\tau}} \quad \tau = RC $$
$$ \boxed{R = \frac{t}{C\ln\!\left(\dfrac{V}{V_c}\right)}} $$
Using logββ
$$ \boxed{R = \frac{0.4343\,t}{C\,\log_{10}\!\left(\dfrac{V}{V_c}\right)}} $$
Leakage Correction
$$ \frac{1}{R_{meas}} = \frac{1}{R_x} + \frac{1}{R_C} $$
$$ R_x = \frac{R_{meas} \cdot R_C}{R_C - R_{meas}} $$
Error Sources
Leakage of capacitor itself
Meter loading during measurement
Temperature effects on C
π Loss of Charge Calculator
R = β
π§ 5.3 Megger (Insulation Resistance Tester)
Electrodynamometer Β· ΞΈ β 1/Rx Β· Guard TerminalDeflection Principle
$$ \theta \propto \frac{1}{R_x} $$
$$ \theta = f\!\left(\frac{V}{R_x}\right) $$
No spring β no restoring torque
At Extremes
$$ R_x \to \infty: \; I_{CC} \to 0,\; \theta \to 0\;\;(\infty\text{ mark}) $$
$$ R_x = 0: \; I_{CC} \to \text{max},\; \theta \to \theta_{max}\;\;(0\text{ mark}) $$
Test Voltages
500V β General wiring
1000V β Motors, transformers
2500V, 5000V β Cables, switchgear
Insulation Standards
R β₯ 1 MΞ©/kV (general rule)
>100 MΞ© = Excellent
1β10 MΞ© = Warning
<1 MΞ© = FAIL
π§ Megger Insulation Grader
Grade: β
π 5.4 Direct Deflection & Carry Foster Bridge
Cable Insulation Β· Guard Ring Β· Slide WireDirect Deflection
$$ R_x = \frac{V}{I_{galv}} $$
$$ R_{insulation} = \frac{V}{I_{volume}} $$
Guard ring removes $I_{surface}$
Current Components
$$ I_{total} = I_{volume} + I_{surface} $$
$$ I_{measured} = I_{volume} \text{ only} $$
Carry Foster Bridge
$$ \boxed{R = S + s(\ell_2 - \ell_1)} $$
$s$ = resistance per unit length
π Direct Deflection & Carry Foster
Rx = V/Ig = β
R = S + s(βββββ) = β
π 5.5 AC Bridges: General Theory
ZβZβ = ZβZβ Β· Magnitude Β· Angle Β· DetectorsBalance Condition
$$ \boxed{\bar{Z}_1 \bar{Z}_3 = \bar{Z}_2 \bar{Z}_4} $$
$$ \text{Magnitude: } |Z_1||Z_3| = |Z_2||Z_4| $$
$$ \text{Angle: } \theta_1 + \theta_3 = \theta_2 + \theta_4 $$
Phase Angles
$R$ only β $\theta = 0Β°$
$L$ only β $\theta = +90Β°$
$C$ only β $\theta = -90Β°$
$R,L$ series β $0Β° < \theta < 90Β°$
$R,C$ series β $-90Β° < \theta < 0Β°$
Detectors
Vibrant Galvanometer: 5 Hz β 1 kHz
Head Phone: 250 Hz β 4 kHz
Tuned Amplifier: 10 Hz β 100 kHz
Most sensitive at null: HEAD PHONE
π AC Bridge Angle Condition Checker
ΞΈβ+ΞΈβ vs ΞΈβ+ΞΈβ β β
π 5.6 Bridge Balance: Worked Examples
Angle Condition Β· Maxwell Β· Hay Β· ScheringMaxwell L-C
$$ \theta_1 = +\phi_x \;(RL) $$
$$ \theta_4 = -\phi_4 \;(RC\!\parallel) $$
$$ \phi_x = \phi_4 \;\checkmark $$
Hay's Bridge
$$ \theta_1 = +\phi_x \;(RL) $$
$$ \theta_4 = -\phi_4 \;(RC\text{ series}) $$
$$ \phi_x = \phi_4 \;\checkmark $$
Schering
$$ \theta_1 = -\phi_1 \;(RC\text{ series}) $$
$$ \theta_2 = -90Β° \;(C) $$
$$ \phi_4 = 90Β° - \phi_1 \;\checkmark $$
Impedance Reference
$Z=R:\;\theta=0Β°$
$Z=j\omega L:\;\theta=+90Β°$
$Z=\frac{1}{j\omega C}:\;\theta=-90Β°$
$Z=R+j\omega L:\;\theta=\arctan\frac{\omega L}{R}$
$Z=R-\frac{j}{\omega C}:\;\theta=-\arctan\frac{1}{\omega CR}$
π 5.7 Self Inductance Measurement: Bridge Selector
Q Factor Β· Maxwell Β· Hay Β· Anderson Β· OwenQ Factor
$$ Q = \frac{\omega L}{R} $$
Bridge Selection
$Q < 1$: Anderson's Bridge
$1 \le Q \le 10$: Maxwell L-C
$Q > 10$: Hay's Bridge
Any Q: Owen's Bridge
| Bridge | $L_x$ Formula | $Q$ Formula | Q Range |
|---|---|---|---|
| Maxwell L-C | $R_2 R_3 C_4$ | $\omega C_4 R_4$ | 1 β 10 |
| Hay's | $\frac{R_2 R_3 C_4}{1+1/Q^2}$ | $\frac{1}{\omega R_4 C_4}$ | > 10 |
| Anderson's | $C[r(R_3\!+\!R_0)+R_2 R_3]$ | β | < 1 |
| Owen's | $R_2 R_4 C_3$ | $\omega R_2 C_2$ | All |
π― Coil Q Estimator & Bridge Selector
Q = ΟL/R = β β Bridge: β