MODULE 4 β High Resistance & AC Bridge Measurements
π 4.1 Loss of Charge Method (High Resistance)
RC Discharge Β· R > 100kΞ© Β· Β±2-5%Capacitor Discharge
$$ \boxed{V_C(t) = V e^{-t/\tau}} $$
$$ \tau = RC, \quad V_C(\tau) = 0.368\,V $$
Resistance Formula
$$ R = \frac{t}{C\ln\!\left(\frac{V}{V_C}\right)} $$
$$ \boxed{R = \frac{0.4343\,t}{C\log_{10}\!\left(\frac{V}{V_C}\right)}} $$
Conversion
$$ \ln x = 2.3026\log_{10}x $$
Suitable for R > 100 kΞ©
Accuracy: Β±2% to Β±5%
π Loss of Charge Calculator
R = β
π 4.2 Megger Method & Direct Deflection
Insulation Test Β· ΞΈ β 1/Rx Β· Guard RingMegger Deflection
$$ \boxed{\theta \propto \frac{1}{R_x}} $$
$$ \frac{T_{deflect}}{T_{control}} = \frac{V/R_x}{V} = \frac{1}{R_x} $$
Insulation Requirements
$$ R_{insul} \geq 1\,\text{MΞ©} $$
$$ R_{insul} \geq \frac{V_{rated}}{1000}\;\text{MΞ©} $$
Carry Foster Bridge
$$ \boxed{R - S = (\ell_2 - \ell_1)\cdot r} $$
r = resistance per unit length of slide wire
Test Voltages & Guard Ring
500V β Low voltage cables
1000V β Medium voltage systems
2500V β HV cables/transformers
$$ I_{true} = I_{total} - I_{surface} $$
π Megger Insulation Check
β
π 4.3 AC Bridge: Fundamentals
ZβZβ = ZβZβ Β· Magnitude + Angle Β· BalanceGeneral Balance
$$ \boxed{\bar{Z}_1\bar{Z}_4 = \bar{Z}_2\bar{Z}_3} $$
Magnitude Condition
$$ \boxed{|Z_1||Z_4| = |Z_2||Z_3|} $$
Angle Condition
$$ \boxed{\theta_1 + \theta_4 = \theta_2 + \theta_3} $$
Phase Angle β Element Type
| Phase ΞΈ | Element |
|---|---|
| 0Β° | R (pure resistance) |
| +90Β° | L (pure inductance) |
| β90Β° | C (pure capacitance) |
| 0Β° to +90Β° | R + L (series/parallel) |
| β90Β° to 0Β° | R + C (series/parallel) |
π AC Bridge Balance Checker
β
π΅ 4.4 Measurement of Self-Inductance
Maxwell Β· Hay's Β· Anderson Β· OwenMaxwell's L/C (1<Q<10)
$$ \boxed{L_x = R_2 R_3 C_1} $$
$$ R_x = \frac{R_2 R_3}{R_1} $$
$$ Q = \omega C_1 R_1 $$
β Frequency independent
Hay's Bridge (Q>10)
$$ \boxed{L_x = \frac{R_2 R_3 C_1}{1+(\omega C_1 R_1)^2}} $$
$$ R_x = \frac{\omega^2 C_1^2 R_1 R_2 R_3}{1+(\omega C_1 R_1)^2} $$
$$ Q = \frac{1}{\omega C_1 R_1} $$
Anderson's (Q<1)
$$ L_x = C[R_4(R_1\!+\!R_3)\!+\!R_1 R_3] $$
$$ R_x = \frac{R_2 R_3}{R_1} $$
Fixed C, no variable C needed
Owen's Bridge & Selection Guide
Owen's: $L_x = R_2 R_4 C_1$, $R_x = \frac{C_1}{C_2}R_2$
| Bridge | Q Range | Key Feature |
|---|---|---|
| Maxwell | 1β10 | Freq. independent |
| Hay's | >10 | High Q coils |
| Anderson | <1 | Low Q, fixed C |
| Owen's | Wide | Independent balance |
π΅ Maxwell's Bridge Calculator
Lx = β | Rx = β | Q = β
π΅ Hay's Bridge Calculator
Lx = β | Rx = β | Q = β
π΅ 4.5 Capacitance & Dissipation Factor
Schering Β· De Sauty Β· tan Ξ΄ Β· QualitySchering Bridge
$$ \boxed{C_x = C_s \cdot \frac{R_4}{R_3}} $$
$$ R_x = R_3 \cdot \frac{C_1}{C_s} $$
$$ \boxed{D = \tan\delta = \omega C_1 R_3} $$
De Sauty's Bridge
$$ \boxed{C_x = C_s \cdot \frac{R_2}{R_1}} $$
Only for ideal capacitors (D = 0)
Dissipation Factor
$$ D = \omega C_x R_x = \frac{P_{loss}}{P_{stored}} $$
$$ Q_C = \frac{1}{D} = \frac{1}{\tan\delta} $$
π΅ Schering Bridge Calculator
Cx = β | Rx = β | D = β
π 4.6 Frequency Measurement β Wien's Bridge
f = 1/2ΟRC Β· Audio Range Β· Complete SummaryWien's Bridge
$$ \boxed{f = \frac{1}{2\pi RC}} $$
$$ R_1 = 2R_2 \;\text{(balance)} $$
Range: 100 Hz β 100 kHz (audio)
General Balance
$$ \frac{R_1}{R_2} = \frac{R_3}{R_4} + \frac{C_4}{C_3} $$
For $R_3\!=\!R_4\!=\!R$, $C_3\!=\!C_4\!=\!C$:
$$ R_1/R_2 = 2, \quad f=1/(2\pi RC) $$
AC Bridge Complete Summary
| Bridge | Measures | Balance Formula |
|---|---|---|
| Maxwell | L (med Q) | Lx=RβRβCβ, Rx=RβRβ/Rβ |
| Hay's | L (high Q) | Lx=RβRβCβ/(1+(ΟCβRβ)Β²) |
| Anderson | L (low Q) | Fixed C, no var C |
| Owen's | L (wide) | Lx=RβRβCβ, Rx=CβRβ/Cβ |
| Schering | C, tan Ξ΄ | Cx=CsRβ/Rβ, D=ΟCβRβ |
| De Sauty | C (ideal) | Cx=CsRβ/Rβ |
| Wien's | Frequency | f=1/(2ΟRC) |
π΅ Wien Bridge Frequency
f = β