MODULE 6 โ Frequency & Instrument Transformers
๐ต 6.1 Wien's Bridge โ Frequency Measurement
AC Bridge ยท f = 1/(2ฯRC) ยท Audio RangeBalance Condition
$$ \boxed{f = \frac{1}{2\pi\sqrt{R_1 R_3 C_1 C_3}}} $$
For $R_1=R_3=R,\;C_1=C_3=C$:
$$ \boxed{f = \frac{1}{2\pi RC}} $$
Arm Impedances
$$ Z_1 = R_1 + \frac{1}{j\omega C_1} \quad \text{(series)} $$
$$ Z_3 = \frac{R_3}{1 + j\omega C_3 R_3} \quad \text{(parallel)} $$
Ratio Arm Condition
$$ \frac{R_2}{R_4} = \frac{C_3}{C_1} + \frac{R_1}{R_3} $$
For equal R, C: $$ \boxed{\frac{R_2}{R_4} = 2} $$
Applications
Audio frequency measurement (20 Hz โ 20 kHz)
Harmonic distortion analyzer
Accuracy: ยฑ0.1% to ยฑ1%
Notch filter / feedback oscillator
๐ต Wien Bridge Frequency Calculator
f = โ
โก 6.2 Instrument Transformers โ Overview
CT ยท PT ยท Isolation ยท Step-downCurrent Transformer (CT)
$$ I_p \cdot N_p = I_s \cdot N_s $$
$$ \text{Nominal Ratio} = \frac{I_p}{I_s} = \frac{N_s}{N_p} $$
Secondary: 5A or 1A standard
Potential Transformer (PT)
$$ V_p \cdot N_s = V_s \cdot N_p $$
$$ \text{Nominal Ratio} = \frac{V_p}{V_s} = \frac{N_p}{N_s} $$
Secondary: 110V or 120V standard
Why Instrument Transformers?
$$ \text{1. Electrical isolation} $$
$$ \text{2. Extend instrument range} $$
$$ \text{3. Standardised meters} $$
Current Transformer
Potential Transformer
๐ 6.3 Transformer Ratios โ Nominal vs Actual
Ratio Correction Factor ยท RCF ยท BurdenCT Ratios
$$ K_n = \frac{I_p(\text{rated})}{I_s(\text{rated})} $$
$$ R = \frac{I_p(\text{actual})}{I_s(\text{actual})} $$
$$ \boxed{RCF = \frac{R}{K_n}} $$
PT Ratios
$$ K_n = \frac{V_p(\text{rated})}{V_s(\text{rated})} $$
$$ R = \frac{V_p(\text{actual})}{V_s(\text{actual})} $$
$$ \boxed{RCF = \frac{R}{K_n}} $$
True Value
$$ I_p = RCF \times K_n \times I_s $$
$$ V_p = RCF \times K_n \times V_s $$
Burden
CT Burden = Z_load on secondary (in VA or ฮฉ)
PT Burden = total VA of instruments on secondary
Higher burden โ larger errors
๐ RCF Calculator
RCF = โ | Ip = โ
๐ 6.4 Current Transformer โ Complete Analysis
Ratio Error ยท Phase Error ยท Phasor DiagramRatio Error
$$ \boxed{\% \text{Ratio Error} = \frac{K_n I_s - I_p}{I_p} \times 100} $$
Phase Angle Error
$$ \boxed{\theta = \frac{180}{\pi}\cdot\frac{I_m \cos\delta - I_c \sin\delta}{n I_s}} $$
$I_m$ = magnetizing, $I_c$ = core loss component
Actual Ratio
$$ R = n + \frac{I_c \cos\delta + I_m \sin\delta}{I_s} $$
$n = N_s/N_p$, $\delta$ = secondary pf angle
Reducing CT Errors
Low core loss material (CRGO, Mumetal)
Large core cross-section โ low B
Small secondary impedance (burden)
Turns compensation: add 1โ2 secondary turns
๐ CT Error Calculator
% Ratio Error = โ
โก 6.5 Potential Transformer โ Complete Analysis
Voltage Error ยท Phase Error ยท Capacitive PTVoltage Ratio Error
$$ \boxed{\% \text{Ratio Error} = \frac{K_n V_s - V_p}{V_p} \times 100} $$
Phase Angle Error
$$ \boxed{\beta = \frac{180}{\pi}\cdot\frac{I_m R_s + I_c X_s}{K_n V_s}} $$
$R_s, X_s$ = secondary R & X referred to secondary
Actual Transformation Ratio
$$ R = K_n\!\left(1 + \frac{I_p(R_p\cos\phi+X_p\sin\phi)}{V_p}\right) $$
Reducing PT Errors
Low flux density in core
Small winding resistance & leakage
High permeability core (grain-oriented)
Capacitor voltage transformer (CVT) for EHV
โก PT Error Calculator
% Ratio Error = โ
โ ๏ธ 6.6 Errors in Instrument Transformers
Ratio Error ยท Phase Error ยท Burden Effect ยท FrequencyCT Ratio Error
$$ \varepsilon_R = \frac{K_n I_s - I_p}{I_p} \times 100\% $$
Positive โ meter reads high
PT Ratio Error
$$ \varepsilon_V = \frac{K_n V_s - V_p}{V_p} \times 100\% $$
Depends on flux density & core loss
Factors Affecting Errors
$$ \text{Burden (VA)} \uparrow \Rightarrow \text{Error}\uparrow $$
$$ \text{Power Factor}\downarrow \Rightarrow \text{Phase Error}\uparrow $$
$$ \text{Frequency change} \Rightarrow \text{magnetizing error} $$
Accuracy Classes (IEC / IS)
| Class | Ratio Error (%) | Phase Error (min) | Application |
|---|---|---|---|
| 0.1 | ยฑ0.1 | ยฑ5 | Precision lab |
| 0.2 | ยฑ0.2 | ยฑ10 | Revenue metering |
| 0.5 | ยฑ0.5 | ยฑ30 | Industrial metering |
| 1.0 | ยฑ1.0 | ยฑ60 | General purpose |
| 3.0 | ยฑ3.0 | โ | Protection only |
๐ 6.7 Summary โ CT vs PT Comparison
Quick Reference ยท Key Differences ยท Safety| Parameter | Current Transformer (CT) | Potential Transformer (PT) |
|---|---|---|
| Purpose | Step down current | Step down voltage |
| Connection | Series with line | Parallel (across line) |
| Primary | Few turns / bar (high I) | Many turns (high V) |
| Secondary | Many turns โ 5A / 1A | Fewer turns โ 110V / 120V |
| Burden | Low Z (ammeter + wires) | High Z (voltmeter) |
| Open Circuit | DANGEROUS โ high voltage! | Safe (no load) |
| Short Circuit | Safe (normal condition) | DANGEROUS โ high current! |
| Core operates at | Low flux density | Near normal flux |
| Accuracy class | 0.1, 0.2, 0.5, 1.0, 3.0 | 0.1, 0.2, 0.5, 1.0, 3.0 |
โ ๏ธ Safety Rules
NEVER open-circuit a CT secondary under load!
NEVER short-circuit a PT secondary!
Always short CT secondary before removing ammeter
Ground one terminal of secondary winding