Circuits, Three-Phase Power and Electrical Measurements — ESE Electrical

Weightage: Circuit theory and measurements open the Electrical papers and feed every later chapter. Questions are short numerical items on theorems, resonance, three-phase power and instrument ranges, and most can be answered in a minute when the standard result is known.

1. Network theorems

Kirchhoff's laws are the basis: the algebraic sum of currents at a node is zero, and the sum of voltages around a loop is zero.

• Thevenin: a linear two-terminal network is equivalent to a voltage source $V_{Th}$ (the open-circuit voltage) in series with $R_{Th}$ (the resistance seen with sources deactivated).
• Norton: the same network is a current source $I_N=V_{Th}/R_{Th}$ in parallel with $R_{Th}$.
• Superposition: in a linear circuit, the response is the sum of responses to each source acting alone. Voltage sources are shorted and current sources opened when inactive. It does not apply to power.
• Maximum power transfer: a load receives maximum power when $R_L=R_{Th}$, and the power is $V_{Th}^2/(4R_{Th})$. For AC the load impedance must be the complex conjugate of the source impedance.

Worked example. A source has $V_{Th}=12$ V and $R_{Th}=4\Omega$. The maximum power is $12^2/(4\times 4)=9$ W at $R_L=4\Omega$, with an efficiency of only 50 percent.

Reciprocity holds for linear passive networks, and Tellegen's theorem says the sum of power absorbed by all branches is zero.

2. Transients

A first-order circuit responds exponentially with time constant $\tau =L/R$ for RL and $\tau =RC$ for RC. After one time constant a charging capacitor reaches 63.2 percent of its final voltage, and after five time constants the response is practically complete.

At $t=0^{+}$: an inductor keeps its current continuous (it acts as an open circuit for a step if the initial current is zero), and a capacitor keeps its voltage continuous (it acts as a short circuit if uncharged).

For a series RLC circuit, the damping condition is set by $R$ compared with $2\sqrt{L/C}$. A larger $R$ gives an overdamped response, equality gives critical damping, and a smaller $R$ gives an underdamped, oscillatory response.

3. Resonance

A series RLC circuit resonates when $X_L=X_C$:

${\omega }_0=\frac{1}{\sqrt{LC}},Q=\frac{{\omega }_{0}L}{R},BW=\frac{{\omega }_0}{Q}=\frac{R}{L}$

At resonance the impedance is a minimum ($R$) and the current is a maximum, in phase with the voltage. The voltages across $L$ and $C$ are each $Q$ times the supply voltage.

Worked example. $R=10\Omega$, $L=0.1$ H, $C=100\mu$F. Then ${\omega }_0=1/\sqrt{0.1\times 10^{-4}}=316$ rad/s, $Q=316\times 0.1/10=3.16$ and $BW=R/L=100$ rad/s.

A parallel resonant circuit has the maximum impedance at resonance and minimum line current.

4. AC power and power factor

For a sinusoidal voltage and current with phase angle $\varphi$:

• Real power $P=VI\cos \varphi$ (watts).
• Reactive power $Q=VI\sin \varphi$ (VAr).
• Apparent power $S=VI$ (VA), with $S^2=P^2+Q^2$.
• Power factor $\cos \varphi =P/S$.

Poor power factor raises current for the same real power, so losses and voltage drop rise. A capacitor bank of reactive size $Q_c=P(\tan {\varphi }_1-\tan {\varphi }_2)$ raises the factor from $\cos {\varphi }_1$ to $\cos {\varphi }_2$. For 100 kW at 0.8 lagging corrected to unity, $Q_c=100\times 0.75=75$ kVAr.

The RMS value of a sine wave is the peak divided by $\sqrt{2}$, and the form factor (RMS over average) is 1.11.

5. Three-phase systems

In either connection, balanced three-phase power is:

$P=\sqrt{3}{V}_L{I}_L\cos \varphi$

A delta load draws three times the power of the same impedances connected in star on the same supply.

The two-wattmeter method measures three-phase power: $P=W_1+W_2$ and $\tan \varphi =\sqrt{3}(W_1-W_2)/(W_1+W_2)$. At unity power factor the readings are equal, at 0.5 one reads zero, and below 0.5 one reads negative.

Worked example. Readings 2000 W and 1000 W give $P=3000$ W and $\tan \varphi =\sqrt{3}\times 1000/3000=0.577$, so $\varphi =30^{\circ }$ and the power factor is 0.866.

6. Measuring instruments

Extending range: a shunt across the meter takes $I_{sh}=I-I_m$, so $R_{sh}=R_m{I}_m/(I-I_m)$. A series multiplier gives $R_s=V/I_m-R_m$.

Worked example. A meter of $100\Omega$ and 1 mA full scale becomes a 100 mA ammeter with $R_{sh}=100\times 1/99=1.01\Omega$.

Instrument transformers let standard meters read high AC quantities: a current transformer keeps its secondary nearly short-circuited and must never be opened under load, and a potential transformer works near open circuit.

7. Bridges

A DC Wheatstone bridge balances when $R_1{R}_4=R_2{R}_3$ and measures medium resistance. AC bridges balance when $Z_1{Z}_4=Z_2{Z}_3$ in both magnitude and phase.

• Maxwell bridge: inductance.
• Hay bridge: inductance of high-$Q$ coils.
• Schering bridge: capacitance and dielectric loss angle.
• Wien bridge: frequency and capacitance.
• Kelvin double bridge: very low resistance.

Errors are classed as gross, systematic and random, and accuracy is stated as a percentage of full-scale reading, so readings near the bottom of the scale carry a larger percentage error. For that reason an instrument should be read in the upper part of its range.

Common traps

• Using superposition for power. It holds for voltage and current only.
• Assuming 100 percent efficiency at maximum power transfer. It is 50 percent.
• Mixing star and delta relations for line and phase quantities.
• Opening the secondary of a current transformer while the primary carries current.
• Reading a moving-iron scale as linear.

Memory aids

• "RL equals Rth": maximum power.
• "Root three in star voltage, in delta current": three-phase relations.
• "Series resonance minimum Z, parallel maximum Z": resonance.

Summary

Network theorems reduce circuits to a source and a resistance, and transients and resonance follow from time constants and the condition $X_L=X_C$. AC power uses the power triangle, and three-phase work uses star-delta relations and the two-wattmeter method.

Measurements revolve around instrument types, range extension and bridge balance conditions.

Exam protocol

• Reduce to Thevenin first when a single element varies.
• Draw the power triangle for any power-factor question.
• Check star or delta before using a three-phase relation.
• Never open a CT secondary in a safety item.