Ôn tập mạch điện và máy điện

Understanding this rule is essential for interpreting power flow diagrams and for correctly sizing protective devices.

2. Quantities that Characterise Power Transfer

The three fundamental quantities that directly describe the rate of energy exchange in any electrical network are:

• Current (I) – the flow of electric charge, measured in amperes (A).
• Voltage (U) – the electric potential difference, measured in volts (V).
• Power (P) – the product of voltage and current, measured in watts (W), indicating how fast energy is transferred.

Other parameters such as resistance, capacitance, or inductance describe the behaviour of components but do not themselves quantify the instantaneous energy exchange.

3. Frequency of a Direct‑Current (DC) Source

Direct‑current sources provide a constant voltage (or current) that does not vary with time. Because there is no periodic variation, the frequency of a DC source is defined as 0 Hz. This contrasts with alternating‑current (AC) sources, where the voltage repeats every cycle and the frequency is typically 50 Hz or 60 Hz depending on the region.

4. Series RLC Circuits – Reactance Dominance

In a series RLC circuit, the total reactance is the difference between inductive reactance (X_L) and capacitive reactance (X_C). The sign of the net reactance determines the circuit’s overall character:

• If X_L > X_C, the circuit behaves as inductive (more opposition to changes in current).
• If X_C > X_L, the circuit behaves as capacitive.
• If X_L = X_C, the circuit reaches resonance, and the net reactance is zero.

5. Calculating the Impedance of a Series RLC Branch

For a series connection with resistance R and reactances X_L and X_C, the magnitude of the total impedance Z is:

Z = \sqrt{R^{2} + (X_{L} - X_{C})^{2}}

Example: with R = 1 Ω, X_L = 2 Ω, and X_C = 1 Ω, the impedance is:

Z = \sqrt{1^{2} + (2-1)^{2}} = \sqrt{2} \approx 1.41 Ω

This calculation is frequently used in power‑system studies to determine voltage drops and losses.

6. Three‑Phase Star (Wye) Connection – Line and Phase Voltages

In a balanced three‑phase system connected in a star (wye) configuration, the relationship between the line voltage (U_d) and the phase voltage (U_p) is derived from vector geometry:

• The line voltage is the vector difference between two phase voltages that are 120° apart.
• Using the law of cosines, the magnitude becomes U_d = √3 U_p.

This factor of √3 is a cornerstone in the design of transformers, motor drives, and distribution networks.

7. Function of an AC Transformer

An alternating‑current transformer changes the magnitude of the voltage while keeping the frequency unchanged. The principle of electromagnetic induction allows the secondary winding to produce a voltage proportional to the turns ratio, but the frequency of the induced voltage equals the primary frequency. Therefore, a transformer does not alter frequency, making it ideal for stepping voltage up or down without affecting the system’s timing.

8. Synchronous Speed of an Induction Motor

For an induction (asynchronous) motor, the rotating magnetic field produced by the stator rotates at the synchronous speed n_s given by:

n_{s} = \frac{120 f}{P}

where f is the supply frequency and P is the total number of poles (twice the number of pole pairs p). For a motor with p = 2 pole pairs (i.e., P = 4 poles) supplied by a 50 Hz source, the synchronous speed is:

n_{s} = \frac{120 \times 50}{4} = 1500 \text{ rpm}

The actual rotor speed will be slightly lower due to slip, but the synchronous speed provides the reference for motor control and performance analysis.

9. Summary of Key Concepts

• Negative instantaneous power indicates a branch is delivering energy.
• Current, voltage, and power are the primary descriptors of energy exchange.
• DC sources have a frequency of 0 Hz.
• In a series RLC circuit, the dominant reactance defines whether the circuit is inductive or capacitive.
• The impedance of a series RLC branch is calculated with Z = √[R² + (X_L‑X_C)²].
• For a star‑connected three‑phase system, line voltage equals √3 times the phase voltage.
• Transformers modify voltage magnitude but keep frequency constant.
• Synchronous speed of rotating machines follows n_s = 120f/P.

Mastering these fundamentals equips you to analyse more complex power‑system problems, design efficient electrical machines, and troubleshoot real‑world installations.