交流電與應用：練習題

An ideal transformer has a primary coil with \(600\) turns and a secondary coil with \(150\) turns. If an alternating voltage of \(220\text{ V}\) (r.m.s.) is applied to the primary coil, what is the r.m.s. voltage across the secondary coil?

A power station transmits \(1.2 \times 10^{5}\text{ W}\) of electrical power through cables with a total resistance of \(10\text{ }\Omega\). What is the reduction in power loss in the cables if the transmission voltage is increased from \(6000\text{ V}\) to \(12000\text{ V}\)?

A composite current is formed by the superposition of a steady direct current (DC) of \(4\text{ A}\) and a sinusoidal alternating current (AC) with a peak value of \(6\text{ A}\). What is the root-mean-square (r.m.s.) value of this combined current?

A purely resistive heater is connected to an AC supply with an r.m.s. voltage of $$220 \text{ V}$$. If the resistance of the heater is $$50 \text{ }\Omega$$, what is the peak power dissipated by the heater?

A power station generates $$100 \text{ MW}$$ of electrical power. The power is transmitted through transmission lines with a total resistance of $$4 \text{ }\Omega$$. Initially, the power is transmitted at an r.m.s. voltage of $$25 \text{ kV}$$. If a step-up transformer is used to increase the transmission voltage to $$200 \text{ kV}$$ (r.m.s.), by what factor does the power loss in the transmission lines decrease? Assume the transformers are ideal and the load is purely resistive.

An alternating current (AC) supply has a peak voltage of \( 311 \text{ V} \). Calculate its root-mean-square (r.m.s.) voltage and briefly explain the physical significance of the r.m.s. value.

When considering an alternating current (AC) supply, explain the primary reason why the root-mean-square (r.m.s.) voltage is more practically significant than the peak voltage, especially when calculating the average power dissipated in a resistive component.

A step-down transformer is designed with a solid soft iron core. If this core is replaced with a laminated soft iron core, explain how this change affects the efficiency of the transformer and the underlying physical reason.

A sinusoidal alternating current (a.c.) supply has a peak voltage of \(311\text{ V}\) and a frequency of \(50\text{ Hz}\).

(a) Calculate the r.m.s. voltage of the supply.
(b) The supply is connected across a resistor of resistance \(100\text{ }\Omega\). Calculate the average power dissipated in the resistor.
(c) Explain why the r.m.s. value of an a.c. is used when calculating power rather than the peak value.
(d) If the frequency of the supply is increased to \(100\text{ Hz}\) while the peak voltage remains unchanged, state and explain any change in the average power dissipated.

A power station produces \(1.2\text{ MW}\) of electrical power at a voltage of \(10\text{ kV}\) (r.m.s.). This power is transmitted to a distant town through transmission cables with a total resistance of \(5.0\text{ }\Omega\). A step-up transformer with an efficiency of \(95\%\) is used at the power station to increase the voltage to \(240\text{ kV}\) (r.m.s.).

(a) Calculate the current in the transmission cables.
(b) Determine the power loss in the transmission cables due to heating.
(c) Calculate the total power reaching the town.
(d) Suggest TWO reasons why a real transformer is not \(100\%\) efficient and state one method used to minimize one of these losses.