7. ALTERNATING CURRENT

We have so far considered direct current (dc) sources and circuits with dc sources. These currents do not change direction with time. But voltages and currents that vary with time are very common. The electric mains supply in our homes and offices is a voltage that varies like a sine function with time. Such a voltage is called alternating voltage (ac voltage) and the current driven by it in a circuit is called the alternating current (ac current). Today, most of the electrical devices we use require ac voltage. This is mainly because most of the electrical energy sold by power companies is transmitted and distributed as alternating current. The main reason for preferring use of ac voltage over dc voltage is that ac voltages can be easily and efficiently converted from one voltage to the other by means of transformers. Further, electrical energy can also be transmitted economically over long distances. AC circuits exhibit characteristics which are exploited in many devices of daily use. For example, whenever we tune our radio to a favourite station, we are taking advantage of a special property of ac circuits – one of many that you will study in this chapter.

Alternating current (AC) is a type of electrical current that changes direction periodically. Unlike direct current (DC) which flows in one direction, AC constantly varies its voltage and current. This alternating nature allows AC to be transmitted efficiently over long distances using transformers, which can step up or step down voltage levels. This characteristic is particularly useful for power companies because it minimizes energy loss during transmission. Most household electrical devices depend on this alternating voltage supplied from mains electricity, which oscillates typically in a sine wave pattern.

Think of AC as a swing moving back and forth, where the highest points represent the peaks of the voltage. Just like the swing returns back to the start position each time, AC current alternates direction, making it suitable for delivering power across vast distances, similar to how a swing can cover a wider distance as it moves back and forth.

Figure 7.1 shows a resistor connected to a source ε of ac voltage. The symbol for an ac source in a circuit diagram is . We consider a source which produces sinusoidally varying potential difference across its terminals. Let this potential difference, also called ac voltage, be given by v = v sin(wt) where v is the amplitude of the oscillating potential difference and ω is its angular frequency. To find the value of current through the resistor, we apply Kirchhoff’s loop rule (∑ ε(t) = 0), to the circuit to get v sin(wt) = iR, or i = v sin(wt)/R. Since R is a constant, we can write this equation as i = i sin(wt) where the current amplitude i is given by i = v/R. The voltage and current are in phase with each other, meaning they reach their maximum and minimum values at the same time.

When an AC voltage is applied across a resistor, the voltage and current through the resistor can be described using sinusoidal functions. The voltage oscillates over time as v = v sin(wt), causing the current to also oscillate in sync, defined as i = i sin(wt) with i = v/R. This relationship indicates that as the voltage reaches a peak, the current does so as well. Both voltage and current averages over a cycle ultimately contribute to the power dissipated as heat in the resistor, which is derived from Joule's law.

Imagine a water hose representing the voltage, where opening the nozzle allows water to flow through (current). If you open the nozzle fully (peak voltage), a maximum amount of water flows through (maximum current). As you close the nozzle (drop in voltage), the flow weakens correspondingly. Similar to how the water flows in sync with the nozzle's position, AC voltage and current are in sync with their oscillations.

The instantaneous power dissipated in the resistor is p = i²R. Following the sinusoidal nature, we represent the average power over a cycle as P = <i²>R, where <i²> is the average of i² over one cycle. Since the oscillating current alternates between positive and negative, the average value becomes positive (meaning energy is indeed consumed). The average power can be expressed as P = (1/2)i²R, utilizing the effective or rms current I for simplicity.

In an AC circuit with a resistor, power is dissipated in the form of heat, and the instantaneous power can be calculated using the expression p = i²R. However, because the current continues to oscillate, it is important to consider the average power over a complete cycle, which ultimately can reduce the mathematical complexity. Using the root mean square (rms) values, we can express the average power as P = I²R, where I represents the effective current that would produce the same average power as the AC current.

You can think of measuring how much water flows through a pipe in terms of total volume over a period of time, even if the flow changes frequently (like current changing with time). By measuring the average flow, you get a clearer understanding of how much will actually reach the tap, similar to how the average power in an AC circuit represents the usable power based on the fluctuating current.

To express the power in a similar framework as in DC circuits, we need to redefine the current and voltage in an AC circuit into effective values called root mean square (rms) values. The rms current I is derived from the maximum current as I = im/√2 (approximately 0.707 im), and similarly, the rms voltage V can be expressed as V = vm/√2. These rms values allow the equations for power in AC circuits to resemble those in DC circuits, which is greatly beneficial for analyzing energy consumption.

RMS values are crucial in AC circuits since they allow comparison and calculation using the same formulas as those used in DC circuits. The rms current provides an effective measure of AC flowing through a resistor since it relates back to heating similarly to a DC current. This method simplifies the complexity of working with sinusoidally varying quantities while still delivering results that reflect real-world characteristics of power consumption.

Think of eating a full cake (peak voltage) vs. eating an average slice every day for a week (rms value). You’d be able to enjoy cake regularly (reflecting the energy usage across time), even though it might seem like an overwhelming quantity if thought of all at once. RMS values are that slice size which helps us understand energy consumption in terms that are practical and manageable.

The average power dissipated in AC circuits varies based on the relationship of the current and voltage introduced by the phase angle. The power factor is defined as cosφ, where φ represents the phase angle between the current and voltage. It plays a critical role in determining how effectively the electrical energy is converted into useful work. In scenarios involving inductive or capacitive circuits, cosφ can be less than 1, leading to a lower effective power usage.

In alternating current circuits, the phase angle between current and voltage impacts how much power is actually used for work versus what is transmitted through the system. The power factor, which is the cosine of that phase angle, quantifies this relationship. Ideally, resistance would dominate the circuit such that the power factor equals 1, meaning all power is used effectively. However, inductive and capacitive elements can introduce phase shifts leading to reduced performance since not all of the power results in useful work.

Consider a performer dancing to a song. If they are perfectly in sync with the rhythm (power factor of 1), all their energy is utilized effectively for the performance. If they are somewhat offbeat (lower power factor), some of that energy goes wasted; they might look great but not achieve the intended dance goals. In the same way, a lower power factor reflects inefficiencies in how energy is used in AC circuits.

Transformers are used to adjust the level of alternating voltage for efficient power transmission. They consist of two coils of wire wound around a magnetic core. When an AC voltage is applied to the primary coil, it creates an alternating magnetic field that induces voltage in the secondary coil. The ratio of turns between the primary (Np) and secondary (Ns) coils determines if the transformer steps up or down the voltage; for instance, if Np < Ns, the output voltage (Vs) is greater than the input (Vp). Transformers are vital for long-distance transmission, allowing high voltage to minimize energy loss.

Transformers utilize electromagnetic induction to manage voltage levels in AC circuits. Their operation relies on the principle that an alternating current in the primary winding generates a fluctuating magnetic field, which in turn induces current in the secondary winding. This mechanism is crucial for the efficiency of power distribution systems, as high voltages reduce losses while transmitting electricity over long distances, thus ensuring reliable power delivery to consumers.

Think of a transformer as a bridge. Just like a bridge can allow cars to travel shorter distances without stopping significantly (keeping high speed), transformers enable electricity to travel around a city or countryside without losing much energy along the way. If the bridges (transformers) are well-designed, they can effectively manage how electricity flows without causing traffic jams (losses).

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