Physics (9702) | Characteristics of alternating currents

Welcome to Alternating Currents (AC)!

Hello future Physicists! In your AS Level studies, you spent a lot of time on Direct Current (DC) circuits, where charge flows steadily in one direction (think of a battery). Now, we are diving into the world of Alternating Current (AC).

AC is crucial because it is how electricity is delivered to your homes and schools. Understanding AC involves dealing with quantities that constantly change direction and magnitude, which requires some new tools and concepts, especially the idea of root-mean-square (r.m.s.) values.

Don't worry if this chapter seems tricky at first. We will break down the constantly changing nature of AC and learn how to deal with it mathematically!

21.1 Characteristics of Alternating Currents

What is Alternating Current?

An Alternating Current (AC) is an electric current or voltage that periodically reverses its direction and continuously changes its magnitude with time. In most practical applications, like the mains supply, the variation is sinusoidal (it follows a sine wave pattern).

Key Definitions for Sinusoidal AC

When observing an AC signal on an oscilloscope, we identify three key characteristics:

• Period (\(T\)): The time taken for one complete cycle of the waveform. Measured in seconds (s).
• Frequency (\(f\)): The number of complete cycles occurring per second. Measured in Hertz (Hz).

  Quick connection: Frequency and period are inverses: \(f = 1/T\). (In many countries, mains electricity has a frequency of 50 Hz or 60 Hz.)

• Peak Value (\(x_0\)): This is the maximum magnitude reached by the current (\(I_0\)) or voltage (\(V_0\)) in either the positive or negative direction.

The Sinusoidal AC Equation

Since the current or voltage is constantly changing, we can describe its instantaneous value (\(x\)) mathematically using a sine function, where \(x\) could be instantaneous current \(I\) or voltage \(V\).

The equation for a sinusoidally alternating current or voltage is:
$$x = x_0 \sin(\omega t)$$

Where:

• \(x\): The instantaneous value (current \(I\) or voltage \(V\)).
• \(x_0\): The peak value (maximum amplitude).
• \(\omega\): The angular frequency (or angular speed), measured in radians per second (rad s\(^{-1}\)).
• \(t\): Time.

Linking Frequency and Angular Frequency:
The angular frequency (\(\omega\)) is related to the frequency (\(f\)) and period (\(T\)) by the relation:
$$ \omega = 2\pi f = 2\pi / T $$

Memory Aid: Angular frequency is just how fast the cycle is repeating, measured in terms of angle (\(2\pi\) radians for one full cycle).

Power in Resistive AC Circuits

In a DC circuit, power \(P = IV\) is constant. In an AC circuit, since both \(I\) and \(V\) are constantly fluctuating, the instantaneous power (\(P\)) also fluctuates.

Instantaneous power is given by \(P = IV\). Since \(I\) and \(V\) are changing sinusoidally, the power oscillates between zero and a maximum value, \(P_0 = I_0 V_0\), but importantly, it is never negative in a purely resistive circuit (because \(P = I^2 R\), and \(I^2\) is always positive).

Mean Power (\(\bar{P}\))

When calculating how much useful work the AC source does, we use the mean power (\(\bar{P}\)) over one full cycle.

• For a sinusoidal AC supply driving a resistive load:
• The mean power is exactly half the maximum (peak) power.
• $$ \bar{P} = \frac{1}{2} P_{max} $$
• Since \(P_{max} = I_0 V_0\), we have: $$ \bar{P} = \frac{1}{2} I_0 V_0 $$

Key Takeaway: The power dissipated by an AC source in a resistor is only half the power calculated using the peak current and voltage, because the current and voltage are only at their maximum for a brief instant during the cycle.

Root-Mean-Square (R.M.S.) Values

Since AC voltage and current are constantly changing, how do we compare the effectiveness of an AC supply to a DC supply? We use R.M.S. values.

Defining R.M.S.

The Root-Mean-Square (r.m.s.) current (\(I_{r.m.s.}\)) is defined as the value of the steady DC current that would dissipate energy at the same rate (produce the same mean power) as the AC current in a resistive load.

Essentially, the r.m.s. value is the equivalent DC value. When you see a mains voltage quoted (e.g., 230 V), this is always the r.m.s. value.

R.M.S. Formulas (For Sinusoidal Waves Only)

For purely sinusoidal AC (which is assumed in this syllabus):

1. R.M.S. Voltage: $$ V_{r.m.s.} = \frac{V_0}{\sqrt{2}} $$
2. R.M.S. Current: $$ I_{r.m.s.} = \frac{I_0}{\sqrt{2}} $$

Since \(1/\sqrt{2} \approx 0.707\), the r.m.s. value is roughly 71% of the peak value.

Example: If the UK mains voltage is 230 V r.m.s., the actual peak voltage hitting your appliance is \(V_0 = 230 \times \sqrt{2} \approx 325\) V. That's a huge difference!

Connecting R.M.S. back to Power

We defined r.m.s. based on equivalent power dissipation. If we use R.M.S. values, we can use the familiar DC power formulas to calculate the mean power:

$$ \bar{P} = V_{r.m.s.} I_{r.m.s.} = I_{r.m.s.}^2 R = \frac{V_{r.m.s.}^2}{R} $$

Did you know? The term "root-mean-square" literally describes the mathematical process used to calculate it: Square the values, find the Mean of the squares, then take the square Root. This process ensures the sign changes don't cancel out the effect of the current.

21.2 Rectification and Smoothing

Most electronic devices (like computers and phone chargers) require Direct Current (DC) to operate, but they plug into the AC wall outlet. The process of converting AC into DC is called rectification.

Rectification is achieved using diodes. A diode is a semiconductor component that allows current to flow easily in one direction (forward bias) but blocks it almost entirely in the reverse direction (reverse bias).

1. Half-Wave Rectification

This is the simplest form of rectification, achieved using only one diode.

The Process:

• When the AC voltage is positive, the diode is forward-biased and conducts. The output voltage equals the input voltage.
• When the AC voltage is negative, the diode is reverse-biased and blocks the current. The output voltage is zero.

The result is a pulsating DC output where only half of the original AC wave is present. The output voltage is not smooth DC; it is pulsating, falling to zero once every cycle.

Visualizing the Graph: If you were to sketch the graph, the negative halves of the sine wave are simply chopped off and replaced by a straight line at \(V=0\).

2. Full-Wave Rectification (The Bridge Rectifier)

Half-wave rectification wastes half the input energy. Full-wave rectification utilizes the entire AC waveform to produce a much more efficient, albeit still pulsating, DC output. This is typically achieved using a circuit called the diode bridge rectifier (four diodes).

The Process (How the Bridge Works):

The four diodes are arranged in a bridge structure such that:

1. During the positive half-cycle of the AC input, current flows through two specific diodes to the load (resistor).
2. During the negative half-cycle of the AC input, the current direction reverses. However, it flows through the other two specific diodes in the bridge.

The result: Current flowing through the load resistor always flows in the same direction, regardless of the input AC polarity. The negative halves of the input sine wave are "flipped up" to become positive.

Distinguishing Graphically:

• Half-wave: Output hits zero for half the time (period of output pulse = \(T\)).
• Full-wave: Output never hits zero, and the frequency of the pulses is doubled (period of output pulse = \(T/2\)).

3. Smoothing the Output (The Capacitor)

The rectified current (whether half-wave or full-wave) is still poor quality DC—it's highly pulsating. It wouldn't power sensitive electronics well. We need to smooth the output.

We achieve smoothing by connecting a large capacitor in parallel with the load resistor.

How Smoothing Works (The Reservoir Analogy):

Think of the capacitor as a small energy reservoir.

1. Charging: As the rectified voltage rises towards its peak, the capacitor charges rapidly, storing electrical energy.
2. Discharging (The key function): As the rectified voltage starts to drop away from its peak (and would normally fall sharply), the diode switches off. The capacitor now acts as the temporary source, discharging its stored energy through the load resistor. This discharge keeps the voltage high until the next pulse arrives.

The result is that the output voltage never drops significantly, reducing the "ripple" (the small fluctuation in the voltage).

Analyzing the Effect of C and Load Resistance (R)

The effectiveness of the smoothing depends on the time the capacitor takes to discharge, which is governed by the Time Constant (\(\tau\)). For an RC circuit, \(\tau = RC\).

• Effect of Capacitance (\(C\)):

  A larger capacitance (C) means a larger time constant (\(\tau\)). The capacitor takes longer to discharge, meaning the voltage drops less before the next pulse arrives. Result: Better smoothing (smaller ripple).

• Effect of Load Resistance (\(R\)):

  A larger load resistance (R) means the load draws less current. The capacitor discharges more slowly through the high resistance. Result: Better smoothing (smaller ripple).

We want the time constant (\(\tau = RC\)) to be much greater than the period of the input AC supply to ensure excellent smoothing.