⚡ Alternating Currents (AC): A Level Physics Study Notes (9702) ⚡
Hello future physicist! This chapter moves us beyond simple DC circuits (where current flows one way) into the fascinating world of Alternating Current (AC). AC is how virtually all electrical energy is transmitted and delivered to your homes—it’s the power that drives modern society! Don't worry if this seems tricky at first; we will break down the key characteristics, how we measure AC effectively, and how we convert it back to DC for electronics.
21.1 Characteristics of Alternating Currents
What is Alternating Current (AC)?
In a Direct Current (DC) circuit (like a battery), the current flows steadily in one direction. In contrast, an Alternating Current (AC) circuit has current that periodically reverses its direction and continuously changes its magnitude.
Analogy: Think of DC as water flowing constantly through a pipe. AC is like water oscillating back and forth in the pipe—it doesn't go anywhere overall, but it transfers energy very effectively.
Sinusoidal Waveform
AC voltage and current usually follow a sinusoidal wave pattern (a sine wave). This means the voltage (or current) starts at zero, increases to a maximum in one direction, returns to zero, reaches a maximum in the opposite direction, and then returns to zero, completing one cycle.
The mathematical equations describing a sinusoidal alternating quantity (\(x\), which could be voltage \(V\) or current \(I\)) are:
$$ x = x_0 \sin(\omega t) $$
- \(x\): The instantaneous value of the current or voltage at time \(t\).
- \(x_0\): The peak value (or maximum amplitude).
- \(\omega\): The angular frequency, measured in radians per second.
Key AC Terms (Period, Frequency, Peak Value)
1. Peak Value (\(V_0\) or \(I_0\))
The peak value is the maximum magnitude reached by the voltage or current during one cycle. It is the amplitude of the sine wave.
- If the equation is \(V = V_0 \sin(\omega t)\), then \(V_0\) is the peak voltage.
2. Period (T) and Frequency (f)
The Period (T) is the time taken for one complete oscillation or cycle, measured in seconds (s).
The Frequency (f) is the number of complete cycles per second, measured in Hertz (Hz).
These two are inversely related: $$ f = \frac{1}{T} $$
3. Angular Frequency (\(\omega\))
Angular frequency relates the time period to the full circle \(2\pi\) radians.
$$ \omega = \frac{2\pi}{T} = 2\pi f $$
Quick Review: The Sine Wave Cycle
If a cycle takes \(T\) seconds:
At \(t=0\): \(x=0\)
At \(t=T/4\): \(x\) reaches maximum peak \(x_0\)
At \(t=T/2\): \(x=0\)
At \(t=3T/4\): \(x\) reaches maximum negative peak \(-x_0\)
At \(t=T\): \(x=0\) (one full cycle complete)
Power in AC Circuits and RMS Values
When dealing with DC, calculating power is easy: \(P = VI\). But since AC voltage and current are constantly changing, the instantaneous power is also changing. If we just averaged the instantaneous voltage or current, the average would be zero (due to the positive and negative halves cancelling out).
However, heat generated by a resistor depends on \(P = I^2 R\), and since squaring always results in a positive value, the resistor heats up regardless of the current direction!
Mean Power in a Resistive Load
For a resistive load powered by a sinusoidal AC source, the instantaneous power fluctuates from zero up to a maximum value, \(P_0\).
The syllabus requires you to recall and use the fact that the mean power (\(P_{mean}\)) dissipated in a resistive load is half the maximum instantaneous power (\(P_{max}\)).
$$ P_{mean} = \frac{1}{2} P_{max} $$
This is a key result: AC power efficiency is often based on this factor of half.
Root-Mean-Square (r.m.s.) Value
To compare AC effectively with DC in terms of power delivery (like heating a kettle or lighting a bulb), we use the Root-Mean-Square (r.m.s.) value.
The r.m.s. value of an alternating current or voltage is the equivalent steady DC value that would dissipate the same amount of power (produce the same heating effect) in a given resistor.
For a sinusoidal alternating current or voltage, the r.m.s. value is calculated using the peak value divided by \(\sqrt{2}\):
r.m.s. Current: $$ I_{r.m.s.} = \frac{I_0}{\sqrt{2}} $$
r.m.s. Voltage: $$ V_{r.m.s.} = \frac{V_0}{\sqrt{2}} $$
Note: Since \(1/\sqrt{2} \approx 0.707\), the r.m.s. value is about 70.7% of the peak value.
Did you know? When we say household mains voltage is 230 V (in many countries), we are referring to the r.m.s. voltage. The actual peak voltage is \(V_0 = 230 \times \sqrt{2} \approx 325 \text{ V}\)!
Key Takeaway for 21.1: AC changes direction sinusoidally. Use \(P_{mean} = \frac{1}{2} P_{max}\) for power calculation, and use RMS values (\(x_0/\sqrt{2}\)) to relate AC to the equivalent DC heating effect.
21.2 Rectification and Smoothing
Most electronic devices (like phones, laptops, and TVs) require steady DC power. Since the mains supply is AC, we need a process called rectification to convert AC to pulsating DC, and then smoothing to make that DC steady.
1. Rectification: AC to Pulsating DC
Rectification uses components called diodes. A diode is a semiconductor device that allows current to flow easily in only one direction.
Half-Wave Rectification
Process: Half-wave rectification uses a single diode.
Result: Current (or voltage) flows only during one half-cycle of the AC input. The negative half-cycle is completely blocked.
When the AC voltage is positive, the diode is forward-biased and conducts. When the AC voltage is negative, the diode is reverse-biased and blocks the current.
- Graphically: The output wave consists of positive pulses separated by gaps of zero voltage.
Full-Wave Rectification (Bridge Rectifier)
Process: Full-wave rectification uses four diodes arranged in a specific configuration called a bridge rectifier.
Result: Both the positive and negative halves of the AC input cycle are used. The negative half-cycle is inverted (turned upside down) so that the current always flows in the same direction through the load resistor.
- Graphically: The output wave consists of continuous positive pulses (no gaps). This provides a more efficient and higher mean voltage output compared to half-wave rectification.
2. Smoothing: Pulsating DC to Steady DC
The output from a full-wave rectifier is still not ideal; it's pulsating DC. We need to "smooth" these pulses out to get a steady voltage, similar to battery output.
The Role of the Smoothing Capacitor
A large capacitor is placed in parallel with the load resistor after the rectifier.
Step-by-Step Smoothing Explanation:
- Charging (On the Rising Edge): As the rectified voltage pulse increases, the capacitor rapidly charges up to the peak voltage \(V_0\).
- Discharging (On the Falling Edge): As the voltage from the rectifier starts to fall after the peak, the diode stops conducting. The capacitor now acts as the energy source and slowly discharges through the load resistor \(R\), maintaining the voltage across the load.
- Ripple Effect: Before the capacitor voltage drops significantly, the next voltage pulse arrives, recharging the capacitor back to \(V_0\). The small fluctuation in voltage across the load is called the ripple voltage.
Analogy: The capacitor acts like a small water reservoir. When the water pressure (voltage) from the pump (rectifier) is high, it fills up. When the pump pressure drops, the reservoir releases water slowly, ensuring a continuous flow to the house (load).
Factors Affecting Smoothing (Ripple Voltage)
The degree of smoothing (the size of the ripple voltage) depends on the time constant of the discharge circuit, \(\tau = RC\). For effective smoothing, the time constant must be much longer than the period between the rectified pulses.
Better smoothing (smaller ripple voltage) is achieved by:
- Larger Capacitance (C): A larger capacitor stores more charge, allowing it to discharge more slowly, thus maintaining the voltage for a longer time during the gap between pulses.
- Larger Load Resistance (R): A larger load resistance means less current is drawn from the capacitor during discharge (since \(V=IR\)), slowing down the discharge process.
A full-wave rectified output is easier to smooth than a half-wave output because the pulses come twice as often, meaning the capacitor has less time to discharge before the next pulse recharges it.
Common Mistake to Avoid:
Do not confuse the mean value of a raw AC signal (which is zero) with the RMS value (which is a measure of power effectiveness). The RMS value is essential for practical AC calculations.
Key Takeaway for 21.2: Diodes convert AC to pulsating DC (rectification). Full-wave rectification (bridge) is superior to half-wave. A capacitor placed in parallel with the load smooths this pulsating DC, and larger \(R\) and \(C\) values lead to a smaller, more desirable ripple voltage.