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Half Wave Rectifier Theory
But the diode is only part of it – a complete halfwave rectifier circuit consists of 3 main parts:
A half wave rectifier circuit diagram looks like this:
We’ll now go through the process of how a halfwave rectifier converts an AC voltage to a DC output. First, a high AC voltage is applied to the to the primary side of the stepdown transformer and we will get a low voltage at the secondary winding which will be applied to the diode.
During the positive half cycle of the AC voltage, the diode will be forward biased and the current flows through the diode. During the negative half cycle of the AC voltage, the diode will be reverse biased and the flow of current will be blocked. The final output voltage waveform on the secondary side (DC) is shown in figure 3 above. This can be confusing on first glance – so let’s dig into the theory of this a bit more. We’ll focus on the secondary side of the circuit. If we replace the secondary transformer coils with a source voltage, we can simplify the circuit diagram of the halfwave rectifier as:
Now we don’t have the transformer part of the circuit distracting us.
For the positive half cycle of the AC source voltage, the equivalent circuit effectively becomes:
This is because the diode is forward biased, and is hence allowing current to pass through. So we have a closed circuit.
But for the negative half cycle of the AC source voltage, the equivalent circuit becomes:
Because the diode is now in reverse bias mode, no current is able to pass through it. As such, we now have an open circuit. Since current can not flow through to the load during this time, the output voltage is equal to zero. This all happens very quickly – since an AC waveform will oscillate between positive and negative many times each second (depending on the frequency). Here’s what the half wave rectifier waveform looks like on the input side (Vin), and what it looks like on the output side (Vout) after rectification (i.e. conversion from AC to DC):
The graph above actually shows a positive half wave rectifier. This is a halfwave rectifier which only allows the positive halfcycles through the diode, and blocks the negative halfcycle. The voltage waveform before and after a positive half wave rectifier is shown in figure below.
Conversely, a negative halfwave rectifier will only allow negative halfcycles through the diode and will block the positive halfcycle. The only difference between a posive and negative half wave rectifier is the direction of the diode. As you can see in figure below, the diode is now in the opposite direction. Hence the diode will now be forward biased only when the AC waveform is in its negative half cycle.
We will now derive the various formulas for a half wave rectifier based on the preceding theory and graphs above.
Ripple Factor of Half Wave Rectifier
The formula for ripple factor is:
Which can also be rearranged to equal:
The ripple factor of half wave rectifier is equal to 1.21 (i.e. γ = 1.21). Note that for us to construct a good rectifier, we want to keep the ripple factor as low as possible. This is why we use capacitors and inductors as filters to reduce the ripples in the circuit.
Efficiency of Half Wave Rectifier
Rectifier efficiency (η) is the ratio between the output DC power and the input AC power. The formula for the efficiency is equal to:
The efficiency of a half wave rectifier is equal to 40.6% (i.e. η_{max} = 40.6%)
RMS value of Half Wave Rectifier
To derive the RMS value of half wave rectifier, we need to calculate the current across the load. If the instantaneous load current is equal to iL = Imsinωt, then the average of load current (IDC) is equal to:
Where I_{m} is equal to the peak instantaneous current across the load (I_{max}). Hence the output DC current (I_{DC}) obtained across the load is:
For a halfwave rectifier, the RMS load current (I_{rms}) is equal to the average current (I_{DC}) multiple by π/2. Hence the RMS value of the load current (I_{rms}) for a half wave rectifier is:
I_{rms} = I_{m}/2
Where I_{m}= I_{ma}_{x} which is equal to the peak instantaneous current across the load.
Peak Inverse Voltage of Half Wave Rectifier
Form Factor of Half Wave Rectifier
Form factor (F.F) is the ratio between RMS value and average value, as shown in the formula below:
F.F = RMS Value/Average Value
The form factor of a half wave rectifier is equal to 1.57 (i.e. F.F= 1.57).
Output DC Voltage
The output voltage (V_{DC}) across the load resistor is denoted by:
Applications of Half Wave Rectifier
Advantages of Half Wave Rectifier
Disadvantages of Half Wave Rectifier
3 Phase Half Wave Rectifier
The threephase voltages are shown below.
The voltage across the resistive load is shown below. The voltage is shown in black.
So we can see from the above figure that the diode D1 conducts when the R phase has a value of thevoltage that is higher than the value of the voltage of the other two phases,and this condition begins when the R phase is at a 30^{o} and repeats after every complete cycle. That is to say, the next time diode DI begins to conduct is at 390^{o}. Diode D2 takes over conduction from D1 which stops conducting at angle 150^{o} because at this instant the value of voltage in B phase becomes higher than the voltages in the other two phases. So each diode conducts for an angle of 150^{o} – 30^{o} = 120^{o}.
Here, the waveform of the resulting DC voltage signal is not purely DC as it is not flat,but rather it contains a ripple. And the frequency of the ripple is 3 × 50 = 150 Hz.
The average of the output voltage across the resistive load is given by
Where,
The RMS value of the output voltage is given by
The ripple voltage is equal to,
And the voltage ripple factor is equal to,
The equation above shows that the voltage ripple is significant. This is undesirable as this leads to unnecessary power loss.
DC output power,
AC input power,
Efficiency,
Even though the efficiency of the 3 phase halfwave rectifier is seemingly high, it is still less than the efficiency provided by a 3 phase full wave diode rectifier. Although three phase half wave rectifiers are cheaper, this cost saving is insignificant compared to the money lost in their higher power losses. As such, threephase halfwave rectifiers are not commonly used in industry.
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