asserted to indicate that the conversion terminated.
By means of this algorithm, the sampled analog input is converted using M steps where M is the resolution of the converter. Other features about ADCs could be treated more in detail, but this brief overview is explaining of the key concepts necessary to understand the topics discussed in this thesis.
0 Ts 2Ts 3Ts 4Ts 5Ts 6Ts 7Ts 8Ts
Time [s]
0 0.5 1 1.5 2 2.5 3 3.5 4
Voltage [V]
VIN VOUT
Figure 2.15: Timing diagram of partial results of a Successive Approximation ADC.
resultant waveform perceived by the load must be as smooth as possible. In the application of this thesis, the PWM will be used to drive the inverter switches to control the motor supply.
The two characteristics that defines a PWM signal are the switching period and duty cycle. The switching period is the time that elapses between two positive edge of the signal, and in general, is maintained constant. The duty cycle is the percentage of time that the signal is on with respect to its period, and in general, is changed according to the control strategy. In Fig. 2.16 is represented a PWM signal showing a complete cycle with its period, and TON and TOF F time intervals.
0 Ts 2Ts 3Ts
Time [s]
0 Vcc
Voltage [V]
Ton
Complete Cycle
Toff
ON OFF ON OFF ON OFF
Figure 2.16: Example of PWM signal with 30% of duty cycle.
PWM can be classified in edge aligned and centered aligned pulse modulation.
In edge aligned PWM, the TON period starts from the left or right edge of the switching period. In leading edge modulation, the TON period is placed at the beginning of the switching period and expands to the right with the increase of the duty cycle, instead trailing edge modulation implements the opposite behaviour.
In centered modulation, the TON period is placed at the center of switching period and grows symmetrically to both edges when the duty cycle increases. This is why it is often called symmetrical PWM. An example of center aligned and edge aligned PWM is represented in Fig. 2.17.
0 Ts 2Ts 3Ts Time
Voltage [V]0
0 Ts 2Ts 3Ts
Time Voltage [V]0
0 Ts 2Ts 3Ts
Time Voltage [V]0
Figure 2.17: Three PWM signals with the same duty cycles and switching frequency - in blue leading edge aligned, in red trailing edge aligned, and in green centered aligned.
For the purpose of this thesis, it is useful to discuss how PWM signals are generated. The base concept of PWM generation is the comparison between the so called carrier wave and modulating wave. In general, the carrier signal is a sawtooth or triangular waveform with a frequency equal to the switching frequency.
The modulating wave can be different depending on the application - sinusoidal waves or constant DC signals are often used. The constantly comparison between the carrier and modulating waveforms gives result to the so called modulated signal. The modulated wave has the characteristics seen in Fig. 2.16. Indeed, the comparison gives an output waveform which is high when the modulating wave is greater than the carrier one and vice versa. In such a way, the resulting duty cycle can be modified according to changes in the modulating wave. An example of this is reported in Fig. 2.18. Different carrier waveforms are used to generate different types of PWM signals - sawtooth carriers are used for edge aligned modulation, triangular ones for center aligned modulation. In leading edge modulation the carrier is a sawtooth leading edge with positive ramp followed by the step decay, instead in trailing edge modulation a sawtooth trailing edge with a vertical rise followed by a negative ramp is used. The triangular carrier can be also seen as a double edge sawtooth wave with a positive ramp followed by the negative one, and
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Time [s]
0 2 4 6
Voltage [V]
Carrier Signal Modulating Signal
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Time [s]
0 Vcc
Voltage [V]
Modulated Signal
Figure 2.18: PWM generation with a sinusoidal modulating wave.
it used for centered modulation. The three different carrier waves are represented in Fig. 2.19.
An application of PWM related to this thesis is the driving of switches (often constitute by thyristors) in power electronics circuitries. As will be seen in section 2.5.1, the turning on and off of these switches is done by means of PWM signals.
There are many cases where two switches must be driven in a complementary fashion to avoid short circuits. For this reasons, complementary PWM signals (represented in Fig. 2.20) are used for the purpose. Their main characteristic is that one signal is high when the other is low and vice versa. From a theoretical point of view, this is enough to avoid shorts due to switches that must be driven in a complementary fashion, but in practice turn-off and turn-on delays can’t be neglected. To take into account delays of real components, a deadtime insertion could be required. In complementary PWM signals, the deadtime is the time interval that elapses from the turn off of one signal to the turn on of its complementary one. An example of deadtime insertion between two complementary PWM signals is reported in Fig.
2.20. The application and the necessity of complementary PWM and deadtime insertions will be more clear in section 2.5.1 where inverters are discussed. More about PWM could be explained, but the topics discussed in this section were
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Time [s]
0 2.5
Voltage [V]
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Time [s]
0 2.5 5
Voltage [V]
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Time [s]
0 2.5 5
Voltage [V]
Figure 2.19: Different carrier waveforms - on top a triangular carrier, in the middle a trailing sawtooth carrier, and on bottom leading sawtooth carrier.
considered the most pertinent for the purpose of this thesis.