Summary of Power Switch Analysis

Switches are the key to electronic conversion of energy. Three major aspects of design – the hardware, software, and interface problems – can be defined and studied. Consider some of the major concepts covered so far:

• The hardware problem (build a switch matrix). Switches can be organized in a matrix. Given information about the source and the load, the switch matrix dimensions (such as 2X2 or 4X3) can be determined. The sources help define the current and voltage polarities that a switch will see. This permits the selection of restricted switches to build a matrix. The restricted switch types correspond to specific kinds of devices.
• The software problem (operate the matrix to provide a desired conversion). Switching function provide a convenient mathematical representation of switch action. Circuit laws constrain switch operation. We need to avoid switching actions that might try to violate Kirchhoff’s Voltage Law, or Kirchhoff’s Current Law. Fourier analysis identified a frequency matching requirement to ensure that energy is transferred successfully in a circuit. The duty cycle, frequency, and phase completely define the switch action. The ultimate objective is to produce a large wanted component while ensuring that unwanted components are small.
• The interface problem (add energy storage to filter energy flow for the application requirements). Theis remains to be studied in-depth in later posts. The frequency matching requirement helps suggest the importance of filters.

A switch matrix will either be direct, meaning that it simply interconnects the input and output, or indirect, meaning that energy storage is included in the matrix. switch matrices are one of the few types of circuits in which a designer can try to break KVL or KCL , but of course this violates the underlying physics. accidental switch action that tries to violate circuit laws is perhaps the most common cause of failure in power converters.

• KVL restriction: a switch matrix must avoid interconnecting unlike voltage sources.
• KCL restriction: a switch matrix must avoid interconnecting unlike current sources.

Voltage sources switch across inductors, or current sources switch int capacitors, will violate KVL or KCL in the long run.

Switching functions, which are functions of time valued at 1 when the corresponding physical switch is on and 0 when it is off, offer a convenient representation of switch action. KVL and KCL restrictions, for instance, can be stated in terms of switching functions. Converter waveforms turn out to be products of switching functions and sources.

The ideal switch can carry any current, block any voltage, switch under any conditions, and change back and forth between on and off instantly. This ideal device is not yet very useful in power conversion, because it is hard to create with real devices. Semiconductor switches exhibit polarity limitations in addition to physical limits on current, voltage, and time. The polarity limitations define restricted switches. These are extremely useful in converter design. Any converter can be defined initially with restricted switches. This ideal representation is translated into corresponding semiconductor hardware once it is fully analyzed and understood. Restricted switches are defined in terms of current and voltage polarity. Five types exist:

1. Forward-conducting reverse-blocking (FCRB) switches, corresponding to ideal diodes.
2. Forward-conducting, forward-blocking (FCFB) switches, corresponding to idealisations of power bipolar junction transistors
3. Forward-conducing bidiretional-blocking (FCBB) switches, corresponding to idealized GTOs, or to series combinations of FCFB and FCRB switches.
4. Bidirectional-conducting forward-blocking (BCFB) switches, corresponding to ideal power MOSFETs.
5. Bidirectional-conducting bidirectional-blocking (BCBB) switches, corresponding to the ideal switch.

There are other important devices, such as the SCR, that add timing properties to the restricted switch behavior. These will be examined in context in later posts.

Diode circuits can be analyzed with a trial method. If a diode is on, its forward current must be positive. If it is off, its forward voltage must be negative. The method begins with circuit configurations, the various possible arrangements defined by switch state, then analyses the plausible configurations to see which one is consistent with both the circuit laws and the diode properties. If a configuration’s voltages or currents contradict diode properties, the switch action to change the configuration would occur in a real circuit. In a diode bridge, the trial method shows that the usual action is for devices to act in diagonal pairs.

Through Fourier analysis, we can resolve periodic signals into individual frequency components. A port in an electrical circuit will have nonzero average power only if the voltage and current at the port have a common Fourier component frequency. Furthermore, the power at a port is the sum of powers contributed by the individual frequencies. Cross-frequency terms do not contribute to average energy flow.

A real periodic function $f(t)$ can be represented with the Fourier series

$f(t) = \sum\limits_{n=0}^{\infty} c_n cos (n \omega t + \theta_n)$

A general switching function with radian frequency $\omega$, duty cycle $D$, and phase (referenced to the center of the pulse) $\phi$ has the Fourier Series

$q(t) = D + \frac{2}{\pi} \sum\limits_{n=1}^{\infty} \frac{sin(n \omega D)}{n} cos(n \omega t - n \phi)$

The three parameters completely define the switching function and hence the switch action. A useful converter must permit some adjustment of its operation. Pulse-width modulation and phase control are the most common adjustment methods in power electronics.