Transistor and PWM-Switch Analogs
The transistor and the PWM-switch, the core circuit of power converters, have interesting analogs. Both have three terminals, hence three configurations. They are active devices, and each has a key parameter. In this article, BJTs, FETs, and PWM-switches - three kinds of active devices - are compared analogously.
For bipolar junction transistors (BJTs) the key parameter is α; for the three-terminal PWM-switch, it is duty-ratio, D. While α is fixed for BJTs, it can be varied for the PWM-switch, and this leads to time-variant circuit behavior in switching converters.
Transistor and Inductive Switch Configurations
A two-port network, such as an amplifier or power converter, has an input and an output port. Each port has a pair of terminals, as shown below.
The relationship between input and output ports is usually expressed as a transfer function or transmittance. A device with three terminals, such as a transistor, has one input, one output, and one common terminal. The common terminal is shared by input and output ports, usually as a common ground (–) terminal. This results in three configurations where each terminal assumes the common position. For the BJT and FET (either JFET or MOSFET) of either polarity (NPN or PNP, n-channel or p-channel), corresponding configurations are listed in the following table.
The analog between emitter-source, base-gate, and collector-drain is familiar and not hard to recognize, given the similarity in the devices.
The inductive PWM-switch is a circuit that can be regarded as a circuit element, shown below.
The inductor is in series with a single-pole, double-throw current switch. It is in the active position (connected to the A terminal) for D×Ts of the time, where Ts is the switching period and D is the duty ratio. The switch is in the passive position for D'×Ts = (1 – D)×Ts.
The PWM-switch can be regarded as a three-terminal active device, like the transistor. Consequently, it too has three configurations. They are, by name:
- common passive (CP) or buck
- common active (CA) or boost
- common inductor (CL) or buck-boost
The voltage transfer function for a buck converter is
where Vs Vo is the secondary voltage (output voltage plus diode drop, if any) and Vg is the input voltage. The steady-state transfer function is easily derived from flux balance of the inductor, so that on-time and off-time flux changes are equal, or
With a net Δ λ = 0 for each cycle, the average current remains constant over multiple cycles in steady-state converter operation. In the CP configuration, the A terminal of the PWM-switch is common to both Vg (input) and Vs (output) circuit loops.
The other two configurations have the following voltage transfer functions:
The current transfer function of a BJT of the CE configuration (iC /iβ) is β , for the CB it is α, and for the CC, it is β + 1. Expressing β in terms of α,
Then a correspondence between α and duty ratio, D, becomes evident, as summarized in the following table for BJTs, FETs and PWM-switches.
The α D analogy for BJTs applies to FETs as a λ D analogy, though the correspondence may be less familiar. Just as β is a current-ratio parameter of BJTs, µ is the corresponding voltage-ratio parameter for FETs;
FET model variations are shown below. The model on the right is like a BJT T model. Gate current remains zero in the FET T model because the dependent current-source current is the same as that through rm, leaving no gate current.
The simple circuit model of a FET is a dependent voltage source between source and drain (negative terminal to drain) for which the voltage is µ ×vGS, in series with a drain resistance, ro. The alternative expression of this parameter has a correspondence with α which is λ;
(As a precedes β , λ precedes μ in the Greek alphabet.) The ratio of source to gate voltage is the voltage gain of the CD source-follower configuration, and is λ. The source-to-drain voltage gain is μ + 1.
The above table relates BJTs, FETs, and switches in their three configurations. The configurations are analogous:
CE CS CL
CB CD CP
CC CG CA
Consequently, the terminals also correspond;
emitter source « inductor (switch common terminal)
base drain passive
collector gate active
The one imperfection in the analogy is that the corresponding BJT-FET terminals of base-gate and collector-drain are reversed. This is caused by the use of a current analogy for BJTs and the dual voltage analogy for FETs. If either a β-based or μ-based model were used for either BJT or FET, then the BJT and FET terminals would again correspond. As it is, the analogous PWM-switch terminals correspond according to the β or μ analogies (for both BJT and FET) as shown in the table.
While these analogies might not produce immediate understanding of PWM-switches from transistor concepts, they can help analog engineers to think analogically about converter circuits. Just as transistor amplifier gain and port resistances vary with α, in converters they also vary with D. Duty ratio is best thought of as a PWM-switch parameter, not a dynamic variable, though it is often varied dynamically in converters as a control variable.
This suggests why deceptively simple-looking converter circuits are actually more complicated and demanding of a designer than linearized analog circuits; imagine designing circuits for which β varies with time. Just as the BJT transistor has a linear model based on constant β and re (at constant IE), the inductive switch can be linearized around a constant-D operating point. Richard Tymerski first worked out the PWM-switch model in the mid-1980s while a graduate student under the supervision of Vatché Vorperian (both at VPI at the time), and it can be used for linear frequency-response analysis of converter (or any switched-inductor) circuits. The incremental (small-signal) PWM-switch model, however, requires additional development.