Chapter 5 — Switches and self-commutating devices

Module 1's networks were linear and fixed: once you chose the topology and the time step, the conductance matrix $\mathbf{G}$ never changed, so it was factorized once and reused (Chapter 3). Switching devices break that comfort. A breaker, diode, thyristor, or transistor changes the network's conductance pattern when it opens or closes — and some of them decide their own state from the very solution you are trying to compute. This chapter builds the switch model, shows how diodes, thyristors, and IGBTs choose their state, and explains the in-step iteration the engine runs to keep everything consistent.

Learning objectives

By the end of this chapter you should be able to:

• Model an ideal switch as a two-state conductance and read the real on/off parameters off a NumaSim device.
• State the conduction rule for a diode, a thyristor, and an IGBT, and say which are self- and which are externally-commutated.
• Explain why a switch operation costs the solver a re-factorization (Chapter 3) and why a self-deciding device needs iteration within a step.
• Read a diode's conduction intervals off a rectifier and a converter waveform.

5.1 The ideal switch as a two-state conductance

The simplest switch model is a resistor with two values. Closed, it is a tiny resistance $R_{\text{on}}$ (a large conductance $G_{\text{on}}$); open, it is a huge resistance $R_{\text{off}}$ (a tiny conductance $G_{\text{off}}$):

$$G_{\text{sw}}=\{\begin{array}{ll}{G}_{\text{on}}=1/R_{\text{on}},& \text{closed},\\ G_{\text{off}}=1/R_{\text{off}},& \text{open}.\end{array}$$

A switch is a conductance that flips between two values: a small on-resistance $R_{\mathrm{on}}$ when closed and a large off-resistance $R_{\mathrm{off}}$ when open. Each state stamps like an ordinary resistor (Chapter 3).

These are not $0$ and $\mathrm{\infty }$. A perfect short would make rows of $\mathbf{G}$ linearly dependent, and a perfect open would disconnect a node — both make the matrix singular. Instead the engine uses a large but finite ratio. In the Module 2 samples the electronic switches use $R_{\text{on}}=0.005\mathrm{\Omega }$ (so $G_{\text{on}}=200\text{S}$) and $R_{\text{off}}={10}^5\text{-}{10}^6\mathrm{\Omega }$ (so $G_{\text{off}}={10}^{-5}\text{-}{10}^{-6}\text{S}$) — a ratio of ${10}^7$ or more, small enough to keep the solve well-conditioned yet stiff enough that the "off" branch carries negligible current.

When a switch changes state, $\mathbf{G}$ changes, so the "factor once" trick from Chapter 3 is interrupted: the engine must re-stamp and re-factorize $\mathbf{G}$. That is the price of every switching event, and it is why a circuit that switches constantly (a PWM converter) does more linear-algebra work than a passive one.

5.2 The diode — state from its own terminals

A diode is the simplest self-commutating device: nothing external tells it when to switch. It conducts when forward-biased and blocks when reverse- biased. In the two-state picture:

• On (forward): anode-to-cathode voltage tries to exceed the small forward drop $v_f$; the diode is $R_{\text{on}}$ in series with $v_f$ and carries current in the forward direction only.
• Off (reverse): the diode is $R_{\text{off}}$ and carries essentially no current.

NumaSim's diode exposes exactly these parameters: v_forward (e.g. $0.01\text{V}$), resistance_on ($0.005\mathrm{\Omega }$), resistance_off ($100\text{k}\mathrm{\Omega }$). The catch is that you do not know the state until you solve: whether the diode is forward-biased depends on the node voltages, which depend on whether the diode is conducting. That circularity is resolved by iteration (§5.5).

The two consistency conditions a solved diode must satisfy are:

$$\text{if assumed ON:}{i}_D\ge 0(\text{current really is forward}),$$$$\text{if assumed OFF:}{v}_{ac}<v_f(\text{it really is reverse-biased}).$$

If either check fails, the diode's assumed state was wrong: flip it and solve again.

5.3 Thyristors and IGBTs — adding a gate

Two more devices add a control terminal, and they differ in a way worth pinning down: how they turn off.

• Thyristor (SCR). Like a diode, but it will not turn on until it is both forward-biased and given a gate pulse. Once latched on, the gate loses control: the thyristor conducts until its current tries to reverse, at which point it turns off at the natural current zero. This is line (natural) commutation — the circuit, not the gate, turns it off. It is the workhorse of phase-controlled rectifiers.
• IGBT (and MOSFET). A fully controllable switch: raise the gate above its threshold and it conducts; drop the gate and it turns off regardless of the current — forced commutation. NumaSim's IGBT uses a gate_threshold (e.g. $0.5$): a gate signal above it closes the switch, below it opens the switch. This is what lets a PWM signal chop a DC bus at kilohertz rates.

Because an IGBT can interrupt current on command, an inductive load needs somewhere for its current to go the instant the IGBT opens — otherwise $v=Ldi/dt$ spikes. That path is a freewheeling diode, and it is why the buck converter of §5.6 pairs an IGBT with a diode.

Device	Turn-on	Turn-off	Commutation
Diode	forward bias	current zero	self (natural)
Thyristor	gate pulse + forward bias	current zero	line (natural)
IGBT / MOSFET	gate above threshold	gate removed	forced

5.4 A switch changes the topology

Every state change re-shapes $\mathbf{G}$. Concretely, closing a switch adds a large $G_{\text{on}}$ between two nodes (often tying them nearly together); opening it removes that coupling. The stamp is the ordinary resistor stamp from Chapter 3 — only the value jumps. Two consequences follow:

1. The matrix is re-factorized at each state change (the cost noted in §5.1).
2. The reference/ground still matters (Chapter 3 §3.5): opening a switch must not orphan an island. If a switch is the only tie between a sub-circuit and ground, opening it leaves that sub-circuit floating and the solve fails — a real gotcha when you build your own switched circuits.

5.5 In-step iteration for state consistency

Because a diode/thyristor/IGBT can decide its own state from the solution, the engine cannot simply stamp and solve once when a device might switch. It runs a small fixed-point loop within the time step:

The in-step iteration a switching solver runs. It assumes a set of device states, stamps and solves the linear network, then checks each device against its conduction rule; any inconsistent device is flipped and the network re-solved, until all states are consistent — then the step is committed.

The loop almost always converges in one or two passes, because only devices near a switching boundary are ambiguous. But it is the reason a power-electronic step can cost several linear solves rather than one — the subject, along with what can go wrong numerically, of Chapter 6.

5.6 Lab: watch devices choose their state

Lab 5A — a single diode's conduction gaps

Open the half-wave rectifier in app →

(Full description: half-wave rectifier.)

This circuit has exactly one diode, which makes its state logic easy to see. Run it and overlay Vac (the source) and Vdc (the capacitor/load). Watch what the diode does:

• Near each positive peak of Vac, the source momentarily exceeds the capacitor voltage, the diode is forward-biased, and it conducts — you see Vdc get pulled up (a charging pulse).
• Everywhere else the source is below Vdc, the diode is reverse-biased and blocks, so the load quietly discharges the capacitor and Vdc sags.

The diode conducts only during a short slice near each peak — those are the intervals where its ON-state consistency check ($i_D\ge 0$) holds. Between them it is a $100\text{k}\mathrm{\Omega }$ open circuit.

Lab 5B — a gated switch and its freewheeling diode

Open the buck converter in app →

(Full description: buck converter.)

Here a PWM gate signal drives an IGBT and a freewheeling diode shares the switch node Vm. Run it and plot Vm and ID (the diode current):

• When the IGBT is on, Vm jumps to the $12\text{V}$ input and the diode is reverse-biased (ID $\approx 0$).
• When the IGBT turns off, the inductor forces its current to keep flowing, so the diode turns on to freewheel it: Vm drops to about $0\text{V}$ and ID carries the inductor current.

Note that ID is never negative — the diode is a one-way device. Because the duty is $0.5$, the diode conducts about half the time, and its average current ($\approx 0.15\text{A}$) is roughly half the inductor's average ($\approx 0.30\text{A}$).

5.7 Summary

• A switch is a two-state conductance — small $R_{\text{on}}$ closed, large $R_{\text{off}}$ open — using a large-but-finite ratio to keep the matrix well-conditioned.
• A diode self-commutates from its terminal bias; a thyristor needs a gate to turn on but line-commutates off at current zero; an IGBT is fully gate-controlled (forced commutation) and needs a freewheeling diode for inductive loads.
• Every state change re-shapes and re-factorizes $\mathbf{G}$, and must not orphan a grounded island.
• Devices that decide their own state force an in-step iteration: assume, solve, check, flip, re-solve — usually converging in a pass or two.

5.8 Problems

Problem 5.1. In the half-wave rectifier at some instant Vac $=6\text{V}$ and the capacitor voltage Vdc $=11\text{V}$. Is the diode conducting? What about when Vac $=14\text{V}$ and Vdc $=11\text{V}$?

Solution 5.1

The diode conducts only when the anode (source side) exceeds the cathode (capacitor side) by more than the forward drop. With Vac $=6<$ Vdc $=11$, it is reverse-biased and blocks. With Vac $=14>$ Vdc $=11$ (by well over $v_f=0.01\text{V}$), it is forward-biased and conducts, topping up the capacitor. This is why charging happens only near the peaks.

Problem 5.2. A NumaSim diode has $R_{\text{on}}=0.005\mathrm{\Omega }$ and $R_{\text{off}}=100\text{k}\mathrm{\Omega }$. Compute the two conductances and their ratio. Why not use $0$ and $\mathrm{\infty }$?

Solution 5.2

$G_{\text{on}}=1/0.005=200\text{S}$ and $G_{\text{off}}=1/{10}^5={10}^{-5}\text{S}$; the ratio is $2\times {10}^7$. A true short ($G=\mathrm{\infty }$) makes two nodes identical and the rows of $\mathbf{G}$ linearly dependent; a true open ($G=0$) can leave a node with no path to ground. Both make $\mathbf{G}$ singular. A large-but-finite ratio keeps the system solvable while making the off branch's current negligible.

Problem 5.3. Why can't the solver stamp a diode's state once and move on, the way it stamps a resistor?

Solution 5.3

A resistor's conductance is fixed and independent of the solution. A diode's state (on/off) depends on its terminal voltage or current — which are part of the unknown solution. The engine must therefore guess the state, solve, and check the guess against the diode's conduction rule; if the guess was wrong it flips the state and re-solves. This in-step iteration is unavoidable whenever a device decides its own state.

Problem 5.4. A thyristor and an IGBT are both conducting a positive current. You remove the gate signal from each. What happens?

Solution 5.4

The IGBT turns off immediately — it is forced-commutated, so removing the gate opens it regardless of current (the inductive load then needs a freewheeling path). The thyristor keeps conducting: once latched, its gate has no control, and it will only turn off when the external circuit drives its current to zero (line commutation).

Problem 5.5. In the buck converter, why is the freewheeling diode essential, and during which part of the switching cycle does it conduct?

Solution 5.5

The output inductor cannot change its current instantaneously. When the IGBT turns off, the inductor current must keep flowing; without a path it would force a large $v=Ldi/dt$ voltage spike. The freewheeling diode provides that path: it conducts during the IGBT's off interval, clamping the switch node Vm near $0\text{V}$ and carrying the inductor current back to the load. With duty $D=0.5$ it conducts roughly half of each cycle.

5.9 References

• H. W. Dommel, Electromagnetic Transients Program (EMTP) Theory Book, Bonneville Power Administration — switches and the in-step network solution.
• N. Mohan, T. Undeland, and W. Robbins, Power Electronics: Converters, Applications, and Design, Wiley — device characteristics and commutation.
• J. Arrillaga and N. R. Watson, Power Systems Electromagnetic Transients Simulation, IET Power and Energy Series 39 — power-electronic device models and switching solutions.

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Previous: Chapter 4 — Sources and control signals · Next: Chapter 6 — Numerical issues of switching.