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Everywhere states that a MOSFET has no current at the gate. However, this is not true since, during the time the gate capacitance charges up to reach the certain voltage threshold, the current is entering the gate capacitance.

My actual question: During the charge-up time of the "gate capacitance", does the current only go to the source? (Meaning current enters the gate capacitance while charging and then leaves through the source for a N channel type?) enter image description here

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    \$\begingroup\$ Yes, but your wording isn't quite right. Not necessarily to ground. Depends on how it's wired. Current circulates out of the source through the supply (or capacitor) and to the gate. \$\endgroup\$ – DKNguyen Jul 25 at 3:13
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    \$\begingroup\$ @DKNguyen Current can also go through the drain, which can cause problems in switching converters that don't have well-designed gate drives. \$\endgroup\$ – Hearth Jul 25 at 3:29
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    \$\begingroup\$ As the channel starts to form, the charge comes from the substrate. Then the minuscule channel links up with the Source which also provides charge. \$\endgroup\$ – analogsystemsrf Jul 25 at 4:35
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Remember, there are two (major) gate capacitances in a power MOSFET: the gate-source capacitance Cgs, and the gate-drain capacitance, Cgd. If the drain voltage never changes, then yes, all gate current goes into the gate-source capacitance.

However, in power MOSFETs, the drain voltage almost always changes. During turn-on, at the point of switching, the drain voltage in an NFET will start to fall. This discharging of Cgd causes current to flow from the gate to the drain. Depending on drain-source voltage Vds, transistor and driver parameters, the gate-source voltage Vgs may have a small inflection point, a Miller plateau, or wild oscillation.

For a better discussion on the Miller plateau, see this answer

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There is capacitance from the gate to every other electrode in the device.

When the gate is charged, some current flows through the source. This terminal is often grounded, especially in a switching application.

Some current flows through the drain. While the gate-drain capacitance is often an order of magnitude smaller than the gate-source capacitance, the effect of the charge in it, because of the voltage swing on the drain when switching, is often an order of magnitude larger than that of the source. This is called the Miller Effect, and causes the long flat area in the gate charge graph, which delays the full device turn-on when current is not supplied fast enough to the gate.

Currents from both drain and source eventually end up at the device driving current into the gate, to complete the circuit.

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