# How can I determine the voltage the resistive load in this relay?

I am hoping someone can help me understand something. I'm an engineering student trying to analyze this circuit, but am finding myself not understanding and going in circles somewhat. I've searched as much as I can but am still stuck.

The circuit is a relay. I did not design it, only analyzing it. The only part of the analysis I am getting hung up on is the voltage across the mosfet (Vds). I feel like I have a good understanding of mosfets and their operating regions. I understand this mosfet is a switching mosfet and must be operating in the linear region. I can confirm this when I measure the voltage in simulation software and get Vds << Vgs - Vth. For the load, I am using a 24V DC LED strip light. The lights dissipate ~3.36W, so I = 3.36/24 = 140mA and Rl = 171 omhms.

Here is my hangup: I want to calculate Vds to determine the current through the mosfet, as well as determine the operating region. The source voltage is 0V, but how can I calculate Vd? I don't know the current through the mosfet, because that is determined by Vds, and I don't know Vd because it depends on the current through the load. I feel like I am missing something. The typical IRF540 $$\I_D\$$-$$\V_{ds}\$$ relationship is shown in this chart from the IRF540 datasheet: This shows that with 10 V $$\V_{gs}\$$, which is about what you could expect to be applied with your circuit, the typical IRF540 will be in triode operation (what we often call "fully switched") with up to about 100 A load.

That means you can essentially treat the MOSFET channel as a resistor when it's in the on state and conducting less than 100 A. The datasheet also guarantees a channel resistance of no more than 77 mΩ with 10 V $$\V_{gs}\$$.

$$I_{L} = \frac{24\ {\rm V}}{R_L + 77\ {\rm m\Omega}}$$