EGR solenoid does what exactly?

In most applications the EGR solenoid acts as the controller for the EGR. The ECM uses electrical pulses to control things in the vehicle, however many EGR valves are vacuum actuated and as such they are not easily controlled by electronics. The EGR solenoid however is a valve that is controllable and acts as an interface between the two parts. The ECM can now send a pulse width modulated signal to the solenoid which uses that signal to control the valving in the vacuum section of the solenoid. So, the ECM decides that the EGR needs to flow at 50%, it sends the signal to the solenoid which then opens the valving so that vacuum can flow and the EGR valve opens 50%. Other makes use of an electronic EGR that doesn't use vacuum but those have been problematic at times because of the heat and corrosive nature of exhaust causing failures and the replacements tend to be very expensive. The EGR solenoid can cause different symptoms depending on how it fails. If it fails with the valve open it can cause stalling and rough running because of the excessive EGR flow. If it fails closed, you can end up with spark knock and poor emissions. By remote mounting the control section they can avoid the heat and still control the EGR.

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EGR is used mainly to reduce the temperatures inside the cylinder to lower emissions and to stop detonation of the fuel charge. The former helps keep the people alive, and the latter keeps the engine alive. How it is controlled is based on the engine design and fuels used as well as sensor data from the engine itself. In the "old days" the EGR was simply controlled by a thermostatic valve that watched coolant temperature and the vacuum levels in the engine. So, you started the engine, and the cold coolant kept the vacuum valve closed so no EGR would happen. When cold you don't want anything to impede the warming of the engine or reducing power as you already have enough problems with fuel atomization at lower temperatures and that slows the flame front and causes pockets of static mix in the cylinder, so you are basically throwing unburnt fuel into the air. You add exhaust gas at that point, and you just make it worse. One the engine was warmed up you could start adding in exhaust gas to lower the emissions, but you needed a way to control that, so it only happened at points when detonation or low fuel use. So, they controlled the valve using engine vacuum and an orifice in the valve. However, today's engines are far more complex than those engines and instead of having a smooth vacuum curve, you can end up with various strategies of valve timing and EGR control needed in those situations is much more than a simple constant orifice will work with. So, you get the modern systems that use solenoids or even cam timing and internal passages to control EGR flow. It gets rather complex when you start looking at the various flow patterns and flame propagation timing in modern engines and are trying to maximize torque and lower emissions while raising MPG and lowering operational costs.