 Hi, Mrs. Veena Sunil Patki, Assistant Professor, Department of Electronics Engineering, Walchand Institute of Technology, Solapur, welcome you for this session. At the end of this session, students can analyze RLC series circuit through AC. So, this is the series circuit, resistor, inductor and capacitors are connected in series, VR is the voltage across resistor, VL is the voltage across inductor and VC is the voltage across capacitor, R is measured in terms of Ohms, L is measured in terms of Henry and capacitance in Farad, V is the total voltage is given by Vm sin omega t that is the AC voltage connected across series combination. So, we can write down VR as I into R, VL equal to I into XL and VC equal to I into XC where XL equal to 2 pi FL and XC equal to 1 upon 2 pi FC both are measured in terms of Ohms. So, total voltage is the phasor addition of voltage across resistor, inductor and capacitor. So, we can add VR, VL, VC as a vector because all that three voltages are not in phase. So, you can see here from these three diagrams that is VR is in phase with the current for resistive circuit, VL is leading to the current by 90 degree for inductive circuit and VC lags current by 90 degree for capacitive circuit. For this RLC series circuit we can draw the phasor diagram. So, here we can draw the current phasor and then the VR is in phase with the current and here VL leads current by 90 degree and VC lags current by 90 degree. So, here this is the basic phasor diagram for RLC series circuit. And if we want to draw the phasor diagram for this RLC series circuit we have to consider the three conditions that is VL is less than VC, VL is greater than VC and VL equal to VC. So, we are going to discuss all these three conditions one by one. So, for this first condition VL is less than VC. So, again we can draw this phasor diagram like this as draw first the current vector then VR is in phase with the current, VL leads this current by 90 degree and VC lags current by 90 degree. Now, when we calculate the phasor addition for these three first we are going to add VL and VC because both are exactly opposite to each other. So, after addition of VL and VC here we will get the VC minus VL because VC is greater than VL. So, here we will get the resultant of that VC minus VL then that VC minus VL is added with this VR and then we will get the total voltage V like this. So, this is the total voltage and the angle between voltage and current is theta. So, you can see here the current leads voltage by theta degree and we can draw the voltage triangle from this phasor diagram and we can write down the voltage and current equation like this V equal to Vm sin omega t and I equal to Im sin omega t plus theta. So, from this voltage triangle we can write down the voltage equation V equal to under root VR square plus in bracket VC minus VL bracket square. Then by putting the values of VR, VC and VL we will get the equation like this and we can write down this as V equal to I under root R square plus in bracket XC minus XL bracket square where under root R square plus XC minus XL bracket square is Z. So, here we will get the equation V equal to I into Z or Z equal to V by I and Z is the impedance measured in ohms. So, we can draw the voltage triangle and impedance triangle like this and Z equal to under root R square plus XC minus XL bracket square and in rectangular form Z equal to R minus J XC minus XL and in polar form Z equal to magnitude of Z angle of minus theta and from this triangles we can write down the tan theta and we can calculate theta as tan inverse of XC minus XL by R. So, power factor is cos theta and cos theta is here leading power factor because the circuit is capacitive in nature. Now pause the video and think for series RLC circuit impedance or voltage total must always be calculated by A adding values vectorially, B graphing the angles, C multiplying the values and D subtracting the values. So, what is the answer? So, answer is A because always we are going to calculate the total voltage and impedance by adding the values vectorially. For the second condition, VL greater than VC. So, draw the first current phasor then VR is in phase with the current, VL leads the current by 90 degree and VC lags current by 90 degree. So, first we are going to add VL and VC so resultant will be VL minus VC and by adding this VL minus VC to VR we will get the total voltage V here like this and the theta is the angle between voltage and current and current is lagging to voltage by theta degree. So we can draw the voltage triangle here like this so we can write down the voltage equation and current equation here V equal to Vm sin omega t and I equal to Im sin omega t minus theta because current is lagging to voltage by theta degree. So here the voltage triangle from that voltage triangle we can calculate the voltage as V equal to under root VR square plus VL minus VC bracket square. By putting the values of VR, VL and VC we will get the equation like this and we can write down that equation as V equal to I under root R square plus in bracket XL minus XC bracket square and V equal to I into Z where Z is the impedance Z equal to under root R square plus XL minus XC bracket square. So Z equal to V by I so Z is measured in ohms. From this voltage triangle we can also draw the impedance triangle and we can write down the formula for Z as under root R square plus XC minus XL bracket square. So in rectangular form Z equal to R plus J XL minus XC and in polar form Z equal to magnitude of Z angle of theta and we can write down the tan theta from this both triangle and we can calculate theta as tan inverse of XL minus XC by R. We are going to discuss about the condition VL equal to VC. So this is the basic phasor diagram. So first you draw the current phasor VR is in phase with the current, VL leads current by 90 degree and VC lags current by 90 degree. Now if we calculate the resultant voltage for this first add VL and VC but here VL and VC both are equal and both are opposite to each other. So resultant of that is 0. So if VL equal to VC then the total voltage is nothing but only the resistive voltage that is given by IN2R or we can also write down that as IN2Z and from that we can write down resistor equal to impedance for this condition. So here we can calculate the power factor equal to R by Z and if R equal to Z then the cos theta equal to 1. We can say that the unity power factor is there for this condition. So here we can write down the voltage and current equation as V equal to Vm sin omega t and I equal to Im sin omega t because voltage and current both are in phase. So the condition for resonance we can summarize all that points as VL equal to VC. So we can write down that as XL equal to Xc and impedance is equal to resistance that is Z equal to R, power factor is unity and in this condition for resonance as VL equal to VC we can write down that as XL equal to Xc that is why we can write down the equation as 2 pi FL inductive reactance XL is given by 2 pi FL and capacitive reactance is given by 1 by 2 pi FC. So we can equate that and from that equation we can write down the frequency equal to 1 by 2 pi root LC. So this frequency is called as resonant frequency and that resonant frequency is measured in terms of hertz. So you can refer the book Electrical Technology by B. L. Therese, thank you.