Derive the expression for series RLC circuit with its circuit and phasor diagram. Also write the power factor of the circuit.
Answers
Answer:
Series RLC Circuit
series rlc circuit analysis
The series RLC circuit above has a single loop with the instantaneous current flowing through the loop being the same for each circuit element. Since the inductive and capacitive reactance’s XL and XC are a function of the supply frequency, the sinusoidal response of a series RLC circuit will therefore vary with frequency, ƒ. Then the individual voltage drops across each circuit element of R, L and C element will be “out-of-phase” with each other as defined by:
i(t) = Imax sin(ωt)
The instantaneous voltage across a pure resistor, VR is “in-phase” with current
The instantaneous voltage across a pure inductor, VL “leads” the current by 90o
The instantaneous voltage across a pure capacitor, VC “lags” the current by 90o
Therefore, VL and VC are 180o “out-of-phase” and in opposition to each other.
For the series RLC circuit above, this can be shown as:
series rlc circuit waveforms
The amplitude of the source voltage across all three components in a series RLC circuit is made up of the three individual component voltages, VR, VL and VC with the current common to all three components. The vector diagrams will therefore have the current vector as their reference with the three voltage vectors being plotted with respect to this reference as shown below.
Individual Voltage Vectors
series rlc circuit vectors
This means then that we can not simply add together VR, VL and VC to find the supply voltage, VS across all three components as all three voltage vectors point in different directions with regards to the current vector. Therefore we will have to find the supply voltage, VS as the Phasor Sum of the three component voltages combined together vectorially.
Kirchhoff’s voltage law ( KVL ) for both loop and nodal circuits states that around any closed loop the sum of voltage drops around the loop equals the sum of the EMF’s. Then applying this law to the these three voltages will give us the amplitude of the source voltage, VS as.
Instantaneous Voltages for a Series RLC Circuit
series rlc circuit instantaneous voltages
The phasor diagram for a series RLC circuit is produced by combining together the three individual phasors above and adding these voltages vectorially. Since the current flowing through the circuit is common to all three circuit elements we can use this as the reference vector with the three voltage vectors drawn relative to this at their corresponding angles.
The resulting vector VS is obtained by adding together two of the vectors, VL and VC and then adding this sum to the remaining vector VR. The resulting angle obtained between VS and i will be the circuits phase angle as shown below.
Phasor Diagram for a Series RLC Circuit
series rlc circuit phasor diagram
We can see from the phasor diagram on the right hand side above that the voltage vectors produce a rectangular triangle, comprising of hypotenuse VS, horizontal axis VR and vertical axis VL – VC Hopefully you will notice then, that this forms our old favourite the Voltage Triangle and we can therefore use Pythagoras’s theorem on this voltage triangle to mathematically obtain the value of VS as shown.
Voltage Triangle for a Series RLC Circuit
Voltage Triangle for a series RLC Circuit
Please note that when using the above equation, the final reactive voltage must always be positive in value, that is the smallest voltage must always be taken away from the largest voltage we can not have a negative voltage added to VR so it is correct to have VL – VC or VC – VL. The smallest value from the largest otherwise the calculation of VS will be incorrect.
We know from above that the current has the same amplitude and phase in all the components of a series RLC circuit. Then the voltage across each component can also be described mathematically according to the current flowing through, and the voltage across each element as.
Instantaneous Voltages for a series RLC Circuit
By substituting these values into the Pythagoras equation above for the voltage triangle will give us:
source voltage equation
So we can see that the amplitude of the source voltage is proportional to the amplitude of the current flowing through the circuit. This proportionality constant is called the Impedance of the circuit which ultimately depends upon the resistance and the inductive and capacitive reactance’s.
Then in the series RLC circuit above, it can be seen that the opposition to current flow is made up of three components, XL, XC and R with the reactance, XT of any series RLC circuit being defined as: XT = XL – XC or XT = XC – XL whichever is greater. Thus the total impedance of the circuit being thought of as the voltage source required to drive a current through it.