Technical dry goods! Understand the power supply technology in one article: RLC circuit analysis

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Technical dry goods! Understand the power supply technology in one article: RLC circuit analysis [Copy link]

In a pure ohmic resistor, the voltage waveform is "in phase" with the current. In a pure inductor, the voltage waveform "leads" the current by 90o, giving the following expression: ELI. In a pure capacitor, the voltage waveform "lags" the current by 90o, giving the following expression: ICE.

This phase difference Φ depends on the reactance values of the components used, and hopefully now we know that if the circuit element is resistive, then the reactance (X) is zero, if it is inductive, then the reactance (X) is positive, and if it is capacitive, then it is negative. This gives them their final impedances as:

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Element impedance

Instead of analyzing each passive component individually, we can combine all three passive components together to form a series RLC circuit. The analysis of a series RLC circuit is the same as the analysis of the dual series RL and RC circuits we studied previously, only this time we need to consider the magnitude of XL and XC in order to find the overall circuit reactance. Series RLC circuits are classified as secondary circuits because they contain two energy storage elements, the inductor L and the capacitor C^. Consider the RLC circuit below.

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Series RLC Circuit

The series RLC circuit above has a loop where the instantaneous current flowing through each loop element is the same. Since the inductive and capacitive reactances, XL and XC, are functions of the supply frequency, the sinusoidal response of the series RLC circuit will vary with frequency . The voltage drops across each circuit element of the R, L, and C elements will then be "out of phase" with each other, defined as follows:

• I(t) = Imax sin(ωt)

• The instantaneous voltage VR on a pure resistor is "in phase" with the current

• The instantaneous voltage VL across a pure inductor "leads" the current by 90o

• The instantaneous voltage VC on a pure capacitor causes the current to "lag" by 90 degrees.

• Therefore, VL and VC are 180° out of phase and opposite to each other.

For the series RLC circuit above, this can be shown to be:

The source voltage magnitude across all three components in the series RLC circuit consists of three individual component voltages VR, VL and VC, with the current common to all three components. Therefore, the vector diagram will have the current vector as its reference, with the three voltage vectors plotted relative to this reference, as shown below.

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Individual voltage vectors

This means that we cannot simply add VR, VL, and VC together to find the supply voltage VS between all three components because all three voltage vectors point in different directions relative to the current vector. Therefore, we must find the supply voltage VS as the phasor sum of the three component voltages combined together as vectors.

Kirchhoff's voltage law (KVL) for loop and node circuits states that around any closed loop, the sum of the voltage drops around the loop is equal to the sum of the EMFs. Applying this law to these three voltages would then give us the magnitude of the supply voltage, Vs.

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Instantaneous voltage of a series RLC circuit

The phasor diagram for a series RLC circuit is obtained by combining the three phasors above and adding them vectorially. Since the current flowing through the circuit is common to all three circuit elements, we can use this as a reference vector and draw the three voltage vectors relative to this at the corresponding angles.

The resulting vector V S is obtained by adding together the two vectors, V L and V C , and then adding this sum to the remaining vector V R . The resulting angle obtained between V S and I will be the phase angle of the circuit as shown below.

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Phase diagram of series RLC circuit

As can be seen from the phasor diagram above, the voltage vector creates a rectangular triangle consisting of the hypotenuse VS, the horizontal axis VR and the vertical axis VL – VC. Hopefully you will notice that this forms the previously favorite voltage triangle, so we can use the Pythagorean theorem on this voltage triangle to mathematically obtain the value of VS as shown in the diagram.

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Voltage Triangle of Series RLC Circuit

Please note that when using the above formula, the final reactive voltage value must always be positive, that is, the minimum voltage must always be subtracted from the maximum voltage, we cannot add a negative voltage to VR, so the correct answer is VL - Vc or Vc - VL. Subtract the minimum value from the maximum value, otherwise VS cannot be calculated.

From the above we know that the current has the same amplitude and phase in all components of the series RLC circuit. Then, we can also mathematically describe the voltage across each component in terms of the current flowing through it and the voltage across each component.

By substituting these values into the voltage triangle in the Pythagorean equation above, we obtain:

Thus, we can see that the magnitude of the supply voltage is directly proportional to the magnitude of the current flowing through the circuit. This constant of proportionality is called the impedance of the circuit, which ultimately depends on the resistance and the magnitude of the inductive and capacitive reactances.

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 and the reactance, XT of any RLC series circuit is defined as: XT = XL – XC or XT = XC – XL whichever is greater. Therefore, the total impedance of the circuit is considered to be the voltage source required to drive current through it.

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Impedance of a Series RLC Circuit

Since the three vector voltages are out of phase with each other, XL, XC, and R must also be "out of phase" with each other, and the relationship between R, XL, and XC is the vector sum of these three components. This will give us the total impedance of our RLC circuit, Z. These circuit impedances can be plotted and represented by an impedance triangle as shown below.

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Impedance Triangle of a Series RLC Circuit

The impedance Z of a series RLC circuit depends on the angular frequency ω, XL is the same as XC. If the capacitive reactance is greater than the inductive reactance XC > XL, then the overall circuit reactance is capacitive, giving a leading phase angle.

Likewise, if the inductive reactance is greater than the capacitive reactance XL > XC then the overall circuit reactance is inductive giving the series circuit a lagging phase angle. If the two reactances are the same, XL = XC then the angular frequency at which this occurs is known as the resonant frequency and produces an effect called resonance which we will look at in more detail in another tutorial.

The magnitude of the current then depends on the frequency applied to the series RLC circuit. When the impedance Z is at its maximum, the current is at its minimum, and similarly, when Z is at its minimum, the current is at its maximum. Therefore, the above impedance formula can be rewritten as:

The phase angle θ between the source voltage VS and the current i is the same as the angle between Z and R in the impedance triangle. The value of this phase angle can be positive or negative, depending on whether the source voltage leads or lags the circuit current, and can be calculated mathematically from the ohm values of the impedance triangle as follows:

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Series RLC Circuit Example 1

A series RLC circuit consisting of a 12Ω resistor, a 0.15H inductor and a 100uF capacitor is connected across a 100V, 50Hz power supply. Calculate the total circuit impedance, circuit current, power factor and draw the voltage phasor diagram.

Inductive reactance, X-Large.

Capacitive reactance, X C.

Circuit impedance, z.

The current in the circuit, I.

The voltages across the series RLC circuit, VR, VL, Vc.

Circuit power factor and phase angle θ.

Phase diagram.

Since the phase angle θ is calculated as a positive value of 51.8 the total reactance of the circuit must be inductive. Since we have the current vector as the reference vector in the series RLC circuit, the current "lags" the supply voltage by 51.8o, so we can say that the phase angle lags, which is confirmed by the mnemonic "ELI".

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Series RLC Circuit Summary

In a series RLC circuit containing a resistor, an inductor, and a capacitor, the supply voltage VS is the sum of the phasors consisting of three components VR, VL, and VC, and these three components have a common current. Since the current is common to all three components, it is used as a horizontal reference when constructing the voltage triangle.

The impedance of a circuit is the direction completely opposite to the flow of current. For a series RLC circuit, the impedance triangle can be drawn by dividing each side of the voltage triangle by its current I. The voltage drop across the resistive element is equal to I*R, the voltage across the two reactive elements is I*X = I*XL - I*XC and the source voltage is equal to I*Z. The angle between VS and I will be the phase angle θ.

When using a series RLC circuit containing multiple pure resistors or resistors, capacitors, or inductors, they can all be added together to form a single component. For example, all resistors added together, RT = (R1 + R2 + R3) ... and so on, or all inductors LT = (L1 + L2 + L3) ... and so on, so that a circuit containing many elements can be easily reduced to a single impedance.