How is the phase difference between capacitance and inductance produced?

For a sinusoidal signal, the phase of the current flowing through a component and the voltage across it is not necessarily the same.

For a sinusoidal signal, the phase of the current flowing through a component and the voltage across it is not necessarily the same.

How is this phase difference produced? This kind of knowledge is very important, because not only the phase of the feedback signal of the amplifier and free-running oscillator must be considered, but also the phase difference must be fully understood, utilized or avoided when constructing a circuit. Let’s explore this issue.

First, we must understand how some components are constructed; second, we must understand the basic working principles of circuit components; third, find and understand the reasons for the phase difference based on this; fourth, use the phase difference characteristics of the components to construct some basic Circuit.

1. The birth process of resistance, inductance and capacitance

After long-term observation and experimentation, scientists have figured out some truths, and some unexpected accidental discoveries often appear, such as Roentgen’s discovery of X-rays and Madame Curie’s discovery of radium radiation. These accidental discoveries have turned out to be great. Scientific achievements. The same is true in the field of electronics.

When scientists let current flow through the wires, they accidentally discovered the heating and electromagnetic induction of the wires, and then invented resistance and inductance. Scientists also got inspiration from the phenomenon of frictional electrification and invented the capacitor. The discovery of rectification and the creation of diodes is also accidental.

2. The basic working principle of components

Resistance-electric energy → thermal energy

Inductance-electric energy → magnetic field energy, & magnetic field energy → electric energy

Capacitance-electric potential energy → electric field energy, & electric field energy → current

It can be seen that resistance, inductance, and capacitance are the components of energy conversion. Resistance and inductance realize the conversion between different types of energy, and the capacitor realizes the conversion between electric potential energy and electric field energy.

1. Resistance

The principle of resistance is: electric potential energy → current → heat energy.

The positive and negative ends of the power supply store potential energy (positive and negative charges). When the potential is applied to both ends of the resistor, the charge flows under the action of the potential difference-forming a current, and its flow rate is much faster than the disorderly free movement when there is no potential difference. The heat generated by collisions in resistors or conductors is also more.

The positive charge enters the resistor from the high potential end, and the negative charge enters the resistor from the low potential end. The two neutralize inside the resistor.

The neutralization effect makes the amount of positive charge present a gradient distribution from the high potential end to the low potential end inside the resistor, and the amount of negative charge presents a gradient distribution from the low potential end to the high potential end inside the resistor, thereby generating a potential difference between the two ends of the resistor. It is the voltage drop of the resistor.

Under the same current, the greater the resistance of the resistance to neutralization, the greater the voltage drop across its terminals.

Therefore, R=V/I is used to measure the resistance of linear resistance (the voltage drop is proportional to the current passed).

For AC signals, it is expressed as R=v

Note: There is also the concept of non-linear resistance. Its non-linearity includes voltage-influenced and current-influenced types.

2. Inductance

The principle of inductance: inductance-electric potential energy → current → magnetic field energy, & magnetic field energy → electric potential energy (if there is a load, then → current).

When the power supply potential is applied to both ends of the inductance coil, the charge flows under the action of the potential difference-a current is formed, and the current transforms into a magnetic field. This is called the “magnetization” process.

If the power supply potential difference between both ends of the magnetizing inductance coil is cancelled, and there is a load external to the inductance coil, the magnetic field energy will be converted into electrical energy during the attenuation process (if the load is a capacitor, it is electric field energy; if the load is a resistance, it is current) This is called the “demagnetization” process.

The unit to measure how much magnetizing the Inductor coil is is the flux linkage-Ψ. The greater the current, the more the inductance coil is impacted by the flux, that is, the flux is proportional to the current, that is, Ψ=L*I. For a given inductance coil, L is a constant.

Therefore, use L=Ψ/I to express the electromagnetic conversion capability of the inductance coil, and call L as the inductance. The differential expression of inductance is: L=dΨ

According to the principle of electromagnetic induction, the change of the flux linkage produces an induced voltage. The greater the flux linkage changes, the higher the induced voltage, that is, v

Combining the above two formulas, we get: v

3. Capacitance

The principle of capacitance: electric potential energy → electric current → electric field energy, electric field energy → electric current.

When the power supply potential is applied to the two metal plates of the capacitor, the positive and negative charges accumulate to the two plates of the capacitor under the action of the potential difference to form an electric field. This is called the “charging” process. If the power supply potential difference between the two ends of the charged capacitor is cancelled, and the capacitor is connected with a load, the charge on both ends of the capacitor will flow away under its potential difference. This is called a “discharge” process.

In the process of accumulating the charge in the capacitor and flowing out from the two plates of the capacitor, the flow of the charge forms a current.

Special attention should be paid to the fact that the current on the capacitor does not actually flow through the insulating medium between the two plates of the capacitor, but only the flow formed by the accumulation of the charge from the outside to the two plates of the capacitor during the charging process, and the charge from the outside during the discharge process. The flow formed by the two plates of a capacitor flowing outwards. In other words, the current of the capacitor is actually an external current, not an internal current, which is different from resistance and inductance.

The unit to measure how much the capacitor is charged is the number of charges-Q. The greater the potential difference between the capacitor plates, the more the capacitor plates are charged, that is, the number of charges is proportional to the potential difference (voltage), that is, Q=C*V. For the specified capacitance, C is a constant.

Therefore, use C=Q/V to express the capacity of the capacitor plate to store charges, and call C the capacitance.

The differential expression of capacitance is: C=dQ

Because the current is equal to the change in the number of charges per unit time, that is, i

4. Summary: v

It shows that the current change forms the induced voltage of the inductance (no induced voltage is formed if the current does not change).

i

Third, the component changes to the signal phase

First of all, we must remind that the concept of phase is for sinusoidal signals, and there is no concept of phase for DC signals and non-periodic signals.

1. The voltage and current on the resistor are in phase

Because the voltage on the resistor v

2. The current on the inductor lags behind the voltage by 90° phase

Because the induced voltage on the inductor v

Intuitive understanding: Imagine an inductance and resistance series magnetization. From the perspective of the magnetization process, the change of the magnetizing current causes the change of the flux linkage, and the change of the flux linkage generates the induced electromotive force and the induced current.

According to Lenz’s law, the direction of the induced current is opposite to the magnetizing current, which delays the change of the magnetizing current, making the phase of the magnetizing current lag behind the induced voltage.

3. The current on the capacitor leads the voltage by 90° phase

Because the current on the capacitor i

Therefore, the current on the capacitor leads the voltage by 90° phase, or the voltage lags the current by 90° phase.

Intuitive understanding: Imagine a capacitor and resistor are charged in series. From the perspective of the charging process, there is always the accumulation of flowing charge (that is, current) before the voltage change on the capacitor, that is, the current always leads the voltage, or the voltage always lags behind the current.

The following integral equation can reflect this intuitiveness:

v

Fourth, the application of component phase difference

Understanding of the RC Wien bridge and LC resonance process: No matter the RC Wien bridge, or the series resonance or parallel resonance of the LC, it is caused by the voltage and current phase difference of the capacitor or/and the inductance and capacitance components, just like mechanical resonance. The beat is the same.

When two sine waves with the same frequency and phase are superimposed, the amplitude of the superimposed wave reaches the maximum value. This is a resonance phenomenon, which is called resonance in the circuit.

When two sine waves with the same frequency and opposite phase are superimposed, the amplitude of the superimposed wave will be minimized, or even zero. This is the principle of reducing or absorbing vibration, such as noise reduction equipment.

When multiple frequency signals are mixed in a system, if two signals of the same frequency resonate, the energy of other vibration frequencies in the system will be absorbed by the two signals of the same frequency and phase, thus acting as a countermeasure. The filtering effect of other frequencies. This is the principle of resonance filtering in the circuit.

Resonance needs to satisfy the two conditions of the same frequency and the same phase at the same time. How the circuit selects the frequency through the amplitude-frequency characteristic was discussed in the RC Wien Bridge before. The idea of ​​LC series-parallel connection is the same as that of RC, so I won’t repeat it here.

Let’s take a look at a rough estimate of the phase compensation in the circuit resonance (a more accurate phase offset needs to be calculated)

1. Resonance of RC Wien Bridge (Figure 1)

If there is no C2, the current of the sinusoidal signal Uo changes from C1→R1→R2, and the Uf output voltage is formed through the voltage drop on R2. Since the branch current is phase-shifted by the capacitor C1 to lead Uo 90°, the current of the lead phase flows through R2 (the resistance does not produce a phase shift!), making the output voltage Uf lead Uo 90°.

Connect C2 in parallel with R2, and C2 gets the voltage from R2. Due to the hysteresis of the capacitor on the voltage, the voltage on R2 is also forced to hysteresize. (But not necessarily 90°, because there is also the influence of C1→R1→C2 current on the voltage on C2 that is Uf, but at the RC characteristic frequency, the output phase of Uf is the same as Uo after C2 is connected in parallel.) Summary: Parallel capacitors make the voltage Signal phase lag is called parallel compensation of voltage phase. 2. LC parallel resonance (Figure 2)

If there is no capacitor C, the sinusoidal signal u is induced by L to the secondary output Uf, and the voltage of Uf leads u by 90°; in the primary parallel capacitor C of L, the voltage on L is also forced to lag 90° due to the hysteresis of the capacitor on the voltage. . Therefore, after paralleling C, the output phase of Uf is the same as u.

3. LC series resonance (Figure 3)

For the input sinusoidal signal u, the capacitor C makes the current phase on the load R in the series circuit lead u 90°, and the inductance L makes the current phase in the same series circuit lag 90° and the phase shifts just cancel out.

Therefore, the output Uf is in phase with the input u.

4. Summary:

(Note that the phase effect is not always 90°, it is related to other parts, and it needs to be calculated.) The series capacitor makes the phase of the series branch current lead, thereby affecting the output voltage phase.

The parallel capacitor makes the voltage phase of the parallel branch lag, thereby affecting the output voltage phase.

The series inductance makes the phase of the series branch current lag, thereby affecting the output voltage phase.

The parallel inductance makes the parallel branch branch voltage lead, thereby affecting the output voltage phase.

More concise memory: Capacitors lead the current phase, and inductors lead the voltage phase. (Both refer to the current or voltage on the component) Capacitance-current leads, inductance-voltage leads.