How does the phase difference between capacitors and inductors occur?

Jacktang

How does the phase difference between capacitors and inductors occur? [Copy link]

For sinusoidal signals, the current flowing through a component and the voltage across it are not necessarily in phase.

How does this phase difference come about? This knowledge is very important because not only does the feedback signal of an amplifier or self-excited oscillator need to consider the phase, but it is also necessary to fully understand, utilize, or avoid this phase difference when constructing a circuit. Let's explore this issue.

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

1. The birth process of resistors, inductors and capacitors

After long-term observation and experimentation, scientists have figured out some principles, and often make unexpected discoveries, such as Roentgen's discovery of X-rays and Marie Curie's discovery of the radiation phenomenon of radium. These accidental discoveries have become great scientific achievements. The same is true in the field of electronics.

When scientists let current flow through wires, they accidentally discovered the phenomenon of wire heating and electromagnetic induction, and then invented resistors and inductors. Scientists also got inspiration from the phenomenon of friction electrification and invented capacitors. The discovery of the rectification phenomenon and the creation of the diode were also accidental.

2. Basic working principle of components

Resistance - Electrical Energy → Thermal Energy

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

Capacitance - Potential energy → Electric field energy, & Electric field energy → Current

It can be seen that resistors, inductors, and capacitors are energy conversion components. Resistors and inductors realize the conversion between different types of energy, while capacitors realize the conversion between electric potential energy and electric field energy.

1. Resistance

The principle of resistance is: potential energy → current → thermal energy.

There is electric potential energy (positive and negative charges) stored at the positive and negative ends of the power supply. When the potential is applied to both ends of the resistor, the charges flow under the action of the potential difference, forming an electric current. The flow speed of the electric current is much faster than the disordered free motion when there is no potential difference, and the heat generated by the collision in the resistor or conductor is also greater.

Positive charges enter the resistor from the end with high potential, and negative charges enter the resistor from the end with low potential, and the two are neutralized inside the resistor.

The neutralization effect causes the number of positive charges inside the resistor to show a gradient distribution from the high potential end to the low potential end, and the number of negative charges inside the resistor to show a gradient distribution from the low potential end to the high potential end, thereby generating a potential difference across the resistor, which is the voltage drop of the resistor.

Under the same current, the greater the resistance of the resistor to neutralization, the greater the voltage drop across it.

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

For AC signals, it is expressed as R=v(t)/i(t).

Note: There is also the concept of nonlinear resistance, whose nonlinearity includes voltage-affected type, current-affected type, etc.

2. Inductor

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 inductor, the charges flow under the action of the potential difference, forming an electric current, which transforms the magnetic field. This is called the "magnetization" process.

If the power supply potential difference across the magnetized inductor is removed and a load is connected to the inductor, 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 resistor, it is current). This is called the "demagnetization" process.

The unit for measuring the magnetization of an inductor is the flux linkage - Ψ. The greater the current, the more flux linkage the inductor will have, that is, the flux linkage is proportional to the current, that is, Ψ=L*I. For a given inductor, L is a constant.

Therefore, L=Ψ/I is used to express the electromagnetic conversion capability of the inductor coil, and L is called the inductance. The differential expression of the inductance is: L=dΨ(t)/di(t).

According to the principle of electromagnetic induction, changes in magnetic flux generate induced voltage. The greater the change in magnetic flux, the higher the induced voltage, that is, v(t)=d dΨ(t)/dt.

Combining the above two formulas, we can get: v(t)=L*di(t)/dt, that is, the induced voltage of the inductor is proportional to the rate of change of the current (the derivative with respect to time). The faster the current changes, the higher the induced voltage.

3. Capacitor

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

When the power supply potential is applied to the two metal plates of the capacitor, the positive and negative charges gather 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 at both ends of the charged capacitor is removed and the capacitor is connected to an external load, the charges at both ends of the capacitor will flow outward under the potential difference. This is called the "discharging" process.

As charges gather toward the capacitor and flow out from the two plates of the capacitor, the flow of charges forms an electric current.

It is important to note that the current on a capacitor is not the charge actually flowing through the insulating medium between the two plates of the capacitor, but only the flow of charge from the outside to the two plates of the capacitor during the charging process, and the flow of charge from the two plates of the capacitor to the outside during the discharging process. In other words, the current of a capacitor is actually an external current, not an internal current, which is different from resistance and inductance.

The unit of measurement for how much a capacitor is charged is the charge number - Q. The greater the potential difference between the capacitor plates, the more charge is applied to the capacitor plates, i.e. the charge number is proportional to the potential difference (voltage), i.e. Q = C*V. For a given capacitor, C is a constant.

Therefore, C=Q/V is used to express the ability of the capacitor plates to store charge, and C is called capacitance.

The differential expression of capacitance is: C=dQ(t)/dv(t).

Because the current is equal to the change in the number of charges per unit time, that is, i(t)=dQ(t)/dt, combining the above two formulas, we can get: i(t)=C*dv(t)/dt, that is, the capacitor current is proportional to the rate of change of the voltage on it (the derivative with respect to time). The faster the voltage changes, the greater the current.

4. Summary: v(t)=L*di(t)/dt

This indicates that the change in current forms an induced voltage in the inductor (no induced voltage is formed if the current remains unchanged).

i(t)=C*dv(t)/dt indicates that the voltage change forms an external current of the capacitor (actually it is a change in the amount of charge. If the voltage remains unchanged, no external current of the capacitor is formed).

3. Changes in signal phase caused by components

First of all, we should remind you that the concept of phase applies to sinusoidal signals. DC signals, non-periodic changing signals, etc. do not have the concept of phase.

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

Because the voltage on the resistor is v(t)=R*i(t), if i(t)=sin(ωt+θ), then v(t)=R* sin(ωt+θ). Therefore, the voltage on the resistor is in phase with the current.

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

Because the induced voltage on the inductor v(t)=L*di(t)/dt, if i(t)=sin(ωt+θ), then v(t)=L*cos(ωt+θ). Therefore, the current on the inductor lags behind the induced voltage by 90°, or the induced voltage leads the current by 90°.

Intuitive understanding: Imagine an inductor and a resistor are connected in series to magnetize. From the magnetization process, the change of magnetization current causes the change of magnetic flux, and the change of magnetic flux generates induced electromotive force and induced current.

According to Lenz's law, the direction of the induced current is opposite to that of the magnetizing current, which delays the change of the magnetizing current and makes the magnetizing current lag behind the induced voltage in phase.

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

Because the current on the capacitor i(t)=C*dv(t)/dt, if v(t)=sin(ωt+θ), then i(t)=L*cos(ωt+θ).

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

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

The following integral equation captures this intuition:

v(t)=(1/C)*∫i(t)*dt=(1/C)*∫dQ(t), that is, the accumulation of charge changes forms voltage, so dQ(t) is ahead of v(t); and the process of charge accumulation is the process of synchronous current change, that is, i(t) and dQ(t) are in phase. Therefore, i(t) is ahead of v(t).

4. Application of Component Phase Difference

Understanding of the RC Wien bridge and LC resonance process: Whether it is the RC Wien bridge or the series resonance and parallel resonance of LC, they are all caused by the voltage and current phase difference of the capacitor and/or inductor components, just like the beat of mechanical resonance.

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

When two sine waves with the same frequency and opposite phases are superimposed, the amplitude of the superimposed wave will be reduced to the minimum or even zero. This is the principle of reducing or absorbing vibrations, 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 these two signals of the same frequency and phase, thus filtering other frequencies. This is the principle of resonant filtering in circuits.

Resonance requires that both the frequency and phase be the same. The method of selecting the frequency through the amplitude-frequency characteristic of the circuit has been discussed before in the RC Wien bridge. The idea of LC series-parallel connection is the same as that of RC, so I will not repeat it here.

Let's look at a rough estimate of the phase compensation in the circuit resonance (a more precise 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 flows from C1→R1→R2, and forms the output voltage Uf through the voltage drop on R2. Since the branch current is phase-shifted by capacitor C1 to lead Uo by 90°, this leading current flows through R2 (resistance does not produce phase shift!), making the output voltage Uf lead Uo by 90°.

C2 is connected in parallel to R2, and C2 obtains voltage from R2. Due to the hysteresis effect of capacitor on voltage, the voltage on R2 is also forced to lag. (But it is not necessarily 90°, because there is also the influence of C1→R1→C2 current on the voltage on C2, i.e. Uf. However, at the RC characteristic frequency, after C2 is connected in parallel, the output phase of Uf is the same as Uo.) Summary: Parallel capacitors cause the voltage signal phase to lag, which 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 to the secondary output Uf through L, and the voltage of Uf is 90° ahead of u; when the capacitor C is connected in parallel to the primary of L, the voltage on L is also forced to lag 90° due to the lag effect of the capacitor on the voltage. Therefore, after connecting C in parallel, the output phase of Uf is the same as that of 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 by 90°, and the inductor L makes the current phase in the same series circuit lag by another 90°. The phase shifts of the two are exactly offset.

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

4. Summary:

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

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

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

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

A simpler way to remember: Capacitors make the current phase advance, and inductors make the voltage phase advance. (Both refer to the current or voltage on the component) Capacitors - current advance, inductors - voltage advance.