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How to Understand Phase Differences Caused by Capacitors and Inductors？

2025-07-16

How to Understand Phase Differences Caused by Capacitors and Inductors
For sinusoidal signals, the phase of the current flowing through a component and the phase of the voltage across its two ends are not necessarily the same.

How do such phase differences arise? This knowledge is very important because phase must be considered not only for feedback signals of amplifiers and self-oscillators, but also when constructing a circuit, as we need to fully understand, utilize or avoid such phase differences. Let's discuss this issue below.

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

I. The Birth Process of Resistors, Inductors and Capacitors

After long-term observation and experiments, scientists have figured out some principles, and there are often some unexpected accidental discoveries, such as Roentgen's discovery of X-rays and Madame Curie's discovery of radium radiation. These accidental discoveries have even become great scientific achievements. The same is true in the field of electronics.
When scientists passed current through wires, they accidentally discovered wire heating and electromagnetic induction, and then invented resistors and inductors. Scientists also got inspiration from the phenomenon of frictional electrification and invented capacitors. The creation of diodes due to the discovery of rectification is also accidental.

II. Basic Working Principles of Components

Resistor — Electrical energy → Thermal energy
Inductor — Electrical energy → Magnetic field energy, & Magnetic field energy → Electrical energy
Capacitor — Electric 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.

Resistor
The principle of a resistor is: Electric 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 an electric potential is applied across a resistor, charges flow under the action of the potential difference — forming a current. Their flow speed is much faster than the disordered free movement without a potential difference, so the heat generated by collisions in the resistor or conductor is also more.

Positive charges enter the resistor from the end with high potential, and negative charges enter the resistor from the end with low potential. The two neutralize inside the resistor. The neutralization makes the number of positive charges in the resistor show a gradient distribution from the high-potential end to the low-potential end, and the number of negative charges in the resistor show a gradient distribution from the low-potential end to the high-potential end, thus 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 a linear resistor (the voltage drop is proportional to the current passing through it).
For AC signals, it is expressed as R = v(t)/i(t).

Note that there is also the concept of nonlinear resistors, whose nonlinearity includes voltage-influenced type, current-influenced type, etc.

Inductor
The principle of an inductor: Inductor — 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 across the inductor coil, charges flow under the action of the potential difference — forming a current, and the current is converted into a magnetic field, which is called the "magnetization" process. If the power supply potential difference across the magnetized inductor coil is removed and the inductor coil is connected to an external load, the magnetic field energy is 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), which is called the "demagnetization" process.

The unit to measure the magnetization of an inductor coil is flux linkage — Ψ. The larger the current, the more flux linkage the inductor coil is magnetized, that is, the flux linkage is proportional to the current, i.e., Ψ = L*I. For a specified inductor coil, L is a constant.

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

According to the principle of electromagnetic induction, the change of flux linkage generates an induced voltage, and the greater the change of flux linkage, the higher the induced voltage, i.e., v(t) = dΨ(t)/dt.

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

Capacitor
The principle of a capacitor: 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, positive and negative charges gather on the two plates of the capacitor respectively under the action of the potential difference to form an electric field, which is called the "charging" process. If the power supply potential difference across the charged capacitor is removed and the capacitor is connected to an external load, the charges across the capacitor flow out under its potential difference, which is called the "discharging" process. The flow of charges during the process of gathering to the capacitor and flowing out from the two plates of the capacitor forms a current.

It should be specially noted that the current on the capacitor does not mean that charges really flow through the insulating medium between the two plates of the capacitor, but only the flow formed by the gathering of charges from the outside to the two plates of the capacitor during charging and the flow formed by the flow of charges from the two plates of the capacitor to the outside during discharging. That is to say, the current of the capacitor is actually an external current, not an internal current, which is different from resistors and inductors.

The unit to measure the charging amount of a capacitor is the number of charges — Q. The greater the potential difference between the capacitor plates, the more charges the capacitor plates are charged, that is, the number of charges is proportional to the potential difference (voltage), i.e., Q = C*V. For a specified capacitor, C is a constant.

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

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

Since current is equal to the amount of change of charge per unit time, i.e., i(t) = dQ(t)/dt, combining the above two formulas, we get: i(t) = Cdv(t)/dt, that is, the capacitor current is proportional to the rate of change of its voltage (derivative with respect to time). The faster the voltage changes, the larger the current.
Summary: v(t) = Ldi(t)/dt
It shows that the change of current forms the induced voltage of the inductor (no induced voltage is formed if the current is constant).
i(t) = C*dv(t)/dt shows that the change of voltage forms the external current of the capacitor (actually the change of charge amount. No external current of the capacitor is formed if the voltage is constant).

III. The Change of Components to Signal Phase

First of all, it should be reminded that the concept of phase is for sinusoidal signals, and there is no concept of phase for DC signals, non-periodic change signals, etc.

The voltage and current on the resistor are in the same phase
Because the voltage on the resistor v(t) = Ri(t), if i(t) = sin(ωt + θ), then v(t) = Rsin(ωt + θ). Therefore, the voltage and current on the resistor are in the same phase.
The current on the inductor lags behind the voltage by 90° in phase
Because the induced voltage on the inductor v(t) = Ldi(t)/dt, if i(t) = sin(ωt + θ), then v(t) = Lcos(ωt + θ). Therefore, the current on the inductor lags behind the induced voltage by 90° in phase, or the induced voltage leads the current by 90° in phase.

Intuitive understanding: Imagine an inductor in series with a resistor for magnetization. From the perspective of the magnetization process, the change of the magnetizing current causes the change of flux linkage, and the change of flux linkage 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, making the phase of the magnetizing current lag behind the induced voltage.

The current on the capacitor leads the voltage by 90° in phase
Because the current on the capacitor i(t) = Cdv(t)/dt, if v(t) = sin(ωt + θ), then i(t) = Ccos(ωt + θ).
Therefore, the current on the capacitor leads the voltage by 90° in phase, or the voltage lags behind the current by 90° in phase.

Intuitive understanding: Imagine a capacitor in series with a resistor for charging. From the perspective of the charging process, the accumulation of flowing charges (i.e., current) always occurs 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 intuition:
v(t) = (1/C)*∫i(t)*dt = (1/C)*∫dQ(t), that is, the accumulation of charge change forms the voltage, so dQ(t) leads v(t) in phase; and the process of charge accumulation is the process of synchronous change of current, that is, i(t) is in phase with dQ(t). Therefore, i(t) leads v(t) in phase.

IV. Application of Component Phase Difference

— Understanding of RC Wien Bridge and LC Resonance Process
Both RC Wien bridge and LC series resonance and parallel resonance are caused by the phase difference between voltage and current of capacitors and/or inductors, just like the rhythm of mechanical resonance.

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

When two sinusoidal waves with the same frequency but opposite phases are superimposed, the amplitude of the superimposed wave will be reduced to the minimum, even zero. This is the principle of reducing or absorbing vibration, such as noise reduction equipment.

When there are multiple frequency signals mixed in a system, if two signals with the same frequency resonate, the energy of other vibration frequencies in the system will be absorbed by these two signals with the same frequency and phase, thus filtering other frequencies. This is the principle of resonant filtering in the circuit.

Resonance needs to meet two conditions: the same frequency and the same phase. The method of how the circuit selects the frequency through the amplitude-frequency characteristic has been discussed in the RC Wien bridge before. The idea of LC series and parallel is the same as that of RC, so it will not be repeated here.

Let's take a rough estimate of the phase compensation in circuit resonance (more accurate phase shift needs to be calculated)

Resonance of RC Wien Bridge
If there is no C2, the current of the sinusoidal signal Uo flows from C1→R1→R2, and the output voltage Uf is formed through the voltage drop on R2. Since the branch current is phase-shifted 90° ahead of Uo by capacitor C1, this current with leading phase flows through R2 (resistor does not produce phase shift!), making the output voltage Uf lead Uo by 90° in phase.

When C2 is connected in parallel with R2, C2 obtains voltage from R2. Due to the lagging effect of the capacitor on the voltage, the voltage on R2 is also forced to lag. (But it may not be 90°, because there is also the influence of C1→R1→C2 current on the voltage on C2, i.e., Uf, but at the RC characteristic frequency, the output phase of Uf is the same as that of Uo after connecting C2 in parallel.)
Summary: Connecting a capacitor in parallel makes the phase of the voltage signal lag, which is called parallel compensation of voltage phase.

LC Parallel Resonance
If there is no capacitor C, the sinusoidal signal u is induced to the secondary side through L to output Uf, and the Uf voltage leads u by 90°; when capacitor C is connected in parallel with the primary side of L, due to the lagging effect of the capacitor on the voltage, the voltage on L is also forced to lag by 90°. Therefore, the output phase of Uf is the same as that of u after connecting C in parallel.
LC Series Resonance
For the input sinusoidal signal u, the capacitor C makes the current on the load R in the series circuit lead u by 90° in phase, and the inductor L makes the current in the same series circuit lag by 90° in phase. The two phase shifts cancel each other exactly. Therefore, the output Uf is in phase with the input u.

Summary:

(Note that the phase influence is not necessarily 90°, which is related to other parts, and the specific needs to be calculated)
A series capacitor makes the current phase of the series branch lead, thus affecting the output voltage phase.
A parallel capacitor makes the voltage phase of the parallel branch lag, thus affecting the output voltage phase.
A series inductor makes the current phase of the series branch lag, thus affecting the output voltage phase.
A parallel inductor makes the voltage phase of the parallel branch lead, thus affecting the output voltage phase.

A more concise memory:
Capacitors make the current phase lead, and inductors make the voltage phase lead. (Both refer to the current or voltage on the component)
Capacitor — current leads, Inductor — voltage leads.