Phase Relationships in AC Circuits

Hello! Welcome to the third lesson in our module on AC circuits.

In our last lesson, we established the distinct behaviors of resistors, inductors, and capacitors in an AC circuit. We saw that while resistors keep voltage and current in lockstep, inductors and capacitors introduce a 90° phase shift, with voltage leading current in inductors (ELI) and current leading voltage in capacitors (ICE).

Today, we'll build on that foundation to achieve this lesson's learning outcome: to analyze the phase relationship between voltage and current in resistive, inductive, and capacitive loads. We'll move from individual components to combined loads (like RL, RC, and RLC circuits) and see how their individual phase shifts combine to produce an overall phase relationship for the entire circuit. To do this, we'll introduce a powerful graphical tool: the phasor diagram.

A Quick Recap and the Challenge Ahead

Let's quickly summarize what we know for series circuits:

• Resistor: Voltage across it, $V_R$, is in phase with the current $I$.
• Inductor: Voltage across it, $V_L$, leads the current $I$ by 90°.
• Capacitor: Voltage across it, $V_C$, lags the current $I$ by 90°.

The challenge is this: if we connect these components in series, the total voltage from the source, $V_S$, is the sum of the individual voltages: $V_S=V_R+V_L+V_C$. However, because these voltages are not in phase, we cannot simply add their magnitudes. We need a way to account for their different directions in time.

Phasors: A Graphical Tool for AC Analysis

This is where phasors come in. A phasor is a vector that rotates counter-clockwise in the complex plane. Its length represents the amplitude (peak or RMS value) of a sinusoidal wave, and its angle relative to the positive real axis represents its phase. The instantaneous value of the wave is the projection of this rotating vector onto the vertical or horizontal axis.

A phasor is a rotating vector. Its projection on an axis gives the instantaneous value of a sinusoidal quantity. Its length represents the amplitude and its angle represents the phase.

Your background in linear algebra will make this concept feel natural. We are essentially treating sinusoidal quantities as vectors in a 2D space. This turns the problem of solving the circuit's differential equation into a more straightforward problem of vector addition.

Building the Phasor Diagram for a Series RLC Circuit

In a series circuit, the current $I$ is the same through all components. This makes it the ideal reference for our phasor diagram. We'll place the current phasor along the positive horizontal axis. Then, we add the voltage phasors for each component, respecting their phase relationships to the current.

This process is best understood by seeing it built step-by-step.

As you saw in the video, the key steps are:

1. Draw the current phasor $I$ as the horizontal reference.
2. Draw the resistor voltage phasor $V_R$ in phase (parallel) with $I$.
3. Draw the inductor voltage phasor $V_L$ leading $I$ by 90° (pointing straight up).
4. Draw the capacitor voltage phasor $V_C$ lagging $I$ by 90° (pointing straight down).

Since $V_L$ and $V_C$ are in direct opposition (180° out of phase), their net effect is a simple subtraction: $V_L-V_C$. The total source voltage $V_S$ is then the vector sum of $V_R$ and this net reactive voltage $(V_L-V_C)$.

This vector addition forms a right-angled triangle, often called the voltage triangle.

The geometry of the voltage triangle is the key to analyzing the overall phase relationship. The angle $\varphi$ between the total voltage phasor $V_S$ and the current phasor $I$ is the phase angle of the entire circuit.

Interpreting the Overall Phase Relationship

The overall behavior of the RLC circuit—whether it acts more like an inductor or a capacitor—depends on which of the two reactances is dominant at the given frequency.

To summarize the three cases:

1. If $X_L>X_C$: The inductive reactance is dominant. This means $V_L>V_C$. The net reactive voltage $(V_L-V_C)$ is positive (points up). The total voltage $V_S$ leads the current $I$. The circuit has an inductive character.
2. If $X_C>X_L$: The capacitive reactance is dominant. This means $V_C>V_L$. The net reactive voltage is negative (points down). The total voltage $V_S$ lags the current $I$. The circuit has a capacitive character.
3. If $X_L=X_C$: The reactances cancel each other out completely. The net reactive voltage is zero. The total voltage $V_S$ is in phase with the current $I$. The circuit behaves as if it were purely resistive. This special condition is called resonance.

Quantifying the Phase Angle

From the geometry of the voltage triangle, we can define the phase angle $\varphi$ using basic trigonometry:
$\tan (\varphi )=\frac{\text{Opposite}}{\text{Adjacent}}=\frac{V_L-V_C}{V_R}$
Since $V_R=IR$, $V_L=I{X}_L$, and $V_C=I{X}_C$, we can substitute these in and cancel the common current $I$:
$\tan (\varphi )=\frac{I(X_L-X_C)}{IR}=\frac{X_L-X_C}{R}$
This gives us a direct way to calculate the phase angle from the resistance and reactances:
$\varphi =\arctan \left(\frac{X_L-X_C}{R}\right)$
A positive angle $\varphi$ indicates an inductive circuit (voltage leads current), while a negative angle indicates a capacitive circuit (voltage lags current).

A Complete Example

Let's walk through a concrete example to see how these concepts are applied to analyze a circuit.

In this example, since $X_L>X_C$, the final phase angle is positive ($+51.8^{\circ }$), confirming the circuit is inductive, and the current lags the source voltage. The phasor diagram drawn at the end provides a perfect visual summary of this result.

Conclusion

Today, we've developed a powerful method for analyzing AC circuits. By representing sinusoidal voltages and currents as phasors, we can use simple vector addition to understand their combined effect.

Key Takeaways:

• Phasors are rotating vectors that allow us to represent the magnitude and phase of AC quantities.
• In a series circuit, the total voltage is the phasor sum of the individual component voltages.
• The voltage triangle is a geometric representation of the relationship between resistive voltage ($V_R$), net reactive voltage ($V_L-V_C$), and total source voltage ($V_S$).
• The overall phase relationship is determined by the balance between inductive reactance ($X_L$) and capacitive reactance ($X_C$). The circuit behaves inductively if $X_L>X_C$, capacitively if $X_C>X_L$, and resistively if $X_L=X_C$.
• The phase angle $\varphi$ can be calculated using $\varphi =\arctan \left(\frac{X_L-X_C}{R}\right)$.

Preview of the Next Lesson:

The phasor diagram is an excellent visualization tool, but it can be cumbersome for complex calculations. In the next lesson, we will formalize this vector approach by using complex numbers to represent impedance. This will allow us to calculate the total impedance of series RLC circuits using simple arithmetic, providing a more powerful and efficient analytical technique. This directly parallels how you might use complex numbers in other areas of mathematics and physics to handle both magnitude and phase simultaneously.