Calculating Impedance and Complex Impedance

Impedance can be defined as the full resistance that an AC circuit presents against the passage of alternating current, and it is measured in Ohms.

AC Circuits and Impedance

In circuits carrying alternating current (AC), the opposition to current flow is called impedance. Its measured in ohms (Ω) and represents the combined effect of all the components within the circuit that limit current flow. These components include:

• Resistance (R): This is the familiar opposition to current present in both AC and DC circuits.
• Inductance (L): This property opposes changes in current by inducing a voltage in the conductor itself.
• Capacitance (C): This property stores electrical energy and can also oppose changes in current.

Impedance vs. Resistance

While resistance (R) solely determines opposition in a direct current (DC) circuit, impedance (Z) accounts for both resistive and reactive elements in an AC circuit.

Reactive elements are inductance (L) and capacitance (C), which influence current flow but dont dissipate energy as heat like resistance.

Impedance Units and Ohm’s Law

Similar to resistance in DC circuits, impedance in AC circuits is also expressed in ohms (Ω). For convenience we can use multiples and submultiples of ohms depending on the impedance value.

These commonly used prefixes include:

• Microhms (μΩ or 10^-6) for very small values
• Milliohms (mΩ or 10^-3) for small values
• Kilohms (kΩ or 10³) for large values
• Megohms (MΩ or 10⁶) for very large values

Impedance can be calculated using Ohms Law, which applies to both AC and DC circuits. The formula is:

Z = V / I, or I = V / Z, or V = I * Z

Where:

• Z is the impedance in ohms (Ω)
• V is the voltage in volts (V)
• I is the current in amperes (A)

We previously established that impedance (Z) combines the effects of resistance (R) and reactance (X) in an AC circuit. However there is another layer to the story: frequency dependence.

Impedance and Phase Angle

Impedance is influenced by the frequency of the AC current. This introduces a concept called the phase angle, which describes how much the voltage and current waveforms are out of sync.

The reactance (inductive or capacitive) always has a phase angle of 90 degrees relative to the resistance.

Why We Cant Simply Add R and X

Since resistance and reactance have different phase angles, simply adding their values (R + X) wouldn’t accurately represent the overall impedance.

Resistors and In-Phase Behavior

It’s important to note that resistors are special. Their opposition to current (resistance) remains constant regardless of frequency, meaning they have no reactance (except for some special cases like wound resistors).

As a result, for resistors, impedance (Z) is directly equal to resistance (R) and they have no phase angle. This translates to voltage and current through a resistor being perfectly “in-phase,” meaning their peaks and troughs occur at the same time.

Frequencys Impact and Impedance Plots

Unlike resistors, inductors (X_L) and capacitors (X_C) exhibit reactance that changes with frequency. This variation causes the circuit’s overall impedance (Z) to change as the AC supply frequency changes.

To emphasize this distinction, terms like “resistive impedance” (for resistors) and “reactive impedance” (for inductors and capacitors) are sometimes used in AC circuit analysis.

Why Direct Addition Doesn’t Work

Since resistance and reactance have a 90-degree phase difference (like values at right angles), simply adding their numerical values (R + X) wouldnt accurately represent the total impedance.

Visualizing Impedance with a Graph

To address this challenge, we can plot impedance on a two-dimensional graph. The horizontal axis (x-axis) represents the resistive or “real” component, while the vertical axis (y-axis) represents the reactive or “imaginary” component.

This method is similar to constructing a right triangle, where the two sides represent resistance and reactance, and the hypotenuse represents the total impedance. The key difference is that the reactive component is plotted on the vertical axis to account for its 90-degree phase difference from resistance.

In the following diagram, the resistance is plotted on the horizontal axis, and the reactance is plotted on the vertical axis. The total impedance of the circuit is then represented by the length of the hypotenuse (the longest side) of the triangle. This approach highlights the fact that resistance and reactance contribute to the overall impedance at a 90-degree angle to each other.

Pythagoras and Impedance Calculations

Since the relationship between resistance, reactance, and impedance forms a right triangle, we can leverage Pythagoras theorem to calculate the missing side. Heres the equation based on Pythagoras theorem applied to impedance, resistance, and reactance:

Z² = R² + X²

where:

• Z is the impedance (hypotenuse)
• R is the resistance (horizontal side)
• X is the reactance (vertical side)

This approach demonstrates that, impedance (Z) is the resultant vector sum of the resistance vector (R) and the reactance vector (XL). Because, resistance has no phase shift, it’s typically represented by a horizontal vector. The resulting impedance (Z) has a positive slope due to the inductive reactance (X_L) contributing a positive imaginary component.

Understanding Impedance in RL Circuits

• R = R
• X_L = 2πfL
• Z² = R² + X²L
• Z = √(R² + X²_L)

The phase angle (φ), represented in degrees, depict the angular difference between the two vectors as illustrated below.

The Angular Shift: Phase Angle in RL Circuits

• tanΦ = X / R = X_L / R
• ∴ Φ = tan^-1(X_L / R)
• sinΦ = X / Z = X_L / Z
• ∴ Φ = sin^-1(X_L / Z)
• cosΦ = R / Z = R / Z
• ∴ Φ = cos^-1(R / Z)

Similar to inductors, capacitors also introduce a complex impedance in AC circuits. We can utilize the same right angled graph to visualize how resistance (R) and capacitive reactance (X_C) combine. The hypotenuse, representing the total impedance, remains the longest side of the triangle.

However for capacitors, the key difference lies in the direction of the reactance vector (X_C). Unlike the positive slope of inductive reactance, capacitive reactance has a negative slope. This graphical representation highlights that the effect of capacitors on an AC circuit opposes that of inductors. In other words capacitors tend to counteract the current limiting effect of inductors.

Impedance with Capacitive Reactance

Just like with inductors, we can again use Pythagoras theorem to analyze circuits with capacitors. The theorem relates the sides of the right-angled triangle representing resistance (R) and capacitive reactance (X_C) to the total impedance (Z), the hypotenuse. The equation based on Pythagoras theorem for AC circuits is:

• Z² = R² + X²_C
• Z = √(R² + X²_C)

where:

• Z is the impedance (hypotenuse)
• R is the resistance (horizontal side)
• X_C is the capacitive reactance (vertical side with a negative slope)

The phase angle (φ) measured in degrees, represents the angular difference between the voltage and current waveforms in an AC circuit. This angle is related to the impedance (Z) and resistance (R) of the circuit.

Calculating RC Circuit Phase Angle

• tanΦ = X / R = X_C / R
• ∴ Φ = tan^-1(X_C / R)
• sinΦ = X / Z = X_C / Z
• ∴ Φ = sin^-1(X_C / Z)
• cosΦ = R / Z = R / Z
• ∴ Φ = cos^-1(R / Z)

Vector diagrams effectively illustrates, how resistance and reactance (both inductive and capacitive) combine to form the overall impedance of an AC circuit.

Notably, these diagrams, along with the ohmic values (resistance R, impedance Z, or reactance X) of the circuit components, allow us to determine the phase angle (Φ) between the applied voltage (V_S) and the current (I) flowing through the circuit.

Solving an Impedance Problem

We have the following data:

• Inductor (L) = 52 mH (milliHenrys) = 0.052 H (Henrys) (converted to Henrys for easier calculation)
• Resistor (R) = 16 Ω (Ohms)
• Frequency (f) = 50 Hz (Hertz)

Calculate, total impedance and phase angle.

Step 1: Calculate Inductive Reactance (XL)

Inductive reactance (X_L) is the opposition to current flow caused by an inductor and is calculated using the following formula:

X_L = 2 * π * f * L

where:

• X_L is the inductive reactance in Ohms (Ω)
• π (pi) is a mathematical constant (approximately 3.14159)
• f is the frequency in Hertz (Hz)
• L is the inductance in Henrys (H)

Plugging in the values:

• X_L = 2 * π * 50 Hz * 0.052 H
• X_L ≈ 16.33 Ω

Step 2: Calculate Impedance (Z)

Since the resistor and inductor are connected in series, their impedances simply add directly.

The total impedance (Z) is:

• Z = √(R² + X²_L)
• Z = √16² + 16.33²
• = 22.86 Ω

Step 3: Calculate Phase Angle (φ)

• tan(φ) = X_L / R
• tan(φ) = 16.33 Ω / 16 Ω
• φ ≈ arctan(1.02) ≈ 45.5°

Answers:

• Total Impedance (Z) ≈ 22.86 Ω
• Phase Angle (φ) ≈ 45.5°

Solving Another Impedance Problem

We have the following data:

• Resistance (R) = 13 Ω (Ohms)
• Current (I) = 4 A (Amperes)
• Voltage (V) = 110 V (Volts)
• Frequency (f) = 1000 Hz (Hertz)

Calculate the inductance of the coil and the power factor..

Step 1: Calculate Impedance (Z)

Ohm’s Law tells us V = I * Z, so we can rearrange the formula to solve for impedance:

• Z = V / I
• Z = 110 V / 4 A
• Z = 27.5 Ω

Step 2: Calculate Inductive Reactance (XL)

We know the total impedance (Z) and resistance (R) of the circuit. Since the coil has inductance, the remaining impedance must be due to inductive reactance (XL). We can use the following formula to find XL:

• X_L = √(Z² – R²)
• X_L = √(27.5² Ω – 13² Ω)
• X_L ≈ √(593.75 Ω² ) ≈ 24.4 Ω

Step 3: Calculate Inductance (L)

Now that we know the inductive reactance (X_L), we can calculate the inductance (L) of the coil using the formula:

X_L = 2 * π * f * L

where:

• π (pi) is a mathematical constant (approximately 3.14159)

• L = X_L / (2 * π * f)
• L = 24.4 Ω / (2 * π * 1000 Hz)
• L ≈ 3.89 x 10^-3 H
• = 3.89 milliHenrys (mH).

Step 4: Calculate Power Factor (PF)

cosΦ = R / Z = 13 / √(13² + 24.4²) = 0.47

Now we know impedance (Z) combines resistance (R) and reactance (X) in AC circuits. Reactance is 90° out-of-phase with resistance (inductors +90°, capacitors -90°).

But the question is, how does a series circuit containing both inductive reactance (X_L) and capacitive reactance (X_C) affect the complex impedance?

Reactance and Impedance Triangles

Inductors have a positive reactance (X_L) that increases as the frequency goes up. We can visualize this, using an impedance triangle with a positive slope.

On the other hand, capacitors have a negative reactance (X_C) that decreases with frequency, represented by an impedance triangle with a negative slope.

Combining Reactances in Series

When we connect an RLC circuit in series, the total reactance (X) is the sum of the individual reactance of the inductor (X_L) and the capacitor (X_C).

This combined reactance along with the inherent resistance (R), determine the overall impedance (Z) of the entire circuit.

• X = X_L + (-X_C) = X_L – X_C
• Z² = R² + X² = R² + (X_L – X_C)²
• Z = √(R² + X² = √[R² + (X_L – X_C)²]

When working with RLC circuits we often need to find the total reactance (X), which is a combination of the inductive reactance (X_L) and capacitive reactance (X_C).

A helpful rule of thumb exists to simplify this process. Regardless of whether X_L is larger than X_C or vice versa, we can subtract the smaller reactance value from the larger one.

This works, because squaring a negative number in mathematics always results in a positive value. In other words, it doesnt matter if we use (X_L – X_C) or (X_C – X_L), the result will be the same in terms of magnitude.

This combined reactance value, after considering the positive or negative sign, is then added to the resistance (R) of the circuit to determine the overall impedance (Z). The resulting impedance triangle will reflect this combined effect with the slope depending on the dominance of inductive or capacitive reactance.

Understanding Impedance with RLC Circuit Triangles

Impedance and Resonance in RLC Circuits

The slope of the impedance triangle in an RLC circuit depends on which type of reactance dominates, whether inductive (X_L) or capacitive (X_C). A positive slope indicates a dominant inductive reactance while a negative slope indicates a dominant capacitive reactance.

We can express the circuits impedance in complex form as Z = R ± jX, where R is the resistance, j is the imaginary unit, and X is the net reactance (X_L – X_C or X_C – X_L).

If a series AC circuit only contains inductance and capacitance, the impedance simplifies to Z = X_L – X_C, or vice versa.

At resonance, where the inductive and capacitive reactances become equal in magnitude, but opposite in sign (since X_L = X_C), the net reactance (X) cancel out resulting in a total impedance of Z = 0.

This is why at resonance only the resistance (R) limits the current flow in the series circuit.

Solving one more Impedance Problem

We have the following data:

• Resistor (R) = 9 Ω
• Capacitor (C) = 100 μF (microFarad)
• Inductor (L) = 0.16 H (Henry)
• Supply Voltage (V) = 220 V (rms)
• Frequency (f) = 50 Hz

Let us calculate the inductive reactance, the capacitive reactance, the circuits complex impedance, and the power factor.

Calculations:

Inductive Reactance (XL):

• X_L = 2 * π * f * L
• X_L = 2 * π * 50 Hz * 0.16 H
• X_L ≈ 50.27 Ω

Capacitive Reactance (XC):

• X_C = 1 / (2 * π * f * C)
• X_C = 1 / (2 * π * 50 Hz * 100 μF)
• X_C = 31.83 Ω

Complex Impedance (Z):In a series circuit, the impedances of resistor, inductor, and capacitor add vectorially.

We can use the following formula to find the magnitude (Z) and angle (θ) of the impedance:

• Z = √[R² + (X_L – X_C)²]
• θ = tan⁻¹[(X_L – X_C) / R]

Since we have a capacitive reactance (X_C) smaller than the inductive reactance (X_L), we will subtract X_C from X_L.

• Z = √[9² + (50.27 Ω – 31.83 Ω)²] ≈ 20.5 Ω
• θ = tan⁻¹[(50.27 Ω – 31.83 Ω) / 9 Ω] ≈ 64° (positive due to dominant inductive reactance)

Therefore, the complex impedance (Z) is 20.5 Ω∠64°.

As reactive components (inductors and capacitors) introduce a phase shift, the power factor becomes less than 1.

PF = cos(θ) = cos(64°) ≈ 0.44

Final Results:

• Inductive Reactance (X_L) ≈ 50.27 Ω
• Capacitive Reactance (X_C) ≈ 31.83 Ω
• Complex Impedance (Z) = 20.5 Ω∠64°
• Power Factor (PF) ≈ 0.44

Impedance in AC Circuits: Series vs. Parallel

This tutorial explored impedance (Z), the total opposition to current flow in an AC circuit. We learned that impedance combine resistance (which just opposes current) with reactance (the frequency dependent opposition from inductors and capacitors).

Importantly impedance is not a simple sum of these values, but rather, a vector sum due to the “out-of-phase” nature of reactance compared to resistance.

In series circuits, complex impedance follows the same Ohms Law principle as purely resistive circuits. We can simply add the individual impedances (Z1, Z2, etc.) to find the total impedance (Z_T) of the series circuit.

However the calculation for parallel circuits differs. This section will now explore how to determine impedance in AC circuits with parallel components.

Finding Impedance in a Two-Component Parallel Circuit

When working with a parallel circuit containing just a resistor (R) and a reactive element (either an inductor or capacitor, represented by X) we need to find the impedance of each branch separately.

However the good news is that for this specific case of only two components, We can utilize a well known formula for calculating the combined resistance of parallel circuits.

This formula is:

R_T = (R₁ * R₂) / (R₁ + R₂)

where:

• R_T – Total Impedance of the parallel circuit
• R₁ – Resistance of the first component
• R₂ – Resistance of the second component (which in this case is the reactance, X)

• Z_T = Z₁ * Z₂ / Z₁ + Z₂
• = R * X / √(R² + X²)

In the above formula, all the terms Z, R and X are measured in Ohms.

Its important to note that, when dealing with AC supplies and frequencies, the resistive component is 90 degrees out-of-phase with the reactive component. Therefore the product is divided by the vector sum of R and X.

Therefore, if “n” branches with complex impedances are connected in parallel, the total impedance is the vector sum of all the parallel branches. Therefore the reciprocal of the circuits total impedance can be expressed as:

• 1 / ZT = 1 / Z1 + 1 / Z2 + 1 / Z3 + 1 / Z4 + ……etc
• ∴ 1 / (1/Z1 + 1/Z2 + 1/Z3 + 1/Z4 + ….)

Introducing Admittance for Parallel Circuits

Admittance (Y), is the reciprocal of impedance (Y = 1/Z). In a parallel AC circuit the total admittance (YT) is just the sum of the admittances of each individual branch (Y1, Y2, Y3, etc.).

This is a much simpler calculation compared to the vector sum of impedances.

Once we have the total admittance (Y_T), we could find the total impedance (Z_T) by taking the reciprocal again (Z_T = 1/Y_T).

This method using admittance avoids complex vector sum calculation with impedances, for most practical parallel AC circuits.

References: Complex Impedances

https://electronics.stackexchange.com/questions/28285/complex-impedances