# 11. Impedance Measurement - Frequency Effects¶

## 11.1. Objective¶

The objective of this activity is to:

1. Measure component impedance and circuit impedance using the Impedance Analyzer applcation.
2. Study the magnitude and phase changes with change in frequency for an RLC circuit.

## 11.2. Notes¶

In this tutorials we use the terminology taken from the user manual when referring to the connections to the Red Pitaya STEMlab board hardware. Impedance analyzer application is used to measure RLC circuit impedane $$Z(f)$$. Alongside Impedance analyzer application for impedance measurement we have used an LCR meter frontend. Although LCR meter frontend is intended when using LCR meter application it can be also used for Impedance analyzer application. The Impedance analyzer application enables measurements of Impedance, Phase and other parameters of the selected DUT (Device Under Test). Measurements can be performed in the Frequency sweep mode with 1Hz frequency resolution or in the Measurements sweep mode with the desired number of measurements at constant frequency. The selectable frequency range is from 1Hz to 60MHz, although the recommended frequency range is up to 1MHz. The impedance range is from 0.1 Ohm to 10 Mohm. When using the Impedance analyzer application with the LCR Extension module, insert 0 in the shunt resistor field.

## 11.3. Background¶

Impedance is the resistance to the flow of alternating current. It is the total opposition that a circuit offers to the flow of current at a particular frequency. Impedance $$Z$$ is expressed as a combination of Resistance $$R$$ and Reactance $$X$$ and is measured in $$\Omega$$. It can be expressed as a complex quantity as:

$Z = R+jX \quad (1)$

For a ideal resistor, the impedance is the same as the DC resistance $$Z=R_{DC}$$. For a capacitor, the impedance (or more specifically, the reactance) $$X_C$$ is imaginary and negative reactive part of the impedance. The reactance of the capacitor depends upon the frequency and is given as:

$X_C = \frac{1}{j\omega C} \quad (2)$

For an inductor, the impedance (or more specifically, the reactance) $$X_L$$ is imaginary and positive reactive part of the impedance. The reactance of the inductor also depends upon the frequency and is given as:

$X_L = j\omega L \quad (3)$

The impedance of a series RLC circuit is the sum of the impedances of respective components.

$Z = R + Z_L + Z_C \quad (4)$

or

$Z = R + jX_L - jX_C \quad (5)$

This can also be represented as a phasor with magnitude $$|Z|$$ and phase $$P$$ where Z = $$|Z|e^{jP}$$

$|Z| = \sqrt{R^2 + (X_L - X_C )^2} \quad (6)$
$P = arctan(\frac{(X_L - X_C )}{R}) \quad (7)$

Figure 1: Series RLC circuit.

## 11.4. Materials¶

• Red Pitaya STEMlab 125-14 or STEMlab 125-10
• Resistor Rs: 1 kΩ
• Capacitor Cs: 0.047 µF
• Inductor Ls: 22 mH

## 11.5. Procedure¶

### 11.5.1. Measuring components¶

With LCR meter application we can measure inductance, resistance and capacitance our elements in the circuit at selected frequency. LCR meter can help you to measure each component separately in order to extract its value if not visible/readable on the packaging:

1. Start LCR meter
2. Connect the measured component to the LCR meter probes
3. On the LCR meter application select measurement mode/parameter
4. Select measurement frequency to 1kHz
5. Repeat steps above for Rs, Ls and Cs

Figure 2: LCR meter application

Note

Actual(measured) values of the components Rs, Cs, Ls are different than marked (color code for resistor and printed values on inductor and capacitor). The difference is due to components values tolerances

### 11.5.2. Measuring series RLC circuit Impedance¶

1. Set up the circuit as shown in figure 1 and figure 2 on your solderless breadboard, with the component values Rs = 1 KΩ, Cs = 0.047 µF, Ls = 22 mH.

Figure 3: STEMlab with LCR meter frontend and series RLC circuit

1. Start the Impedance analyzer application.

Note

Impedance analyzer is community application and it needs to be downloaded from Application marketplace (bazaar). Click on Application marketplace icon and select Install for Impedance analyzer.

1. Start Impedance analyzer and:
• under Measurement settings menu set number of steps: 20
• under Frequency sweep set Start frequency to 1kHz and End frequency to 50kHz
• select Start measurement

Figure 4: Graph of the RLC circuit impedance magnitude taken with Impedance analyzer application

4.Plot mesured Phase
• under Plot settings menu for Y-axis select P[deg]

Figure 5: Graph of the RLC circuit impedance phase taken with Impedance analyzer application

Note

The frequency at which this occurs (Phase = 0) is called resonant frequency. At resonant frequency the total reactance is zero and the circuit is purely resistive.

For

$Z = R + j(X_L - X_C ) \quad (8)$

If

$X_L - X_C = 0 \quad (9)$

then

$Z = R \quad (10)$

Resonant frequency can be mathematically derived using equation to be:

$f_0 = \frac {1}{2 \pi \sqrt{LC}} \quad (11)$

## 11.6. Questions¶

1. Compute the resonant frequency fo for the series RLC using equation (11) and compare it to the measured value. What is the percentage error between the two?
2. Give your conclusions from the observations made in step 3 of the procedure.
3. Compute the magnitude and phase for the series RLC circuit, when the reactive component equals the resistive component.