Section3.md

Analysis and Design of Circuits Lab

Part 1: Autumn Term weeks 4--6

Section 3: Resonance and Filters

Capacitors and inductors exhibit resonance when they are connected together — at a certain frequency their impedances are equal and opposite resulting in an overall impedance that is (theoretically) either zero or infinity. In practice, impedance never reaches zero or infinity due to parasitic resistance. In this section you will measure the impedance of resonant networks and make filter circuits that use complex impedance to block certain frequencies.

Before the lab

A resistor, capacitor and inductor can be combined to make a second-order RLC filter, meaning that the transfer function $T(f)=V_\text{out}(f)/V_\text{in}$ has a relationship $T\propto f^2$ or $T\propto 1/f^2$ for low or high frequencies.

Use your lecture notes to design a RLC filter to achieve the shape and corner frequency listed for your pair in the table at the end of this page. Choose a value of $R$ that will make your system critically-damped ( $\zeta=1$ ). Pick component values from the list of available parts below to get a corner frequency within 10% of the specification.

Use LT SPICE to produce the magnitude and phase responses of your filter (do this after LT SPICE has been introduced in problem classes)

Resistors ( $\times10^0\Omega$ to $\times10^6\Omega$ )

1.0	1.1	1.2	1.3
1.5	1.6	1.8	2.0
2.2	2.4	2.7	3.0
3.3	3.6	3.9	4.3
4.7	5.1	5.6	6.2
6.8	7.5	8.2	9.1

Capacitors (multilayer ceramic)

1nF	2.2nF	4.7nF	10nF
22nF	33nF	47nF	68nF
100nF	220nF	470nF	1μF

Inductors

1mH	2.2mH	3.3mH
4.7mH	10mH	22mH
33mH	47mH	100mH

Resonant networks

Series LC network

Use the same method as Section 1 and 2 to measure the combined impedance of a 100nF capacitor and 100mH inductor in series. The series combination of the inductor and capacitor forms $Z$, the impedance to be tested.

Resonance occurs at the frequency, $\omega_0$, when the magnitudes of the reactances of the capacitor and inductor are equal, i.e. $\omega_0 L = \frac{1}{\omega_0 C} $. Rearranging gives $\omega_0=\sqrt{\frac{1}{LC}}$

For an ideal capacitor and ideal inductor in series, $Z(\omega)=Z_L(\omega)+Z_C(\omega)=j\omega L+\frac{1}{j\omega C}$

At resonance, $Z(\omega) = j\omega_0 L + \frac{1}{j\omega_0 C} = 0$

Plot your impedance measurements to find $\omega_0$, the frequency where impedance is at a minimum. You will need to take extra measurements around the resonant frequency to see the characteristic in enough detail, so plot your measurements as you take them to see where you need to try intermediate frequencies. Compare the results to theory and also try to explain the exact value of $Z(\omega_0)$, which isn't zero as predicted by theory.

• Plot the impedance of the series LC network as it varies with frequency and compare the measurements to theory.

Parallel LC network

A parallel LC network also exhibits resonance.

Using the equation for impedances in parallel, $Z(\omega)=\frac{Z_L(\omega)Z_C(\omega)}{Z_L(\omega)+Z_C(\omega)}$

Since $Z_L(\omega_0)=-Z_C(\omega_0)$

$Z(\omega_0)=\frac{Z_L(\omega_0)Z_C(\omega_0)}{0}$, which tends to positive infinity as $\omega \longrightarrow \omega_0$.

Repeat the process of measuring impedance, once again with extra measurements to add detail around $\omega_0$. Use the same values for $L$ and $C$. Find the maximum magnitude of the impedance and explain why that is the maximum.

• Plot the impedance of the parallel LC network as it varies with frequency and compare the measurements to theory. Overlay the plot with your results of the series LC network.

First-order filter

A filter is a circuit that transforms a signal by amplifying or attenuating certain constituents, and a linear filter selects those constituents according to frequency. The most common type of linear filter made from passive components is a resistor-capacitor (RC) filter, which can be either low-pass (attenuates high frequencies) or high-pass (attenuates low frequencies). In an audio system, the subwoofer would be driven by the output of a low-pass filter while the tweeter would use a high-pass.

A filter is characterised by its transfer function $T(\omega)$, which is the the relationship between the input and output signals and its dependency on frequency: $T(\omega)=V_\text{out}(\omega)/V_\text{in}$. To measure it, you'll need to record the ratio of output voltage to input voltage over the range of frequencies you are interested in. That means, compared to the earlier exercises where you measured impedance, you can discard the math channel measurement and just measure $V_\text{in}$ with oscillscope CHA and $V_\text{out}$ with CHB.

Note that the transfer function is complex: it has a magnitude and a phase. The magnitude is the ratio of the voltage amplitudes and the phase is the difference in phase angle between the output and input sine waves. The phase response of an RC filter was used to make the phase shifter in the lab skills experiment. Use the oscilloscope cursors to measure the phase difference between input and outpu.

Measure and plot the transfer functions of RC high-pass and low-pass filters made from a 1μF capacitor and 1kΩ resistor. Frequency and $|T(f)|$ should be plotted on logarithmic axes and $\arg(T(f))$ should be plotted on a linear axis. Measure the transfer function at the same frequency points that you used to measure impedance.

• Plot the magnitude and argument of transfer function of the high-pass and low-pass filters.

Challenge: test your second-order filter

• Build the filter you designed in the preparation activity. Measure the magnitude and phase responses between 1Hz and 100kHz. Produce a plot comparing the magnitude and phase responses with the results of your simulation. Explain any discrepancies.

Specifications for RLC filters

Each lab pair has a specified filter type and corner frequency for their RLC filter.

Pair	Type	$f_c$
A01	LP	1900
A02	HP	8700
A03	HP	7400
A04	HP	1800
A05	HP	8800
A06	HP	1900
A07	HP	8800
A08	HP	5000
A09	HP	2800
A10	LP	2700
A11	HP	2300
A12	LP	1900
A13	HP	4900
A14	HP	5000
A15	HP	5300
A16	LP	5900
A17	LP	7400
A18	LP	7300
A19	HP	4100
A20	HP	1700
A21	LP	7400
A22	HP	5900
A23	LP	3400
A24	LP	1600
A25	HP	8700
A26	HP	8800
A27	HP	3400
A28	LP	8800
A29	LP	8700
A30	LP	970
A31	HP	730
A32	LP	3400
A33	HP	3400
A34	HP	5900
A35	LP	4000
A36	HP	3400
A37	LP	1700
A38	LP	8700
A39	LP	5100
A40	HP	4000
A41	LP	3400
A42	HP	5900
A43	HP	4000
A44	HP	1900
A45	HP	4900
A46	HP	5900
A47	LP	2800
A48	LP	6600
A49	HP	2300
A50	HP	6100
A51	LP	5000
A52	LP	5900
A53	LP	7300
A54	HP	8900
A55	LP	2800
B01	LP	4500
B02	LP	7400
B03	LP	3400
B04	HP	1400
B05	LP	6100
B06	LP	1900
B07	HP	1600
B08	LP	1600
B09	LP	4400
B10	HP	1700
B11	HP	1100
B12	LP	4900
B13	LP	5900
B14	LP	1900
B15	LP	2300
B16	HP	2300
B17	LP	4400
B18	HP	5900
B19	HP	8900
B20	HP	7400
B21	HP	8800
B22	LP	5300
B23	HP	1100
B24	HP	4900
B25	HP	7300
B26	LP	2800
B27	LP	4400
B28	LP	2000
B29	LP	2300
B30	LP	5000
B31	HP	1500
B32	HP	5100
B33	HP	2300
B34	LP	2300
B35	LP	3400
B36	LP	4900
B37	HP	3400
B38	HP	3300
B39	LP	2400
B40	LP	3300
B41	HP	2300
B42	LP	5100
B43	LP	2700
B44	LP	5000
B45	LP	3400
B46	LP	4800
B47	LP	5900
B48	HP	8700
B49	HP	1100
B50	HP	7300
B51	HP	7300
B52	LP	2100
B53	LP	8800
B54	LP	1400
B55	HP	5900

HP: High-pass, LP: Low-pass