$35.00

## Description

**1 **** ****Objectives**

The purpose of the lab is to investigate the frequency response of first-order circuits and learn the fundamentals about circuit analysis and design in the frequency domain.

**2 **** ****Introduction**

The frequency response is a representation of the system’s response to sinusoidal inputs at varying frequencies; it is defined as the magnitude ratio and phase difference between the input and output signals. If the frequency of the source in a circuit is used as a reference, it is possible to have a complete analysis in the frequency and time domains. Frequency domain analysis is easier than time domain analysis because differential equations used in the time domain are mapped into linear equations that are function of the complex frequency variable s . It is important to obtain the frequency response of a circuit because we can predict its response to any input signal.

In this laboratory experiment we will plot the frequency response of first order RC circuits. We can characterize the circuits by two features of the frequency response:

**1.**** **The difference between the magnitude of the output and input signals (given by the amplitude ratio)

**2.**** **The time lag or lead between input and output signals (given by the phase shift)

Toplot thefrequencyresponse,value ofthetransferfunctionH(s)=V_{o} (s )/V_{i} (s ) is calculated within a range of input frequencies. A particularly important method of displaying frequency response is the Bode plot, which is the representation of the magnitude (in decibels) and phase (in degrees) of H (j ω) as a function of ω or f in logarithmic scale, where ω and f are the frequency variables in radians per second (rad/s) and hertz (Hz), respectively.

As an example, assume that the transfer function is given as

H (s ) =

_{V}_{o}

_{(}_{s} _{)} _{=}

^{V}_{i}

__ ____ ____ ____1____ ____ __

__ ____ ____ ___{s}__ ____ __

_{1} _{+}

^{2}^{π}^{1000}

_{1}

(1)

^{} _{|}_{H} _{(}_{j} _{ω}_{)}_{|} _{=} r__ ____ __

H (j ω) =

__ ____ ____ ___{1}__ ____ ____ __ ^{}^{}

_{j} _{ω } ^{⇒}

_{1} _{+ } _{}

_{1} _{+} __ ____ __^{ω }__ __ ^{2}

^{2}^{π}^{1000}

(2)

2π1000

^{}^{ } _{∠}_{H} _{(}_{j} _{ω}_{)} _{=} _{−} _{tan}

_{−}_{1} __ ____ ____ __^{ω}__ ____ __

2π1000

If the input signal is V_{i} (t ) = 0.2 sin(2π3000t ), which is a sinusoidal signal with 0.2 V amplitude and 3 kHz (or 2π3000

rad/s) frequency, then the output signal can be calculated as follows:

_{1 } ^{}

^{|}^{H} ^{(}^{j} ^{2}^{π}^{3000)}^{|} ^{=} ^{s}__ ____ __ ^{=} ^{0.316 }

_{2}_{π}_{3000} ^{2 } ^{}

_{1} _{+ } ^{}^{}

^{2}^{π}^{1000}

^{− } ^{− } ^{−}

^{}

_{∠}_{H} _{(}_{j} _{2}_{π}_{3000)} _{= } _{tan}^{−}^{1} ^{2}^{π}^{3000} _{= } _{71.6}^{◦} _{= } _{1.25} _{rad } ^{}

^{2}^{π}^{1000}

⇒ V_{o} (t ) = 0.0632 sin(2π3000t − 1.25) (3)

The outputvoltage V_{o} (t ) calculated above can be verified using either time-domain (transient), or frequency- domain (AC) simulation as follows:

V(vi) V(vo)

Figure 1: Transient Simulation: V^{ˆ}_{o} = 63.2mV , |H | =

Figure 2: AC Simulation: |H | = −9.995dB = 10^{−}^{9.995}^{/}^{20} = 0.316, ∠H = −71.5^{◦} = −1.25 rad

As you can see from Figs. 1 and 2, the values of |H | and ∠H can be verified using transient or AC simulation, however AC simulation provides easier comparison and includes data for any input frequency. On the other hand, transient simulation shows the actual voltage output for a specific set of input signals, so it includes more informa- tion such as initial transients and nonlinearity, as well as any DC component.

**3 **** ****Calculations**

**1.**** **For the circuits in Figs. 3(a) and 3(b), derive the transfer functions in the following forms:

H_{LP} (s ) =

^{V}__LP__

V_{i}

(s ) = K_{L}

__ ____ ____ ____1____ ____ __

__ ___{s}

_{1} _{+}

(4)

H_{HP} (s ) =

^{V}__HP__

V_{i}

^{ω}_{L}

__ ____ ____ ___{s}__ ____ __

H

^{(}^{s} ^{)} ^{=} ^{K}^{H } _{s} _{+} _{ω}

(5)

and express ω_{L} , ω_{H} , K_{L} and K_{H} in terms of resistors and capacitors.

_{V}_{i } ^{R}^{1 } _{C}_{1 } _{R}_{2}

^{V}LP

_{V}_{i } ^{C}^{2 } _{R}_{3 } _{C}_{3}

^{V}HP

(a) (b)

Figure 3: First order (a) lowpass filter (b) highpass filter

The frequencies ω_{L} and ω_{H} are known as:

• Pole frequency, defined as the root of the denominator of H (s )

• Corner frequency, defined as the frequency at which the gain is 0.707 times the passband value

• Half power frequency (compared to passband), since 0.707^{2} = 0.5

• 3dB frequency (compared to passband), since 20 log_{10} (0.707) = −3dB

**2.**** **Find R_{1} , R_{2} , R_{3} , C_{1} , C_{2} , and C_{3} , such that f_{L} = f_{H } = 5 kHz and K_{L} = K_{H } = 0.5.

**3.**** **Sketch the magnitude and phase Bode plots for H_{LP} (s ) and H_{HP} (s ).

**4.**** **Calculate the output voltages V_{LP} (t ) and V_{HP} (t ) for V_{i} (t ) = 0.4 sin(2π4000t ).

**5.**** **Repeat step 4 for V_{i} (t ) = 0.3 sin(2π6000t ).

**4 **** ****Simulations**

**For**** ****all**** ****simulations,**** ****provide **** ****screenshots**** ****showing**** ****the**** ****schematics **** ****and**** ****the**** ****plots**** ****with**** ****the**** ****simulated**** ****values **** ****prop- ****erly**** ****labeled.**

Draw the schematics for the circuits in Fig. 3 with the calculated component values. Perform the following simula- tions for each circuit:

**1.**** **Obtain the magnitude and phase **Bode**** ****plots**** **of the transfer function using **AC**** ****simulation**, and measure the

3-dB frequencies and passband gains. Also measure the magnitude and phase of the transfer function at 4kHz and 6kHz.

**2.**** **Apply the input V_{i} (t ) = 0.4 sin(2π4000t ) and obtain the **time-domain**** ****waveforms **for the input and the output voltages using **transient**** ****simulation**. Measure the magnitudes of the input and the output voltages, and the phase difference between them.

**3.**** **Repeat step 2 for V_{i} (t ) = 0.3 sin(2π6000t ).

**5 **** ****Measurements**

**For**** ****all**** ****measurements,**** ****provide **** ****screenshots**** ****showing**** ****the**** ****plots**** ****with**** ****the**** ****measured**** ****values **** ****properly**** ****labeled.**

Build the circuits in Fig. 3 with the with the simulated component values. Perform the following measurements for each circuit:

**1.**** **Obtain the magnitude and phase **Bode**** ****plots**** **of the transfer function using the **network**** ****analyzer**, and measure the 3-dB frequencies and passband gain. Also measure the magnitude and phase of the transfer function at

4kHz and 6kHz.

**2.**** **Apply the input V_{i} (t ) = 0.4 sin(2π4000t ) and obtain the **time-domain**** ****waveforms**** **for the input and the out- put voltages using the **scope**. Measure the magnitudes of the input and the output voltages, and the phase difference between them.

**3.**** **Repeat step 2 for V_{i} (t ) = 0.3 sin(2π6000t ).

**6 **** ****Report**

**1.**** **Include calculations, schematics, simulation plots, and measurement plots.

**2.**** **Prepare a table showing calculated, simulated and measured results.

**3.**** **Compare the results and comment on the differences.

**7 **** ****Demonstration**

**1.**** **Build the circuits in Fig. 3(a) and (b) on your breadboard and bring it to your lab session.

**2.**** **Your name and UIN must be written on the side of your breadboard.

**3.**** **Submit your report to your TA at the beginning of your lab session.

**4.**** **For both circuits in Fig. 3(a) and (b):

• Show the frequency response using the network analyzer.

– Measure the 3-dB frequency.

– Measure the magnitude and phase at a frequency f_{x} determined by your TA.

• Show the time-domain input and output waveforms using the scope at the frequency f_{x} .

– Measure the gain.

– Measure the phase difference between the input and the output.