Chapter 7
AC Analysis

m7.1 Impedance Transformations

Determine the equivalent impedance Z looking into terminals (a,b) for the circuit of Figure m7.1 at the following frequencies: 100 Hz, 500 Hz, 1000 Hz, and 2000 Hz. Report your results in polar form. Use these component values: R1 = 100 Ω, R2 = 90 Ω, C = 1.0 μF and L = 33 mH.

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Figure m7.1: Circuit for Problem m7.1

NI Multisim Measurements

Figure m7.1a describes a laboratory technique to measure impedance of the circuit at terminals A-B. The sinusoidal voltage source excites the circuit with a known voltage v(t) and causes the current i(t). The impedance of the circuit at a given operating frequency is Z = V/I where V and I are the equivalent phasor representations of the time-domain voltage and current. The magnitude of V is the same as the amplitude of v(t). Similarly, the magnitude of I is the same as the amplitude of i(t). The sinusoidal voltage source amplitude can easily be measured with the oscilloscope, but what about the current?

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Figure m7.1a: Circuit for Problem m7.1

Observe that the current i(t) entering Terminal A also flows through resistor R1. The voltage v1(t) that appears across this resistor is directly proportional to the current i(t) due to Ohm’s Law, consequently the current can be indirectly measured as i(t) = v1(t)/R1.

The procedure to measure impedance therefore requires the following steps:

  1. Apply a sinusoidal voltage at the desired frequency to establish v(t); typically a unit-amplitude voltage is convenient,
  2. Measure the voltage magnitude (also called amplitude and peak value) V M of v(t),
  3. Measure the voltage magnitude across the resistor R1 and divide by the measured resistance R1 to determine the current magnitude IM,
  4. Calculate the impedance magnitude as V M/IM, and
  5. Calculate the impedance phase by measuring the time difference tD between the two sinusoids; convert this time difference to degrees by multiplying by the sinusoidal frequency in Hz and then multiply by 360 degrees. If the current waveform is delayed compared to the voltage waveform then the phase sign is positive; if advanced, then the phase sign is negative.

Use the above procedure to measure the impedance of the circuit at frequencies 100 Hz, 500 Hz, 1000 Hz, and 2000 Hz. Activate the circuit with the Simulate → Instruments → Function Generator and observe the two voltage signals with the Simulate → Instruments → Oscilloscope .

Additional helpful tips:

NI Multisim video tutorials:

NI myDAQ Measurements

The same impedance measurement technique described in the previous Multisim section works well for the physical circuit, too. Activate the circuit with the NI ELVISmx Function Generator on AO0, and place an op amp voltage follower between AO0 and the circuit; the voltage follower is necessary to boost the current drive of the analog output beyond its limit of 2 mA. Do not insert a physical resistor for R2, because this resistor simply models the finite wire resistance of the physical inductor.

Use the oscilloscope to display the voltage v(t) on AI1 and the resistor voltage v1(t) on AI0. Measure the circuit impedance at the frequencies 100 Hz, 500 Hz, 1000 Hz, and 2000 Hz.

Additional helpful tips:

NI myDAQ video tutorials:

Further Exploration with NI myDAQ

The NI ELVISmx Bode Analyzer instrument provides a quick and effective way to study circuit behavior over a range of frequencies, all within a single measurement step. The Bode Analyzer applies a sinusoidal signal (also called a tone pulse) as the circuit stimulus on AO0, measures the actual circuit stimulus on AI0, and measures the circuit response on AI1. The Bode Analyzer applies a series of tone pulses from low to high frequency and then plots the circuit response – “gain” and “phase” – as a function of frequency. This is an automated form of the method you used earlier in this problem.

The “Gain” display plots the ratio of the circuit response to the stimulus. With the myDAQ connections described in the previous section the “Gain” plot therefore displays the magnitude of the circuit impedance because the current v(t) serves as the “stimulus” and the voltage proportional to i(t) serves as the “response.” Simply multiply the “Gain” plot by the value of R1 to see the impedance in ohms.

Set the following values:

Try running the Bode Analyzer with its default step size of 5, and then increase the step size until the plot is reasonably smooth and captures the interesting features of the impedance curve.

Use the cursors to read specific values at the frequencies specified earlier, and compare to your previous measurements.

The impedance plot as a function of frequency offers a “birds eye” view of the circuit behavior for a wide range of frequencies. For what frequency range does the circuit appear inductive? Over what range does it appear capacitive? What do you observe about the phase when the impedance reaches its maximum value?