Circuits Problem Method

From The Tronically Hip

Welcome to the Circuits Solution Guide. We're putting this together as a studying exercise for ourselves, and also to build a reference for everyone. If you see vague or incorrect information, please don't hesitate to make suitable adjustments.

We've drawn attention to specific areas of uncertainty with the blue question boxes.

The purpose of the Guide is specifically to address the Sensors and Instrumentation midterm and final, but there's some fairly significant overlap here with Signals, so don't feel that it has to be exclusive to any particular topic.

1 Circuit Types
2 Kirchoff and Transfer Function
3 Zeros and Poles
4 General Form
5 Bode Plot
6 Amplitude and Phase
7 Further Reading

[edit] Circuit Types

Given a circuit, try to determine what it does at first glance, and what sort of circuit it is just by looking at it (e.g. a filter, what sort of filter, etc. Suggestions on how to do this are GREATLY appreciated).

Low-pass

High-pass

Question: Can you switch either of these to an inductance-based filter just by swapping the the resistor for an inductor, and the capacitor for a resistor? Based on the guidelines below, it would seem this to be the case.

Answer: Yes you can, caps are normally smaller and cheaper than inductors which is why this isn't normally done.

Capacitors generally block low frequencies and admit high frequencies. (A frequency of zero is just direct current, and to DC, a capacitor is an open circuit.)
Inductors generally admit low freqencies and block high ones. (To DC, a coil is simply a short circuit, but the magnetic field opposes rapid oscillations.)

[edit] Kirchoff and Transfer Function

Identify the nodes, unknowns, and which nodes you'll need to write KCL for. Set up your KCL matrix and solve to obtain the Transfer Function: T(s) = Out/In

Note: Remember that T(s) isn't necessarily Vout/Vin. The transfer function is simply the (Output function)/(Input function), regardless of it being voltage or current.

Using Impedences of Components:

Component	Time-domain Behaviour	Frequecy-domain Behaviour
Capacitor	$i = C\frac{dV}{dt}$	$V = \frac{i}{sC}$
Resistor	$V = iR\,$	$V = iR\,$
Inductor	$v = L\frac{di}{dt}$	$V = isL\,$

Question: What about ideal op-amps? The two input terminals are considered to be the same voltage, what about the third terminal?

Answer: The output terminal's voltage is indeterminate and is usually solvable deductively from the input voltages and currents, and remember that current is allowed to flow in/out of the third terminal (unlike the inputs), so watch out when doing your KCL.

Question: Which nodes do I write KCL for?

Answer: Nodes that are not connected to voltage sources. So you cannot write KCL for independent sources (such as Vin) and your opamp output (which acts as a dependent VCVS). Also, you cannot write KCL for the ground node.

Note: Impedance is ratio of the phasor voltage to the phasor current, denoted by Z. It can be written in polar ( $|Z|\angle\theta\,$ ), exponential ( $Ze^{j\omega}\,$ ), or rectangular ( $R + jX\,$ ) forms.

[edit] Zeros and Poles

Once you have your Transfer function in the form $T(s)=\frac{N(s)}{D(s)}$ , solve for the Zeros (roots of the numerator) and the Poles (roots of the denominator).

Note: For poles, if you have complex pairs of roots, then for Bode Plot purposes, the poles are at the distance given by the magnitude or amplitude of the complex root. i.e. $\sqrt{(\mathrm{real part})^2+(\mathrm{coefficient of j})^2}$

[edit] General Form

You can relate the values to the general form to obtain values such as frequency, resistance, inductance or capacitance that is being sought for in the question:

$H(s) = \frac{N(s)}{s^2+\frac{w_0}{Q_0}s+w_0^2}$

$w 0$ — Resonant Frequency of the circuit.
$Q 0$ — quality factor, a dimensionless ratio. [1]
$B W$ - Bandwidth, $BW = \frac{w_0}{Q_0}$
$N (s)$ — A function of s in the numerator that defines what kind of filter the circuit is
$H (s)$ — The transfer function for the circuit $H(s) = \frac{output}{input}$

Note: $H (s)$ and $T (s)$ are the same thing.

$s$ — function parameter in frequecy domain ( $s = j ω$ when converting from frequency domain to phasor domain).

[edit] Bode Plot

With these Poles and Zeros, identify where they will be on the Bode plot; their position should be the distance of the point from the origin (if lying on the Real axis, it is just the absolute value of the root/pole. If floating out in space, use pythagorean).

Now that you have your poles/zeroes on the Bode plot, start sketching. Remember, Zeroes == +20db/dec; Poles == -20db/dec.

Rearrange the T(s) to resemble the general form of T(s). You can relate the values to the general form to obtain values such as frequency, Resistance, Inductance or Capacitance that is being sought for in the question. From the general form, you can find the values of Resonant Frequency ( $W 0$ ), BandWidth and Quality Factor ( $Q 0$ ).

A link is provided in the Further Reading Section at the bottom of this wiki with steps on how to draw Bode Plots.

[edit] Amplitude and Phase

Calculate what the amplitude and phase would be at low and high frequencies.

Low-frequency amplitude: $20\log(\lim_{\omega\to0}T(j\omega))$

High-frequency amplitude: $20\log(\lim_{\omega\to\infty}T(j\omega))$

For the phase plot, it helps to rearrange T(s) so that in both numerator and denominator, you have a form of [Real part + j(imaginary part)]. For low freq, phase of T(jW) is [arctan(numerator imaginary part/numerator real part) - arctan(denominator imaginary part/denominator real part)]. Then take 20log(phase of T(jW)) to get phase at low freq. (You could do the same for high freq, but it is not entirely necessary. Just make sure your Bode plot flattens out after the last Zero or Pole on the plot)

Now that you have your poles/zeroes on the Bode plot, start sketching. For the amplitude plot, Zeroes == +20db/dec; Poles == -20db/dec. For the phase plot, Zeroes == +90 degrees rise over 2 decades; Poles = -90deg drop over 2 decades.

Also remember that it is +/- 20db/dec per zero or pole that occurs at that point. So, if there are 2 poles, it is -40db/dec and -180 degrees drop.

Watch out for the case where two complex poles occur at resonance. In this case, you have a spike of 20log(Qo) on the amplitude plot and a sudden drop of 180 degrees on the phase plot.

Note: Two complex roots in the denominator (this is what we've seen) implies a spike of +20log(Qo). If there are complex conjugate roots in the numerator, that implies a spike of -20log(Qo). I am not sure about phase but, i'm sure it jumps by 180 degrees on phase plot (someone confirm?) )

The width of the resonance peak at -3dB from the top of the peak is the BW , (Wo/Qo). For the phase plot, put a point 45° below the previous level at Wo - BW/2, and a point 135° below the previous level at Wo + BW/2. Draw your curve so it passes through these two points (normally very steep). Level out the curve at 180° below the previous (this is only for complex poles in the denominator).

[edit] Further Reading

As this list of steps is just starting out, there may be some missing parts or things that are incorrect. Please feel free to edit and change them to a more correct/descriptive version. Hopefully as we do so we will obtain a better method of approach as we get closer to the midterm. Feel free to add your name if you contributed. Your input is greatly welcome! Thanks.

Ab.
Arjun.
Mike.
Jerry.
Ryan (J)
Brandon
Mark M.

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Circuits Problem Method

From The Tronically Hip

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