# Steady-state analysis and impedance-admittance calculations

Thread Starter

#### lemon

Joined Jan 28, 2010
125
Hi:
Please assume I know nothing of the topic title above and have to complete the following question. To start with, could somebody please look at the question and guide me to the relevant reading material on this site so that I don't waste time sifting through pages and pages of unrelated material.
Thank you.

The question is given as:
Using the complex plane, use all the voltages and currents for the circuit shown in Figure Q3-1 (please see attached image).

#### Zazoo

Joined Jul 27, 2011
114
Volume II (AC), chapters 1-5 will help you solve this.
Depending on your familiarity witch circuit analysis in general, Volume I (DC), chapters 2, 5, 6, 7 and 10 are also important.
It's hard to narrow it down any further if we assume that you have no knowlede of the topic (since so many concepts build upon other more basic concepts.)

"phasor circuit analysis" would be the most relevant search term to use.

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Thread Starter

#### lemon

Joined Jan 28, 2010
125
wow! That was a good read. These pages are written extremely well, I must say.
I can see that I need to solve a Series-parallel R, L, and C circuit. But there are a couple of things I am confused about. So, my first order of business must be to determine values of impedance (Z) for all components based on the frequency of the AC power source. But where is the frequency here. How do I get it from the trig. equation given for voltage?

Also, the question says to use the complex plane. Does that mean I should use rectangular notion and not complex numbers in my calculations?

#### Zazoo

Joined Jul 27, 2011
114
wow! That was a good read. These pages are written extremely well, I must say.
I can see that I need to solve a Series-parallel R, L, and C circuit. But there are a couple of things I am confused about. So, my first order of business must be to determine values of impedance (Z) for all components based on the frequency of the AC power source. But where is the frequency here. How do I get it from the trig. equation given for voltage?
For a sinusoid, the form is: A cos (ωt ± θ)

A is amplitude
ω is frequency in radians/sec
θ is phase

Also, the question says to use the complex plane. Does that mean I should use rectangular notion and not complex numbers in my calculations?
Points on the complex plane can be represented in both polar and rectangular form. The y axis is the imaginary (j) axis, and the x axis is the real axis.

Rectangular form: x + jy

"Using the complex plane" refers to solving the problem using a frequency domain transformation (phasors, Laplace, etc.) vs. leaving the circuit elements in the time domain and solving the circuit with a differential equation.

Thread Starter

#### lemon

Joined Jan 28, 2010
125
yeah I'm struggling with this one - could somebody please get me started.

the image attached is what i think the 8cos(t+pi/4) but not sure what i am doing

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#### Zazoo

Joined Jul 27, 2011
114
yeah I'm struggling with this one - could somebody please get me started.
Start by converting all circuit elements and quantities into phasors and complex impedances.

For capacitors, impedance is: 1/(jωC)
For inductors, impedance is: jωL
For resistors, impedance is simply: R (i.e. no change)

You can represent the unknown currents and volatges as phasors by writing them in uppercase (e.g. v2 becomes V2)

Known AC voltage and currents (e.g. sources) can be converted to phasors:

Acos(ωt+θ) in the time domain becomes A<θ. (here the less than sign is actually meant to be the "angle" symbol)

Once everything is in phasor form, you can solve the circuit using any the standard network analysis theorems and tools (e.g. nodal/mesh, Thevenin, etc.)

After solving for unknown phasor quantites (currents and voltages) you can convert them back to sinusoids by just reversing the process used to conver them to phasors. Note that the frequency (ω) for every quantity in the circuit will always be the same, it is only the phase and magnitude that can change.

Searching "phasor circuit analysis" on google gives some good example problems. This one walks you through each step:
http://www.usna.edu/MathDept/CDP/ComplexNum/Module_6/EXERCISE6.htm

• lemon
Thread Starter

#### lemon

Joined Jan 28, 2010
125

#### Zazoo

Joined Jul 27, 2011
114
But - there is no frequency(w) with the voltage source given to me, i.e., 8cos(t+pi/4)v
The frequency, in radians, is whatever factor is multiplying "t" when in the form Acos(ωt+θ). In this case ω=1.

Thread Starter

#### lemon

Joined Jan 28, 2010
125
Before I solve the circuit using any the standard network analysis theorems and tools (e.g. nodal/mesh, Thevenin, etc.), could you please check to see if I have done the conversions to phasor quantities correctly, please?

I have these:
Resistors are just as they are - no change
Reactive capacitance(Xc) = 0.3183Ω
Reactive inductance(Xl) = 6.2832Ω
Conversion of capacitor to phasor domain, Zc = 0-j2.0Ω
Conversion of inductor to phasor domain, Zl = 0+1.0Ω
Voltage source V1 = 8<π/4° or 8<0.7854°

thanks

#### Zazoo

Joined Jul 27, 2011
114
I have these:
Resistors are just as they are - no change
Conversion of capacitor to phasor domain, Zc = 0-j2.0Ω
Conversion of inductor to phasor domain, Zl = 0+1.0Ω
Voltage source V1 = 8<π/4° or 8<0.7854°
These all look good. I'm guessing Zl was a typo (meant to be 0+j1.0Ω)

Reactive capacitance(Xc) = 0.3183Ω
Reactive inductance(Xl) = 6.2832Ω
It looks like you used 2∏ in calculating these. The frequency is already in radians (1 radian/sec), so you wouldn't use the 2∏ factor.
If frequency is in radians (ω), then Xl = ωL
if frequency is in hertz (f), then Xl = 2∏fL
In otherwords, ω=2∏f

Xc and Xl are just the imaginary parts of the complex impedances Zc and Zl, i.e. Xc = -2.0 and Zl = 1.0

Thread Starter

#### lemon

Joined Jan 28, 2010
125
these calculations are criminal and i don't know how to do them with a calculator. could someone please check and let me know I'm on track

thank you

#### Zazoo

Joined Jul 27, 2011
114
Complex algebra can be very tedious without a calculator. The TI graphing calculators can easily handle these if your instructor doesn't require you to show all the detail.

Since the question asks for all unknown currents and voltages, it's easier to leave the parallel branches as is (rather than combining them.) I would combine the R3 and L1 components as you did however.
With R3 and L1 joined you can set up a node equation with only one unknown node (and thus only one equation to solve for V1.) Once you have V1 you can just use Ohms law to get all of the branch currents (V/Z=I)

#### thatoneguy

Joined Feb 19, 2009
6,359
HP RPN Scientific calculators can do math with polar and rectangular coordinates without a problem as well.

Though the TI-89 seems to be the calculator of the decade.

Thread Starter

#### lemon

Joined Jan 28, 2010
125
you mean use the Node Voltage Method? I hope not. I'm not understanding that very well but I suppose it will be good practice

Thread Starter

#### lemon

Joined Jan 28, 2010
125
check this out!! Free online graphing calculator ti
http://rentcalculators.org/stheli.html

• Georacer

#### Zazoo

Joined Jul 27, 2011
114
You mean use the Node Voltage Method? I hope not. I'm not understanding that very well but I suppose it will be good practice
Yes. We can help you anywhere you get stuck or make a mistake.

Thread Starter

#### lemon

Joined Jan 28, 2010
125
@Zazoo
Is V1 the voltage across (r3 -- L)?
Or the 8cos(t + pi/4)?

#### Zazoo

Joined Jul 27, 2011
114
@Zazoo
Is V1 the voltage across (r3 -- L)?
Or the 8cos(t + pi/4)?
If you choose the bottom of the circuit as your reference point (your zero, or ground), then V1 is the node voltage at the node where I, I1, I2, and I3 meet. V1 was just an arbitrary variable I used since the node isn't named in the schematic.

Thread Starter

#### lemon

Joined Jan 28, 2010
125
ok. I'm confused by the first step. In the page explaining Node Voltage Method the first step says:
A voltage source in series with a resistance must be replaced by an equivalent current source in parallel with the resistance. We will write KCL equations for each node. The right hand side of the equation is the value of the current source feeding the node.
No problem. If you have a voltage source that is an integer not a trig equation. How do I divide 8cos(t+pi/4) by 2.0 ohms?

Thread Starter

#### lemon

Joined Jan 28, 2010
125
oh wait wait!! I think we covered that right. V1 is 8V.
So:
I1 = E1/R1 = 8/2 = 4A
Am I good?

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