Alternating current analysis

Solution

Since this is the first AC problem we will solve, I will explain every step, including the manual ones, in full detail.

Step 1: sinusoids to phasors. We have to express the value of the source in terms Symbulator can understand, so we convert it from a sinusoid to a phasor: 10∠0°. We can pass this as (10∠0°) or, since the angle is zero, simply as 10.

Step 2: note the frequency. The circuit has a capacitor with a value given in farads. Because of this, Symbulator will need the frequency, given by the problem as 4 rad/s, to convert that value into an impedance in ohms.

Step 3: describe the circuit.

"e,1,0,10:r1,1,2,5:c,2,0,.1"→cir
s\ac(cir,4)

Notice that, in our circuit description, the order we give to the nodes of the resistor and the capacitor is chosen to be convenient for the answers we will ask for.

Step 4: read the answers. Once Symbulator is done, you should take a look at what it found:

• the usual voltages in the nodes, voltage drops in the elements, and currents through the elements, in variables that should be familiar by now
• the average power consumed in the source and the resistor, in ape and apr1. None is given for the capacitor, since capacitors and inductors do not consume real power.
• the complex power consumed in all elements, in sc, se and sr1
• the equivalent impedance of the rest of the circuit as seen by the source, in ze

To get the answers we need for this problem in particular, we ask for ir1 and vc. It is likely that the calculator will give you the answers in rectangular form, with a real part and an imaginary part.

If we want to see them as an amplitude and angle, we can use Symbulator's aa tool:

s\aa(ir1)

1.789∠26.57°

This is correct. You manually convert it to a sinusoid, by putting it back in the same terms the input was given in:

Asking for the capacitor voltage the same way gives 4.472∠-63.43°, that is, \(v(t) = 4.472 \cos(4t - 63.43°)\) V, which is correct.

Future solved problems will not include this level of detail in the solution, only the circuit description and the commands we give.

Solution

"e,1,0,(20∠30°):r1,1,2,4:l,2,0,.2"→cir
s\ac(cir,10)

We ask for the current in the resistor and get 4.472∠3.43°, and for the voltage in the inductor and get 8.944∠93.43°. Both are correct.

Solution

Since this is a passive circuit, to reduce it we will use the er tool.

"ca,1,2,2'm:r1,2,3,3:cb,3,0,10'm:l1,2,4,.2:r2,4,0,8"→cir
s\er(cir,1,0)

When asked what type of analysis, choose AC. Since there are capacitors in farads and an inductor in henries, Symbulator will ask you for the frequency; enter 50. Once the program is done, ask for the equivalent impedance, zeq.

3.22–𝐢11.07

That is 3.22 − j11.07 Ω, which is correct.

Solution

The capacitor is in farads and the inductor in henries, so Symbulator needs the frequency, and the source is entered as its phasor: 50 cos 200t is 50 at an angular frequency of 200.

"e,1,0,50:r,1,2,10:c,2,3,5'm:l,3,0,20'm"→cir
s\ac(cir,200)

Choose AC. Symbulator asks for the frequency because of the farads and henries; enter 200. Then ask for s\aa(ir).

s\aa(ir)

We get 4.789∠-16.7° A.

That is 4.789 A at an angle of −16.7°, which is correct.

Solution

Since the values are in ohms, these are all impedances. This is a passive circuit, so we could use the er tool. However, the structure of this circuit is so simple that we can reduce it by hand. The problem asks for the equivalent admittance, so our answer will be the inverse of the equivalent impedance:

1/(s\pr({4,𝐢8,-𝐢10}))

0.25-0.025𝐢

That is 0.25 − j0.025 S, which is correct.

Solution

The frequency the question gives is superfluous: every value is already in ohms. Nothing here needs a full simulation either; the equivalent impedance is one expression, using the parallel shorthand:

4+𝐢20+s\pr({16,-𝐢14+𝐢25})→zeq

9.135+𝐢27.47

That is 9.135 + j27.47 Ω. The current is the source voltage divided by it:

s\aa(12/zeq)

We get 414.5∠-71.6° mA.

That is 414.5 mA at an angle of −71.6°, which is correct.

Solution

Eight impedances and no source: a job for Find equivalent rather than a solve. The names carry the two nodes each element bridges, a convenience only.

"r10,1,0,𝐢6:r20,2,0,𝐢8:r30,3,0,𝐢8:r40,4,0,𝐢12:r12,1,2,2:r23,2,3,-𝐢6:r34,3,4,4:r14,1,4,-𝐢4"→cir
s\er(cir,1,0)

0.3794+𝐢1.46

That is 0.3794 + j1.46 Ω, which is correct.

Solution

The answer given by the book is 25 µF. But I think this is one of those unfortunately common instances where there is a mistake in the book. I believe they meant to ask "at 2k rad/s", because that is the frequency at which the answer they give is right. Let me tell you how I solved this problem and I will let you be the judge.

"c,1,0,c:r1,1,2,10:l,2,0,5'm"→cir
s\er(cir,1,0)

When asked, specify AC. Then we are asked for the frequency. The problem gives 2000 Hz; Symbulator asks for radians per second, so I enter 2π2E3. I could also have entered 12566.37.

Notice two things. First, I did not use an SI prefix: the shorthand only works in the circuit description string. Second, I used 2E3 rather than 2000 because I wanted an approximate value — with exact values, cSolve took too long.

Once the simulation finishes, solve for the value of c that makes the imaginary part of the equivalent impedance zero:

solve(imag(zeq)=0,c)

The answer is c = 0.000001235, or 1.235 µF. This is not the answer the book gives.

I first used cSolve here, which also works, but my friend Qifan Wang — who verified my answers — pointed out, correctly, that it is not necessary: focusing on the imaginary part removes every reference to the complex operator, so we are solving an equation in real terms only.

Let's now repeat the process, using what I suspect is the frequency they meant: 2000 rad/s. I also want to show you something cool. Now that we know the answer is in the range of µF, we can declare the value of the capacitor as c'µ, so that the value of c we get will be in that scale:

"c,1,0,c'µ:r1,1,2,10:l,2,0,5'm"→cir
s\er(cir,1,0)
solve(imag(zeq)=0,c)

The answer is c = 25, that is to say 25 µF. This is the right answer in the book. So they asked the wrong question — they asked it using the wrong frequency unit.

Solution

"e,1,0,vs:c,1,2,c:r1,2,o,r:o,0,2,o"→cir
s\ac(cir,ω)

We ask for vo/vs and get −c·ω·r·𝐢, which is correct.