Frequency in a circuit using complex notation

[hockeynut]

Homework Statement

The frequency, ω, of the source in the circuit of Figure 2 is adjusted until ig is in phase with vg.

(a)using complex notation, determine the value of ω (rad/sec)
(b)if vg = 20 sin ω t [where ω is the frequency found in 2(a)], determine an expression for the output voltage, v0

Homework Equations

The Attempt at a Solution

See attached attempt on Question A & B - can someone advise if it looks right?

 

[gneill]

I think your frequency ended up being one order of magnitude too high. Check your math.

 

[hockeynut]

Thanks gneill.
Ah so where I have 10 to the power of 8, it should be to the power of 6?

 

[gneill]

Thanks gneill.
Ah so where I have 10 to the power of 8, it should be to the power of 6?

Well, you've got both 10⁶ and 10⁸ in the relevant area of your work, so you really should be more specific. But yes, that is the issue.

 

[DeeNos]

I would like to commend you on your attempt and provide some feedback and clarification on complex notation and its use in circuit analysis.

Firstly, complex notation is a mathematical tool used to represent sinusoidal signals in a more compact and efficient way. It is based on the use of complex numbers, where the real part represents the amplitude of the signal and the imaginary part represents the phase shift.

In order to use complex notation in circuit analysis, we use the concept of phasors. A phasor is a vector that represents the magnitude and phase of a sinusoidal signal. In the case of a circuit, we use phasors to represent the voltage and current signals at a given frequency.

In the given circuit, we are asked to determine the value of ω (rad/sec) at which ig is in phase with vg. This means that the phase difference between the two signals is 0 degrees, or in complex notation, the imaginary parts of their phasors are equal.

To determine the value of ω, we can use the admittance method. This method involves finding the admittance of each component in the circuit and then equating them to each other. In this case, we can equate the admittance of the capacitor and the inductor, as they have equal but opposite reactance. This will give us the value of ω.

Once we have determined the value of ω, we can use it to find the expression for the output voltage, v0. This can be done by converting the given voltage source, vg, into its complex notation form and then using Kirchhoff's voltage law to find the voltage across the resistor, which is also the output voltage.

In conclusion, complex notation is a useful tool in circuit analysis, especially for sinusoidal signals. It allows us to simplify calculations and find solutions more efficiently. I hope this explanation helps with your understanding and future use of complex notation in circuit analysis. Keep up the good work!