Phase constant for high pass filter

AI Thread Summary
The discussion focuses on deriving the phase constant δ for a high-pass filter output voltage given an input voltage expressed as Vin = Vin peak cos ωt. The output voltage is defined as Vout = VH*cos(ωt - δ), with VH depending on the frequency, resistance, and capacitance. A participant suggests substituting the capacitor with its impedance to simplify the circuit into a voltage divider, allowing for the calculation of Vout in terms of Vin. The phase constant can then be determined using the arctan function based on the circuit's real and imaginary components. The final expression for δ is arctan(-1/(ωRC)), considering the negative sign in the phase definition.
dstdnt
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Homework Statement



29-p-044.gif


If the input voltage is given by Vin = Vin peak cos ωt, then the output voltage is
Vout = VH*cos(ωt - δ) where VH = Vin peak/(1 + (ωRC)^(−2)) (Assume that the output is connected to a load that draws only an insignificant amount of current.)

Find an expression for the phase constant δ in terms of ω, R and C.


Homework Equations




The Attempt at a Solution


My teacher specifically told us to solve this problem without using phasors, but the only examples in the book of how to solve this kind of problem use phasors. I know that the answer is arctan(-1/(ωRC)), but I have no idea how to derive that!
 
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Hello dstdnt,

Welcome to physics forums!

The way I would do it is, in the diagram, replace the label "C" with the capacitor's impedance. In other words, replace "C" with "1/(jωC)". Then the circuit is just a simple voltage divider. Solve for Vout in terms of Vin.

The phase is arctan([imaginary part]/[real part]). That's almost the same thing as δ. [Notice the problem statement defined Vout = VH*cos(ωt - δ), where there is a minus sign attached to δ. You'll have to take that into account] :wink:
 
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Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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