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HighPass Filters 

The basic firstorder highpass filters use the same components as the lowpass filters we just studied. However, their positions are swapped. Thus, the RC highpass filter has the capacitor in series with the signal and the resistor across the output, as shown in the first diagram to the right. At high frequencies, C has very low impedance, and the signal passes through unhindered. As the frequency decreases, however, X_{C} becomes significant, until at the cutoff frequency, X_{C} = R, just as with the lowpass filter. At still lower frequencies, X_{C} increases, and less of the signal reaches the output.
One useful feature of the RC highpass filter is that the capacitor serves to block direct current between v_{IN} and v_{OUT}. Thus, two circuits that operate at different DC voltages can be connected by this type of highpass filter without encountering any problems with dc component bias voltages as a consequence.
The RL version of the highpass filter uses a series resistor and a shunt inductor to accomplish its purpose. At high frequencies, X_{L} is large, so the inductor is nearly an open circuit for such signals. At low frequencies, X_{L} is very small, and effectively connects those signals directly to ground. As before, the cutoff frequency occurs where R = X_{L}, so that L is just beginning to have a significant effect on the signal.
As shown to the right, the frequency response of a basic highpass filter is actually a mirror image of its lowpass counterpart. At the cutoff frequency where R = X_{L} or R = X_{C}, the attenuation is only 3 db, so the signal voltage is still 70.7% of its higherfrequency value.
Below the cutoff frequency, attenuation increases at the rate of 20 db per decade, which is the same rolloff as for the lowpass filter. Above the cutoff frequency, attenuation rapidly decreases to nothing, and all higher frequencies pass with ease.
The green line in the graph is the straight line extension of the constant slopes of the actual frequency response. As with the lowpass filter, the intersection point is the cutoff frequency. This straightline approximation of the real frequency response curve is very easy to draw, and is sufficiently accurate for some kinds of applications. Of course, the actual curve near the cutoff frequency is understood.
As we have already seen, in a firstorder lowpass filter, v_{OUT} always lags v_{IN} by some phase angle betweeen 0 and 90°. Exactly the reverse is true for a firstorder highpass filter: as shown in the vector diagrams to the right, v_{OUT} is always taken from across the component whose voltage leads v_{IN} by some phase angle, φ.
For the RL filter, v_{OUT} is taken from across L, so its phase angle necessarily leads v_{IN} as shown in the upper vector diagram. For the RC filter, v_{OUT} is taken from across R, which again leads v_{IN} as shown in the lower vector diagram.
Of course, the actual phase angle by which v_{OUT} leads v_{IN} depends on the specific frequency of the signal, as compared to the cutoff frequency of the filter. As shown in the phase diagram to the right, signals more than 10 times the cutoff frequency show little or no appreciable phase shift, while signals less than 0.1 times the cutoff frequency are shifted close to 90°. Most of the change in phase occurs within a factor of 0.1 to 10 times ω_{CO}.
As with the lowpass filter, nonsinusoidal signals with frequency components at or near ω_{CO} will be distorted when passing through the highpass filter.
The equations for the highpass filter are very similar to the ones for the lowpass filter, as you might expect. Certainly the basic comparisons are the same. Thus, for the RC circuit at the cutoff frequency,
R  =  X_{C} 
=  1  
ω_{CO}C  
ω_{CO}  =  1 
RC 
This much is the same as for the RC lowpass filter. However, because the components have been swapped, the equation for attenuation over a frequency range has become:
v_{OUT}  =  R  
v_{IN}  Z  
=  R  
(R² + X_{C}²)^{½}  
=  R  
(R² + (1/ωC)²)^{½}  
=  1  
1/R × (R² + (1/ωC)²)^{½}  
=  1  
(1 + (1/RωC)²)^{½} 
In this equation, the higher the value of ω, the less effect it has on v_{OUT}. This is exactly the reverse of the lowpass filter, where high values of ω would seriously reduce v_{OUT}. This is in fact the essential difference between the lowpass filter and the highpass filter.
The RL filter behaves the same way, except that at the cutoff frequency:
X_{L}  =  R 
ω_{CO}L  =  R 
ω_{CO}  =  R 
L 
Then, over the frequency spectrum,
v_{OUT}  =  X_{L}  
v_{IN}  Z  
=  X_{L}  
(R² + X_{L}²)^{½}  
=  ωL  
(R² + (ωL)²)^{½}  
=  1  
1/ωL × (R² + (ωL)²)^{½}  
=  1  
((R/ωL)² + 1)^{½} 
At high frequencies, the fraction R/ωL becomes very small and has negligible effect on the signal. At very low frequencies, this fraction becomes very large and blocks nearly all of the signal. And at cutoff, ω = ω_{CO} = R/L, so that v_{OUT}/v_{IN} = 1/ = 0.707 as expected. Thus, the frequency response of the RL filter is exactly the same as the frequency response of the RC filter.
As with the lowpass filters, the phase response for both highpass filters near ω_{co} is exactly the same, and may be calculated as:
φ  =  arctan  X_{C}  =  arctan  R 
R  X_{L}  
=  arctan  1  =  arctan  R  
ωRC  ωL 


 
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