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Unformatted text preview: EE201.3 Frequency Effects – Series & Parallel Resonance Denard Lynch Page 1 of 13 Nov 2008 R – L in Series: Z = R + j ω L at very low frequencies, ( ω ⇒ 0, j ω L ⇒ 0) Z = R ∠ 0 Ω at very high frequencies, ( ω ⇒ ∞ , j ω L ⇒ ∞ ) Z = ∞∠ 90 Ω Define ω c (f c ) when Z R  = Z L  ⇒ R = ω c L; ω c = R/L = τ1 when ω = ω c = R/L Z = R + j(R/L)L = R(1 +j1) = 1.414R ∠ 45 Ω when ω = 0.1 ω c Z = R + j(R/10L)L = R(1 +j0.1) = 1.005R ∠ 5.7 Ω when ω = 10 ω c Z = R + j(10R/L)L = R(10 +j1) = 10.04R ∠ 84 Ω For instance… for a 10K Ω resistor and 10mH inductor: EE201.3 Frequency Effects – Series & Parallel Resonance Denard Lynch Page 2 of 13 Nov 2008 R – L in Parallel Z = R ( ) j ω L ( ) R + j ω L ∗ 1 j ω L 1 j ω L ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ = R R j ω L + 1 = R 1 R j ω L + 1 ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ = R 1 1 − j 1 ω L R ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ When ω is very low (as ω⇒ 0, 1/ ω⇒∞ ) Z ≅ R/(1 j ∞ ) = ∠ 90 Ω . When ω is very high (as ω⇒∞ , 1/ ω⇒ 0) Z ≅ R/(1j0) = R ∠ Ω . Again, cutoff frequency, ω c , is when ⎢ Z R ⎢ = ⎢ Z L ⎢ and ω c = R/L ; = τ1 and Z = R/(1j1) = 0.707R ∠ 45 Ω At ω = 0.1 ω c , Z = R/(1j10) ≅ 0.1R ∠ 84 Ω , At ω = 10 ω c , Z = R/(1+j10) ≅ 0.995R ∠ 5.7 Ω . For instance… for a 10K Ω resistor and 10mH inductor: EE201.3 Frequency Effects – Series & Parallel Resonance Denard Lynch Page 2 of 13 Nov 2008 R – L in Parallel Z = R ( ) j ω L ( ) R + j ω L ∗ 1 j ω L 1 j ω L ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ = R R j ω L + 1 = R 1 R j ω L + 1 ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ = R 1 1 − j 1 ω L R ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ When ω is very low (as ω⇒ 0, 1/ ω⇒∞ ) Z ≅ R/(1 j ∞ ) = ∠ 90 Ω . When ω is very high (as ω⇒∞ , 1/ ω⇒ 0) Z ≅ R/(1j0) = R ∠ Ω . Again, cutoff frequency, ω c , is when ⎢ Z R ⎢ = ⎢ Z L ⎢ and ω c = R/L ; = τ1 and Z = R/(1j1) = 0.707R ∠ 45 Ω At ω = 0.1 ω c , Z = R/(1j10) ≅ 0.1R ∠ 84 Ω , At ω = 10 ω c , Z = R/(1+j10) ≅ 0.995R ∠ 5.7 Ω . For instance… for a 10K Ω resistor and 10mH inductor: EE201.3 Frequency Effects – Series & Parallel Resonance Denard Lynch Page 3 of 13 Nov 2008 R – C in Series: Z = R − jX C = R − j 1 ω C ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = R + 1 j ω C ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = R j ω C ( ) + 1 j ω C = 1 + j ω RC j ω C at very low frequencies, ( ω ⇒ 0, X C ⇒ ∞ ) Z ≅ ∞∠90 Ω at very high frequencies, ( ω ⇒ ∞ , X C ⇒ 0) Z = R ∠ Ω Define ω c (f c ) when Z R  = Z C  ⇒ R = 1/( ω c C); ω c = 1/RC = τ1 when ω = ω c = 1/RC, Z = R  j(R) = R(1 j1) = 1.414R ∠45 Ω when ω = 0.1 ω c Z = R  j(10R) = R(1 +j10) = 10.04R ∠84 Ω when ω = 10 ω c Z = R + j(0.1R) = R(1 +j0.1) = 1.005R ∠5.7 Ω For instance… for a 10K Ω resistor and 10mH inductor: For instance… for a 10K...
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This note was uploaded on 09/25/2010 for the course EE 201 taught by Professor Linch during the Spring '10 term at University of Saskatchewan Management Area.
 Spring '10
 Linch
 Frequency, Parallel Circuit

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