ECE 202 Lesson 10 - 29 29 ∑ ∞ = = odd 1 sin 1 4 n n m g...

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1 Circuit Theory II Lesson 10 The RC Integrator An Application of Fourier Series Response of RC Circuit to Square Wave Effect of the RC Time Constant on the Output Section 16.5, pp. 670-674
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2 Triangle Wave + + + - 2 2 2 5 5 cos 3 3 cos 1 cos 4 2 x x x π π
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3 Application of Fourier Series Application of Fourier series: response of a circuit to a periodic signal ( 29 ( 29 = = odd 1 0 sin 4 n n m g t n n V t v ϖ π
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4 Response to a Sinusoid g g V θ = g V Input phasor Phasor impedance of resistor o 0 = R R Phasor impedance of capacitor o 90 1 1 - = C C j ϖ ϖ o o o V θ = V Output phasor
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5 Computation of Output Phasor RC j C j R C j ϖ ϖ ϖ + = + = 1 V V 1 1 V g g o ( 29 ϖ τ θ τ ϖ ϖ τ 1 2 2 g o tan 1 1 V V - - + = + = g g V j RC = τ Time constant for a sinusoid of frequency ϖ
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6 Comparing the Input and Output Sinusoids ( 29 ( 29 ϖ τ θ ϖ τ ϖ θ ϖ 1 2 2 tan sin 1 sin - - + + + g g g g t V t V input output
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7 n-th Harmonic of Response to Square Wave ( 29 ( 29 τ ϖ ϖ τ ϖ π ϖ π 0 1 0 2 2 0 2 0 tan sin 1 4 sin 4 n t n n n V t n n V m m - - + n-th harmonic of square wave n-th harmonic of response to square wave
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Unformatted text preview: ( 29 ( 29 ∑ ∞ = = odd 1 sin 1 4 n n m g t n n V t v ϖ π ( 29 ∑ ∞ =- - + = odd 1 1 2 2 2 tan sin 2 1 1 4 n n m o T n t n T n n V t v τ input output 9 Effect of Time Constant on Output: Small τ ( 29 ( 29 ( 29 ∑ ∞ =-= odd 1 sin 1 4 n n m o t n n V t v τ ϖ π ( 29 ( 29 ∑ ∞ = = odd 1 sin 1 4 n n m g t n n V t v Square wave Delayed square wave τ << T 10 Effect of Time Constant on Output: Large τ ( 29 ( 29 ∑ ∞ = = odd 1 sin 1 4 n n m g t n n V t v ϖ π Square wave Triangle wave τ T ( 29 ( 29 ∑ ∞ =-= odd 1 2 cos 1 4 1 n n m o t n n V t v τ...
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