lect27_phasors

lect27_phasors - Phasors Part II / Review Prof. Niknejad...

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Department of EECS University of California, Berkeley Phasors Part II / Review Prof. Niknejad
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EECS 40/100 Spring 2010 Prof. A. Niknejad Department of EECS University of California, Berkeley Review of LTI Systems Since most periodic (non-periodic) signals can be decomposed into a summation (integration) of sinusoids via Fourier Series (Transform), the response of a LTI system to virtually any input is characterized by the frequency response of the system: Any linear circuit With L,C,R,M and dep. sources Amp Scale Phase Shift
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EECS 40/100 Spring 2010 Prof. A. Niknejad Department of EECS University of California, Berkeley “Proof” for Linear Systems For an arbitrary linear circuit ( L , C , R , M , and dependent sources), decompose it into linear sub- operators, like multiplication by constants, time derivatives, or integrals: For a complex exponential input x this simplifies:
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EECS 40/100 Spring 2010 Prof. A. Niknejad Department of EECS University of California, Berkeley “Proof” (cont.) Notice that the output is also a complex exp times a complex number: The amplitude of the output is the magnitude of the complex number and the phase of the output is the phase of the complex number
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EECS 40/100 Spring 2010 Prof. A. Niknejad Department of EECS University of California, Berkeley Complex Transfer Function Excite a system with an input voltage (current) x Define the output voltage y (current) to be any node voltage (branch current) For a complex exponential input, the “transfer function” from input to output: We can write this in canonical form as a rational function:
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EECS 40/100 Spring 2010 Prof. A. Niknejad Department of EECS University of California, Berkeley Impede the Currents ! Suppose that the “input” is defined as the voltage of a terminal pair ( port ) and the “output” is defined as the current into the port: The impedance Z is defined as the ratio of the phasor voltage to phasor current (“self” transfer function) + Arbitrary LTI Circuit
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EECS 40/100 Spring 2010 Prof. A. Niknejad Department of EECS University of California, Berkeley Admit the Currents! Suppose that the “input” is defined as the current of a terminal pair ( port ) and the “output” is defined as the voltage into the port: The admmittance Z is defined as the ratio of the phasor current to phasor voltage (“self” transfer function) + Arbitrary LTI Circuit
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lect27_phasors - Phasors Part II / Review Prof. Niknejad...

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