We look at the free oscillating circuit State equation is or The characteristic

# We look at the free oscillating circuit state

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We look at the free-oscillating circuit: State equation is: or The characteristic equation is
12 Prof. C.K. Tse: Dynamic circuits: Frequency domain analysis Example Solving the characteristic equation gives Clearly, if R, L and C are positive, then all λ ’s are negative.
13 Prof. C.K. Tse: Dynamic circuits: Frequency domain analysis Natural frequencies Natural frequencies = eigenvalues, λ k Roots of characteristic equation For an n th order system, there are n natural frequencies. The system solution is complex Exponential decay Exponential growth (unstable) Oscillatory Oscillatory with decaying amp Oscillatory with growing amp (unstable) dynamical modes
14 Prof. C.K. Tse: Dynamic circuits: Frequency domain analysis The complex frequency plane j ω σ t Exponential decay t Exponential growth x x sine wave s-plane x x x x
15 Prof. C.K. Tse: Dynamic circuits: Frequency domain analysis Sources Suppose a voltage has a complex frequency s . v ( t ) = V o e st If s = ±j ω , then it is pure sinusoidal since Sine waves have pure imaginary frequencies, ±j ω rad/sec. cos ω ω ω t e e j t j t = + 2
16 Prof. C.K. Tse: Dynamic circuits: Frequency domain analysis Impedances We can also extend the concept of complex frequency to impedance. Recall: in last lecture, we said that • the impedance of an inductor is j ω L when it is driven by a sine source. • the impedance of a capacitor is 1/j ω C when it is driven by a sine source. Now, we imagine the driving frequency is s . Thus, we have Z L = sL Z c = 1/ sC V L = sL I L I C = sC V C
17 Prof. C.K. Tse: Dynamic circuits: Frequency domain analysis Example Consider the impedance: Thus, we can write: The free-oscillating circuit will have The characteristic equation is The natural frequency (eigenvalue) is λ = 1/ CR which is 1/ τ .
18 Prof. C.K. Tse: Dynamic circuits: Frequency domain analysis Transfer functions Transfer function — ratio of two quantities 1. Voltage gain: v 2 /v 1 2. Current gain: i 2 /i 1 3. Trans-admittance: i 2 /v 1 4. Trans-impedance: v 2 /i 1 + V 1 driven at s I 2 F ( s ) = I 2 (s)/ V 1 (s) in the s domain all voltages and currents are set to frequency s linear circuit
19 Prof. C.K. Tse: Dynamic circuits: Frequency domain analysis Example Suppose the circuit is driven by V 1 at frequency s . So, the inductor impedance = sL the capacitor impedance = 1/ sC We can redraw the circuit as The transfer function from V 1 to V 2 is
20 Prof. C.K. Tse: Dynamic circuits: Frequency domain analysis Getting characteristic equation from transfer function The transfer function from V 1 to V 2 is Clearly, the free-oscillating circuit can be formed by setting V 1 =0. Thus, the characteristic equation is just = 0 In general, F ( s ) = Char. Eqn. is D ( s ) = 0
21 Prof. C.K. Tse: Dynamic circuits: Frequency domain analysis Transfer function on complex plane What do we mean by transfer function? It is the ratio of V 2 to V 1 at complex frequency s .

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• Summer '16
• Martin Chow
• Signal Processing, Complex number, Complex Plane, Prof. C.K. Tse

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