Week 11 Force Response of a Sinusoidal Input and Phasor Concept Network Analysis Week 11 Force Response of a Sinusoidal Input and Phasor Concept

Two Types of Analysis non-periodic electric source (Transient response analysis of a step input) (Steady state response analysis of a sinusoidal input)

Forced Response of Sinusoidal Input In this part of the course, we will focus on finding the force response of a sinusoidal input.

Start oscillate from stop input Period that have transient displacement

Have oscillated for a long time input displacement We will only be interested in this case for force response (not count the transient)

Theory Force response of a sinusoidal input is also a sinusoidal signal with the same frequency but with different amplitude and phase shift. v2(t) Sine wave v1(t) Sine wave Sine wave vL(t) Sine wave

Phase shift Input Amplitude Output

What is the relationship between sin(t) and i(t) ? Phase shift sin(t) i(t)

R circuit Find i(t) Note: Only amplitude changes, frequency and phase still remain the same.

L circuit Find i(t) from

ωL เรียก ความต้านทานเสมือน (impedance) Phase shift -90

Phasor Diagram of an inductor Phasor Diagram of a resistor v v i i Note: No power consumed in inductors i lags v 90o

C circuit Find i(t) ความต้านทานเสมือน (impedance) Phase shift +90

Phasor Diagram of a capacitor Phasor Diagram of a resistor i v v i Note: No power consumed in capacitors i leads v 90o

Kirchhoff's Law with AC Circuit KCL,KVL still hold. vR i v(t) i vC

This is similar to adding vectors. Therefore, we will represent sine voltage with a vector. 3 5 4

Vector Quantity Complex numbers can be viewed as vectors where X-axis represents real parts Y-axis represents imaginary parts There are two ways to represent complex numbers. Cartesian form 3+j4 Polar form 5∟53o Operation add, subtract, multiply, division?

Complex Number Forms (Rectangular, Polar Form) θ a Interchange Rectangular, Polar form

บวก ลบ คูณ หาร vector ?? Rectangular form: 4 + j3 s = 4 + j3 3 σ 4 Rectangular form: 4 + j3 Polar form magnitude=5, angle = 37 บวก ลบ คูณ หาร vector ??

Rectangular form Add, Subtraction Polar form Multiplication Division

Impedance Compare to ohm’s law, impedance is a ratio of V/I in when V and I is in the vector format. Inductor

Capacitor

Note: Impedance depends on frequency and R,L,C values Example: Find impedance in form of polar value for ω = 1/3 rad/sec

Rules that can be used in Phasor Analysis Ohm’s law KVL/KCL Nodal, Mesh Analysis Superposition Thevenin / Norton

Summary of Procedures Change voltage/current sources in to phasor form Change R, L, C value into phasor form Use DC circuit analysis techniques normally, but the value of voltage, current, and resistance can be complex numbers Change back to the time-domain form if the problem asks.

Example Find i(t), vR(t), vL(t) Phasor form

V I

Example Find i(t), vL(t)

Phasor Diagram VL V I VR