INC 112 Basic Circuit Analysis Week 9 Force Response of a Sinusoidal Input and Phasor Concept

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 displacement Period that have transient

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

Input Output Phase shift Amplitude

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 Sine wave v1(t) Sine wave v L (t) Sine wave

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

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

Find i(t) from

ωL เรียก ความต้านทานเสมือน (impedance) Phase shift -90

Phasor Diagram of an inductor v i Power = (vi cosθ)/2 = 0 Phasor Diagram of a resistor v i Power = (vi cosθ)/2 = vi/2 Note: No power consumed in inductors i lags v

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

Phasor Diagram of a capacitor v i Power = (vi cosθ)/2 = 0 Phasor Diagram of a resistor v i Power = (vi cosθ)/2 = vi/2 Note: No power consumed in capacitors i leads v

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

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

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 form3+j4 Polar form5∟53 o Operation add, subtract, multiply, division?

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

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

Rectangular form Add, Subtraction Polar form Multiplication Division

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

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

Example Find i(t), v R (t), v L (t) Phasor form

V I

In an RLC circuit with sinusoidal voltage/current source, voltages and currents at all points are in sinusoidal wave form too but with different amplitudes and phase shifts.

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), v L (t)

V I VR VL Phasor Diagram Resistor consumes power Inductor consumes no power P = 0