lecture 201 Phasor Relationships for Circuit Elements (7.4) Prof. Phillips April 16, 2003

lecture 202 Phasor Relationships for Circuit Elements Phasors allow us to express current-voltage relationships for inductors and capacitors much like we express the current-voltage relationship for a resistor. A complex exponential is the mathematical tool needed to obtain this relationship.

lecture 203 I-V Relationship for a Resistor Suppose that i(t) is a sinusoid: i(t) = I M e j(  t+  Find v(t) R v(t) + – i(t)

lecture 204 Computing the Voltage

lecture 205 Class Example

lecture 206 I-V Relationship for a Capacitor Suppose that v(t) is a sinusoid: v(t) = V M e j(  t+  Find i(t) C v(t) + – i(t)

lecture 207 Computing the Current

lecture 208 Phasor Relationship Represent v(t) and i(t) as phasors: V = V M   I = j  C V The derivative in the relationship between v(t) and i(t) becomes a multiplication by j  in the relationship between V and I.

lecture 209 Example v(t) = 120V cos(377t + 30  ) C = 2  F What is V? What is I? What is i(t)?

lecture 2010 Class Example

lecture 2011 I-V Relationship for an Inductor V = j  L I L v(t) + – i(t)

lecture 2012 Example i(t) = 1  A cos(2  t + 30  ) L = 1  H What is I? What is V? What is v(t)?

lecture 2013 Class Example

lecture 2014 Circuit Element Phasor Relations (ELI and ICE man)

lecture 2015 Phasor Diagrams A phasor diagram is just a graph of several phasors on the complex plane (using real and imaginary axes). A phasor diagram helps to visualize the relationships between currents and voltages.

lecture 2016 An Example – 1F1F VCVC + – 2mA  40  1k  VRVR + + – V  = 377

lecture 2017 An Example (cont.) I = 2mA  40  V R = 2V  40  V C = 5.31V  -50  V = 5.67V  

lecture 2018 Phasor Diagram Real Axis Imaginary Axis VRVR VCVC V

lecture 2019 MATLAB Exercise Let’s use MATLAB to plot an ac current and voltage, and then to graphically determine the lead-lag relationship Start MATLAB on your computer We begin by creating a time vector >> t = 0 : : 0.025; Next, we create the voltage and current >> vt = 170 * cos(377*t+10*pi/180); >> it = 100 * cos(377*t-65*pi/180);

lecture 2020 MATLAB Exercise Now we will graph v(t) and i(t) >> plot(t,vt,'b',t, it,'r--'); >> xlabel('Time (sec)'); >> ylabel('Voltage (Volts) or Current (Amps)'); >> title('Household AC Voltage-Current'); >> legend('v(t)=170cos(377t+10)', 'i(t)=100cos(377t-65)');

lecture 2021 MATLAB Exercise From the graphs created: –Determine whether the current leads the voltage, or vice versa –Determine the amount of lead by the current or voltage Compare the voltage-current lead-lag relationship obtained by graphical means above to an analytic solution which you should be able to compute