## Chapter 24. Analysis of Signalized Intersections

Chapter objectives: By the end of this chapter the student will:         Understand the conceptual framework for the HCM 2010 method, including Critical lane group, v/s ratio, saturation flow rate, capacity of a lane group, v/c ratio for lane group, approach and intersection v/c, LOS, and effective green times and lost time Have general ideas of the modules of the Highway Capacity Manual 2010: Input data, Define movement groups, Compute lane group flow rate, Input or compute phase duration, Compute capacity, and Compute performance measures Will be able to explain how Arrival Type is determined Know how to enter input data into the Highway Capacity Software (HCS2010) Explain the terms of delay models (including Incremental Queue Analysis) Know how to deal with initial queues Understand how the permitted left turns are modeled by the HCM 2010 Understand how the left-turn adjustment factor for compound (protected/permitted) phasing is modeled.

Chapter 24 1

### 24.1 Introduction

What’s new in HCM 2010 as compared with HCM 2000 1.

2.

3.

The model has been set up to handle actuated signal analysis directly.

The estimation of delay is now partially modeled using Incremental Queue Analysis (IQA). IQA allows a more detailed analysis of arriving and departing vehicle distributions.

The definition of lane groups has been altered. Lane groups are identified and separately analyzed as part of the methodology.

“This text focuses on the analysis of pretimed signals because it is more straight forward to present basic modeling theory for fixed time signals.” Chapter 24 2

Chapter 24 3

v/s

Chapter 24 4

### 24.2.1 The Critical-Lane Group Concept

Critical lane analysis (Section 17.3) vs. Critical lane group analysis Critical lane analysis compares actual flow (

v)

with the saturation flow rate (

s

) and capacity (

c

) in a single lane. Critical lane group analysis compares actual flow (

v

) with the saturation flow rate (

s

) and capacity (

c

) in a group of lanes operating in equilibrium. In either case, the ratio of

v

to

c

is the same (when traffic is Exclusive right- or left turn lanes must be separately analyzed because they are separate lane groups.

Lane utilization is considered in computing saturation flow rate.

5 This applies to shared lanes, also.

24.2.2 The

v/s

ratio as a measure of demand & 24.2.3 Capacity and saturation flow rate concepts * The simple method in Chapter 21 (as a comparison – V c is adjusted by converting into tvu (through vehicle unit) & saturation flow is given):

C des

 1 

Nt L V c PHF

(

v

/

c

)( 3600 /

h

) * In the HCM model, demand flow rates are not converted to tvu. It uses veh/hr (though adjusted for PHF). A key part of the HCM 2010 model is a methodology for estimating the saturation flow rate of any lane group based on known prevailing traffic parameters.

s i

s

0

N

i f i

We may not be able to compare directly lane groups because their conditions are different. So HCM use the

flow ratio

,

v/s

, a dimensionless value for comparison purposes. This process is called “normalization.” Chapter 24 6

24.2.3 Capacity (continued)  In the simple timing method in Chapter 21, the capacity of the intersection as a whole was considered.

V c

T G h

 1

h

  3600 

Nt L

 HCM 2010 as well as HCM 2000 gives the capacity of each lane group.

 Demand does not necessarily peak at all approaches at the same time.

 Capacity may change for each approach during the day. (like the effect of curb side parking, bus blocking, etc.)  Capacity is provided to movements to satisfy movement demands. (Note: the critical capacity ratio v/c (for the intersection as a whole) is still calculated in HCM 2010 just like HCM 2000).

Chapter 24

c i

 3600

C

 

s i

7

g i C

### “degree of saturation”

Three issues: (1) Capacity is practically always estimated (because it is difficult to measure.) (2) In existing cases demand is often measured by “departure flows” although it should be “arrival flows.” (3) For future cases, predicted arrival volumes are given (by a planning model) instead of actually counted volumes.

Case 1 & 2: v/c > 1.0 resulted in a HCM analysis for an existing signalized intersection.

If demand is measured by a departure flow (assuming it was correct), this cannot be accepted because max value v/c = 1.0. If arrival flows are measured, v/c > 1.0 may occur – this becomes obvious because queue forms). Capacity must have been underestimated if queue is not formed despite the fact v/c > 1.0 results. Capacity underestimation is possible because HCM models are national average models.

Chapter 24 8

Case 3: v/c > 1.0 resulted in an analysis for a planned signalized intersection.

In a planning case, both demand and capacity are estimates. But, it may indicate that the forecast demand flow exceeds the estimated capacity of the lane group, and a problem will likely occur. Demand is an arrival flow for a predicted case because those values come from a planning model.

Computation of a v/c ratio (

degree of saturation

) for a given lane group (this model does not change among different HCM versions:

X i

v c i i

s i v i g i C

v i g i s i C

Flow ratio/Green ratio Chapter 24 9

Computation of a v/c ratio for an intersection as a whole (p.576): The critical v/c ratio for the intersection  defined as the sum of the critical lane group flows divided by the sum of the lane group capacities available to serve them (compare this one with the Simple Method in Ch 20).

X c

  

v ci g ci s ci C

  

g ci C ci

 

C

L ci

   

ci C C

L C X c

   

ci C C

L X c

min    

ci C C

max  max

L

If the X c > 1.0, then the physical design, phase plan, and cycle length specified do not provide sufficient capacity for the anticipated or existing critical lane group flows.  Do something to increase capacity: (1) longer cycle lengths (less number of cycles, less lost time) , (2) better phase plans (improved LT treatment) , and (3) add critical lane group or groups (meaning change approach layouts  increase capacity) Chapter 24 10

Computation of a v/c ratio for an intersection as a whole (Additional comments):  If the critical v/c ratio is less than 1.00, the cycle length, phase plan, and physical design provided are sufficient to handle the demand and flows specified.

 But, having a critical v/c ratio under 1.00 does not assure that every critical lane group has v/c ratios under 1.00. When the critical v/c ratio is less than 1.00, but one or more lane groups have v/c rations greater than 1.00, the green time has been misallocated. Chapter 24 11

24.2.4 Level of service concepts and criteria  All the HCM delay models assume random arrivals. Hence, the delay model produce delays for approaches with random arrivals. Urban signals are coordinated; hence, many do not have random arrivals. This is corrected by the “quality of progression” factor called “Arrival Type” factor. See Table 24.3 and 24.4. There are 6 arrival types: 1 = poor coordination, 6 = exceptionally good coordination.

 For uninterrupted facilities, like freeways, v/c has a direct connection with the performance of the facility. So, if v/c = 1.0, the facility is at the capacity.

 For signalized intersections (interrupted facilities), this is not necessarily true – especially when delay is used as the MOE.

 You may get LOS=F even if v/c is well below 1.0. For instance LT vehicles may have a long stopped delay even if its v/c is low.

 HCM 1994 delay model focuses on the first 15-min interval. So, even if it is over-saturated (v/c > 1.0), we get a relatively smaller delay. HCM 2010 has 3 study approaches: Single analysis period for 15 min and 1 hour, and multiple 15-min analysis periods.

Chapter 24 12

 The 2010 HCM uses “total control delay” consisting of three terms Total control delay per vehicle = time in queue delay + acceleration deceleration delay New to HCM 2010: Any lane group operating at a v/c ratio greater than 1.00 is also labeled as LOS F.

Because delay is difficult to measure in the field and because it cannot be measured for future situations, delay is estimated using analytic models. The delay models are discussed in section 24.3.7. Chapter 24 13

A B C D

### 24.2.5 Effective green times and lost times

l 1

G

e

y

l 2

ar R

R t L g R r g r

A. Actual signal indications B. Actual use of green and yellow; e is extended green, i.e. part of the yellow used as green C. Lost times l

1

and l

2

are added and placed at the beginning of the green for modeling purposes D. Effective green and effective red

l 1

= 2 sec/phase

e

= 2 sec/phase Default by HCM2010 Chapter 24

l Y t L

2 

Y

y

e

ar

l

1 

l

2

L

i n

  1

t Li

14

Effective green times and the application of the lost times:  HCM delay models use “effective green time” and “effective red time.”  HCM 2010 models assume that all lost times happen at the beginning of the phase.

g i

G i

r g i

 

C G i

 

i g i l

1

y i

 

e ar i

l

1 

l

2 Watch out where t L takes place, especially when an overlap phase exists. That’s where you must add

y

and

ar

in the phase section of the HCS input module.

Chapter 24 15

### 24.3 The Basic Model

24.3.1 Model structure The HCM 2010 signalized intersection analysis consists of 6 modules. Chapter 24 16

Chapter 24 17

### 24.3.2 Analysis time periods

   The peak 15 minutes within the analysis hour (no over saturation exists, no v/c > 1.0. Use PHF.) The full 60-min analysis hour (OK, but masks the peak.) Sequential 15-min periods for an analysis period of one hour or greater (Most comprehensive. PHF = 1.0 is used.

v p

V PHF

Chapter 24 18

### 24.3.3 Input

Input Module: Many parameters are considered. See Table 24-2 in the text). Geometric, traffic, and signalization conditions are considered  Some of them are self-explanatory. See pages 582 – 585 for details and default values.

 Area type: CBD intersections have lower saturation flow rates (in general). Saturation flow rates for CBD is about 10% less than for non CBD.

 Parking conditions and parking activity: Parking activity within 250 ft of the stop line is considered. Parking activities interfere traffic flow  Conflicting pedestrian flow (for RT vehicles): Pedestrian flow between 1700 to 2100 ped/hr completely blocks right-turn vehicles. HCM 2010 considers bicycles as well. Also, check pedestrian min green times  Local bus volume: Buses must stop to be considered in this parameter. If they pass through the intersection, not stopping for passengers, they are considered as heavy vehicles.

 Arrival type: The single most important factor influencing delay Chapter 24 predictions. 19

More discussion on Arrival type (Table 24.3, p.583 text): 1 2 3 4 5 6 Dense platoon, containing over 80% (P) of the lane group volume, arriving at the start of the red phase  very poor progression About 40 to 80% arriving at the start of the red phase  unfavorable progression Main platoon contains less than 40% of the lane group volume  random arrival 40 to 80% arriving throughout the green time  favorable progression Over 80% arriving at the start of the green phase  highly favorable progression Exceptional progression, with minimal or negligible side-street entries.

AT

 3

g P C

Chapter 24 P = Proportion of vehicles arriving on green.

20

### More discussion on Arrival type:

Need to compute a platoon ratio,

R p

: R p = 1.00, when the proportion of vehicles arriving on green is equal to the g/C ratio. P = Proportion of vehicles arriving on green.

R p

g i P C

Chapter 24 Table 24.4 (They accidentally (?) forgot two columns for platoon ratio.

21

### 24.3.4 Movement Groups, Lane Groups, and Demand Volume Adjustment

1. Conversion of hourly demand volumes to peak 15-min flow rates needs to be done first.

2. Establish analysis lane groups (6 types)

v

V PHF

Chapter 24 3. Determination of total lane group demand flow rates,

v gi

These two rules are new in HCM2010.

22

### Figure 24.5 Common Movement and Lange Groups on a Signalized Intersection Approach

Chapter 24 New in HCM2010 23

### 24.3.5 Estimating the Saturation Flow Rate for Each Lane Group

The saturation flow rate module is the most important part of HCM2010. The prevailing total saturation flow rate for each lane group is estimated.

s

s

0

Nf w f HV f g f p f bb f a f LU f RT f LT f Rpb f Lpb

f w

= 0.96 Lane width < 10ft •

f w

= 1.00 10 ft ≤ Lane width ≤ 12.9 ft •

f w

= 1.04 Lane width ≥ 12.9 ft Chapter 24 24

f HV

 1 

P HV

 1

E HV

 1 

E HV

= 2.00

f g

 1 

G

200

P

 Adjustment for Parking Conditions: 0 .

9    18

N m

3 , 600  

f p

 

N

 1

N

 

P f p

N

 0 .

10    18

N m

3 , 600    0 .

05

N

Limitations: • 0 ≤

N m

≤ 180; if

N m

f p

(min) = 0.05

f p

> 180, use 180 mvts/h (no parking) = 1.00 Chapter 24 25

B

Adjustment for Local Bus Blockage:  1 .

0    14 .

4 3 ,

N

600

B

 

f bb

 

N

 1 

N

B f bb

N

   14 .

4

N

3 , 600

B

   0 .

05

N

Limitations: • 0 ≤

N B

≤ 250; if

N B

f bb

(min) = 0.05 > 250, use 250 b/h Adjustment for Type of Area: •

CBD location: f a = 0.90

Other location: f a = 1.00

f LU

v g v g

1

N

See Table 24.7 for default values (next page).

Chapter 24 26

Adjustment for Lane Utilization (continued): Chapter 24 27

Adjustment for Right Turns: • • • From an exclusive RT lane ( From a shared lane From a single-lane approach

f RT

= 0.85) Adjustment for Left Turns (discussed in section 24.5 of the text): • • • • Case 1: Exclusive LT lane with protected phasing (

f LT

• • = 0.95) Case 2: Exclusive LT lane with permitted phasing Case 3: Exclusive LT lane with compound phasing Case 4: Shared lane with protected phasing Case 5: Shared lane with permitted phasing Case 6: Shared lane with compound phasing Chapter 24 28

Adjustment for Pedestrian and Bicycle Interference with Turning Vehicles (There are seven steps; see pages 589-591): Chapter 24 29

Pedestrian and Bike Interference Adjustment (continued):

g ped

= walk + clearance interval • • 2,000 = 3,600/1.8 sec a ped occupies the ped conflict area.

10,000 = 3,600/0.36 sec a ped occupies the ped conflict area walking parallel. 0.4 = 40% occupied. Chapter 24 30

Pedestrian and Bike Interference Adjustment (continued): Because bikes follow the same rule as cars Chapter 24 • • 2,700 = 3,600/1.33 sec a bike occupies the bike conflict area.

0.02 = 2% occupied 31

Pedestrian and Bike Interference Adjustment (continued): 4 Joint probability (Venn diagram) Chapter 24 32

Pedestrian and Bike Interference Adjustment (continued): Meaning “After turning from an exclusive RT lane, there is only one lane in the receiving side” 40% less impact Chapter 24 33

Pedestrian and Bike Interference Adjustment (continued): Chapter 24 34

## 24.3.6 Determine Lane Group Capacities and v/c Ratios

1. The v/s ratio for each lane group is computed.

2. Relative v/s ratios are used to identify the critical lane group in the phase plan; the sum of critical lane group v/s ratios is computed.

3. Lane group capacities are computed (Eq. 24-2).

4. Lane group v/c ratios are computed (Eq. 24-3) 5. The critical v/c ratio for the intersection is computed (Eq. 24-5).

c i

s i g i C

Eq. 24-2

X i

v c i i

s i v i g i C

v i g i s i C

Eq. 24-3 Chapter 24

X c

 

i

i

 

ci

g ci C

  

i

Eq. 24-5

ci C C

L

35

These values are flow ratios (

v/s

).

Finding critical lane groups is similar to the simple method; the only difference is that HCM uses

v/s

to find critical lane groups.

Chapter 24 36

## 24.3.7 Estimating Delay and Level of Service

d = d 1 + d 2 + d 3

In HCM2010, AT is part of the uniform delay,

d 1

, computation (see slide #39).

Where,

d = average control delay per vehicle, s/veh d 1 =average uniform delay per vehicle, d 2 = average incremental delay per vehicle, d 3 = additional delay per vehicle due to a preexisting queue d

d

1

PF

d

2 

d

3 In HCM2000, arrival type factor was multiplied to d 1 as shown on the right.

d

1 0 .

5

C

 1    min

d

2 Chapter 24

d

3 1 1 , 2

g C

g C

    900

T

   (

X

1800

Q b

 1   1 )

u

t

cT

X

 1  2  8

kIX cT

37   

### Incremental Queue Accumulation (IQA) (Note: what’s in the textbook is for pre-timed signals)

By HCM2000 By HCM2010 Difference?

Chapter 24 38

## Incremental Queue Accumulation (IQA)

The effect of progression is built in into the methodology in IQA. First, we need to find out the P value, which is the portion of platoon arriving during the green.

If you don’t have field data for it, you may estimate it by equation 24-26.

P

AT

3 

g C

AT

 3

P g

/

C

P = Proportion of vehicles arriving on green.

Chapter 24 39

### Incremental Queue Accumulation (IQA) Steps

Step 1: Determine the arrival rate during the effective red,

V r V r

  1 

P

r

*

V

*

C

V = average arrival flow rate, veh/hr: The numerator gives the number of vehicles arriving during the effective red in a cycle.

Step 2: Determine the queue at the end of the red time,

q 2 q

2 

q

1 

v s

3600 * 

t q

2  0 During red time, r,

v = V r s = 0

Chapter 24

q 1 = 0 for a single interval analysis Δt = Elapsed time, during red = r

40

### Incremental Queue Accumulation (IQA) Steps

Step 3: Determine the uniform delay during the effective red time

d r

 

t

* 2

q

2

q 1 = 0 for a single interval analysis Δt = r

Step 4: Determine the arrival rate during the effective green time

V g

 

V g

*

C P

 

V

*

P

*

C g

Chapter 24 41

### Incremental Queue Accumulation (IQA) Steps

Step 5: Determine

Δt 2

, the time needed to dissipate the queue

Δt 2 q

2   

V g

3600   * 

t

2 

s

3600 * 

t

2 

t

2  3600

s

* 

V g q

2 Step 6: Determine the uniform delay during the effective green

d g

 

t

2 *

q

2 

q

3 2

q 3 = 0 for Δt 2 ≤ g, undersaturated flow

Chapter 24 42

### Incremental Queue Accumulation (IQA) Steps

Step 6: Determine the uniform delay

q

3 during the effective green (continued) 

q

1    

V g

*

g

  

V r

*

r

  (

s

*

g

)   3600 Step 7: Determine the uniform delay,

d 1

s/veh

d

1  

d

q r

2  

d g n a

 

n a

V g

*

g

3600

n a

= the number of vehicles arriving on green Chapter 24 43

### The incremental delay term, d

2

d

2  900

T

   (

X

 1 )  

X

 1  2  8

kIX cT

   Eq. 24-35 Chapter 24 44

# Upstream Filtering or Metering Adjustment Factor, I

I 0.4

0.922

Degree of Saturation at Upstream Intersection, Xu 0.5

0.858

0.6

0.769

0.7

0.650

0.8

0.500

0.9

0.314

≥ 1.0

0.090

 An I-value of 1.0 is used for an isolated intersection (i.e., one that is 1 mile or more from the nearest upstream signalized intersection). This value is based on a random number of vehicles arriving per cycle so that the variance in arrival equals the mean.)  An I-value of less than 1.0 is used for non-isolated intersections. This reflects the way that upstream signals decrease the variance in the number of arrivals per cycle at the subject intersection. As a result, the amount of delay due to random arrivals is reduced. Chapter 24 45

## Aggregating Delay (p.597)

d A d I

  

i

 

A

A d d v i v i A A v v i A

Average delays are weighted by the number of vehicles experiencing delays.

Computing total control delay,

d I

, per vehicle for the intersection as a whole (This computation is not recommended according to the HCM2010, p.597) Chapter 24 46

### 24.3.8 Interpreting the results of signalized intersection analysis

    

v/c

ratio for every lane group Critical

v/c

ratio (

X c

) for the intersection as a whole Delays and levels of service for each lane group Delays and levels of service for each approach Delays and levels of service for the overall intersection (not recommended by HCM2010.) The following scenarios are possible:  Scenario 1:

X c

≤ 1.00, all

X i

≤ 1.00. No capacity deficiency  Scenario 2:

X c

≤ 1.00, some

X i

> 1.00. As long as

X c

current conditions can handle; reallocate green times ≤ 1.00, the  Scenario 3:

X c

> 1.00, some or all

X i

> 1.00. Need to change timing and phasing and if necessary physical layout changes Chapter 24 47

Chapter 24 48

LT

Chapter 24 49

### 24.5.1 LT Permitted Left Turns from a shared lane

Interaction between LT vehicles and opposing vehicles No gaps are available for LTs when the standing queue is released right after the signal turns green.

If a LT vehicle arrives during this time, it must wait, blocking the left most lane, until the opposing queue has cleared.

After the opposing queue has cleared the intersection, LTs may be made through gaps in the unsaturated opposing flow.

LTs have no impact on the subject approach

until the first LT vehicle arrives (for a shared LT lane).

g q = avg. amount of green time required for the opposing standing queue to clear the intersection, sec.

g f (g f = avg. amount of green time before the arrival of the 1 st = 0.0 sec for an exclusive LT lane) LT vehicle, sec g u = avg. amount of green time

not blocked after

the arrival of the 1 st LT vehicle that is by the clearance of the opposing standing queue, sec Chapter 24 50

### Figure 24.10 Portion of the Green Phase Illustrated

g u g u

g

g q

g

g f if if g q g f

g f

g q

Opposing queued vehicles clearing Green not blocked by the opposing clearing queue, usable by LT vehicles The first LT vehicle must wait, i.e., TH is blocked.

Opp. Queued vehicle clearing First LT veh has not arrived yet Green not blocked by the opposing clearing queue, usable by LT vehicles Chapter 24 51

Fig. 24.11Queue Accumulation Polygon for a Shared Lane with Permitted Left Turns

g q g diff

Int. 0 Sth = saturation flow rate for through veh.

Ssl3 = saturation flow rate for shared-lane interval 3 Int. 1 Int. 2 Int. 3 Int. 4 Chapter 24 52

## Modeling permitted left-turns (cont)

Summary for Permitted Left Turns from a shared lane (general concept) • For multilane opposing approaches

f LT

g f g

( 1 .

0 ) 

g diff g f LT

g f g

g u

(

F

1 )

g

g u g

(

F

1 )

g diff = g q - g f

• For single lane opposing approaches

f LT

g f g

( 1 .

0 ) 

g diff g

  2 

g u

(

F

1 )

g f LT

g f g

g diff g

  2 

g u g

(

F

1 ) 1.00 >= F 1 >= 0.0

F 2 = 0 when opposing approach is multilane. Why?

Chapter 24 53

Basic structure of the permitted LT model (cont): multilane opposing approaches

f LT

g f g

g u g

(

F

1 )

F

1  1 

P L

 1

E L

1  1  But the minimum is (sn = sneakers):

f sn

 2  1 

g P L

 Observations show at least one vehicle can turn during the clearance interval and may be two if the second vehicle is a LT vehicle. 2 = 2 second headway, 1 = minimum 1 LT sneaker, and P L is the proportion (probability, that is) of LT vehicles in the left lane.

Chapter 24 54

Basic structure of the permitted LT model (cont): single-lane opposing approaches When the opposing flow is in a single-lane approach, a LT vehicle on that approach creates a gap in the opposing flow through which a subject LT vehicle may move. We need to consider this available gap (2 nd term below), which does not exist if there are multiple lanes in the opposing approach.

f LT

g g f

( 1 .

0 ) 

g q

g f g

(

F

2 ) 

g g u

(

F

1 ) Proportion of LT vehicles in the opposing single lane approach, decimal

F

2  1 

P L

 1

E L

2  1  Proportion of through and RT vehicles in the opposing single lane approach, decimal.

E L

2  1 

n P THo P LTo

n = No. of opposing vehicles in the period g q – g f , about (g q – g f )/2. n can be zero. 2 is 2 sec/veh headway and n is for joint Chapter 21 probability. The numerator is the probability that one or more vehicles are LT vehicles.

v/s

## Ratio

After v/s ratios are computed, we may need to make adjustments – either reallocation of green time , modifying cycle length , or modifying the intersection layout . For the first two cases, v/s ratios can be used to reduce the amount of trial-and-error computations. (Assume the phase design does not change.) First, we solve X c for C:

C

X c

LX

c

 

ci X c

  

v ci g ci s ci C

   /

g ci C ci

   

ci C C

L

When X c = 1.0, it is like C equation for simple signal timing (eq. 20-13).

Suppose

sum(v/s)

= 0.9, and we desire to achieve

X c

= 0.95. What would be the cycle length to serve this (assume max C = 90 sec)?

C

 9 ( 0 .

95 ) 0 .

95  0 .

90  171 sec 

C

max  90 sec

X c

= 0.95 cannot be achieved in this case. C = 171 sec is too long.

Chapter 24 56

## Modifying signal timing based on v/s ratios (cont)

C needs to be contained within the common cycle lengths (171 sec in the previous example is too long). Typically C = 120 sec is the maximum cycle length accepted. Hence,

X c

   

i C C

L

 120 0 .

90 120  9  0 .

973 Where, Σ

g i = C – L

.

With sum(v/s) = 0.90 and C = 120 sec,

X c

= 0.973 is the minimum that can be achieved (although Xc = 0.95 cannot be achieved). Once C is determined, we can compute new effective green times, then new actual greens for the critical lane groups and for the next trial-and-error analysis.

X i

v c i i

s i v i g i C

 (

v

/

s

)

i g i C

 (

v

/

s

)

i C g i

Chapter 24

g i C G i

  

i C X i

 

g i

g i

t L

L Y i